Electromagnetic

Theory

Syllabus

Electromagnetic Theory

UNIT I: Electrostatics

Laplace and Poisson equation – Boundary value problems - boundary conditions

and uniqueness theorem – Laplace equation in three dimensions – Solution in

Cartesian and spherical polar co ordinates – Examples of solutions for boundary

value problems- Polarization and displacement vectors - Boundary conditions -

Dielectric sphere in a uniform field –Molecular polarisability and electrical

susceptibility -Electrostatic energy in the presence of dielectric – Multipole

expansion.

UNIT II : Magnetostatics

Biot - Savart Law - Ampere's law - Magnetic vector potential and magnetic field

of a localised current distribution - Magnetic moment, force and torque on a

current distribution in an external field - Magnetostatic energy - Magnetic

induction and magnetic field in macroscopic media - Boundary conditions -

Uniformly magnetised sphere.

UNIT III : Maxwell Equations

Faraday's laws of Induction - Maxwell's displacement current - Maxwell's

equations – free space and linear isotropic media - Vector and scalar potentials -

Gauge invariance - Wave equation and plane wave solution - Coulomb and

Lorentz gauges - Energy and momentum of the field - Poynting's theorem -

Lorentz force - Conservation laws for a system of charges and electromagnetic

fields.

UNIT IV : Electromagnetic Waves

Plane waves in non-conducting media - Linear and circular polarization,

reflection and refraction at a plane interface - Fresnel’s law, interference,

coherence and diffraction -Waves in a conducting medium - Propagation of waves

in a rectangular wave guide - Inhomogeneous wave equation and retarded

potentials - Radiation from a localized source - Oscillating electric dipole.

Subject Introduction

UNIT I: Electrostatics

Laplace and Poisson equation – Boundary value problems -

boundary conditions and uniqueness theorem – Laplace equation

in three dimensions – Solution in Cartesian and spherical polar

co ordinates – Examples of solutions for boundary value

problems- Polarization and displacement vectors - Boundary

conditions -Dielectric sphere in a uniform field –Molecular

polarisability and electrical susceptibility -Electrostatic energy in

the presence of dielectric – Multipole expansion.

Classical Electrodynamics

Classical electrodynamics deals with electric and magnetic fields and interactions

caused by macroscopic distributions of electric charges and currents. This means

that the concepts of localised charges and currents assume the validity of certain

mathematical limiting processes in which it is considered possible for the charge

and current distributions to be localised in infinitesimally small volumes of space.

This is in obvious contradiction to electromagnetism on a microscopic scale,

where charges and currents are known to be spatially extended objects. However,

the limiting processes yield results which are correct on a macroscopic scale.

In this Chapter we start with the force interactions in classical electrostatics and

classical magnetostatics and introduce the static electric and magnetic fields and

find two uncoupled systems of equations for them. Then we see how the

conservation of electric charge and its relation to electric current leads to the

dynamic connection between electricity and magnetism and how the two can be

unified in one theory, classical electrodynamics, described by one system of

coupled dynamic field equations.

Electrostatics

The theory that describes physical phenomena related to the interaction between

stationary electric charges or charge distributions in space is called electrostatics.

Coulomb’s law

It has been found experimentally that in classical electrostatics the interaction

between two stationary electrically charged bodies can be described in terms of a

mechanical force. Let us consider the simple case depicted in figure where F

denotes the force acting on a charged particle with charge q located at x, due to

the presence of a charge q_located at x_. According to Coulomb’s law this force

is, in vacuum, given by the expression

where we have used results from Example. In SI units, which we shall use

throughout, the force F is measured in Newton (N), the charges q and q_in

Coulomb (C) [= Ampere-seconds (As)], and the length in meters (m). The

constant Farad per meter (F/m) is the

vacuum permittivity and m/s is the speed of light in vacuum.

In CGS units and the force is measured in dyne, the charge in stat coulomb, and

length in centimeters (cm).

The electrostatic field

Instead of describing the electrostatic interaction in terms of a “force action at a

distance,” it turns out that it is often more convenient to introduce the concept of a

field and to describe the electrostatic interaction in terms of a static vectorial

electric field Estat defined by the limiting process

where F is the electrostatic force, as defined in equation on the facing page, from

a net charge q_on the test particle with a small electric net charge q. Since the

purpose of the limiting process is to assure that the test charge q does not

influence the field, the expression for Estat does not depend explicitly on q but

only on the charge q_and the relative radius vector x _x_. This means that we can

say that any net electric charge produces an electric field in the space that

surrounds it, regardless of the existence of a second charge anywhere in this

space.1 we find that the electrostatic field Estat at the field point x (also known as

the observation point), due to a field-producing charge q_at the source point x_, is

given by

In the presence of several field producing discrete charges q_i, at x_i, i

_1_2_3____, respectively, the assumption of linearity of vacuum2 allows us to

superimpose their individual E fields into a total E field

If the discrete charges are small and numerous enough, we introduce the charge

density r located at x_and write the total field as

where, in the last step, we used formula equation. We emphasize that equation

(1.5) above is valid for an arbitrary distribution of charges, including discrete

charges, in which case r can be expressed in terms of one or more Dirac delta

functions.

I.e., Estat is an irrotational field.

Taking the divergence of the general Estat expression for an arbitrary charge

distribution, equation, and using the representation of the Dirac delta function,

equation we find that

which is Gauss’s law in differential form.

Magnetostatics

While electrostatics deals with static charges, magnetostatics deals with stationary

currents, i.e., charges moving with constant speeds, and the interaction between

these currents.

Ampère’s law

Experiments on the interaction between two small current loops have shown that

they interact via a mechanical force, much the same way that charges interact. Let

F denote such a force acting on a small loop C carrying a current J located at x,

due to the presence of a small loop C_carrying a current J_located at x_.

According to Ampère’s law this force is, in vacuum, given by the expression.

Here dl and dl_are tangential line elements of the loops C and C_, respectively,

and, in SI units H/m is the vacuum

permeability. From the definition of e0 and m0 (in SI units) we observe that

which is a useful relation.

At first glance, equation above appears to be unsymmetrical in terms of the loops

and therefore to be a force law which is in contradiction with Newton’s third law.

However, by applying the vector triple product “bac-cab” formula on page, we

can rewrite in the following way

Recognising the fact the integrand in the first integral is an exact differential so

that this integral vanishes, we can rewrite the force expression, equation above, in

the following symmetric way

This clearly exhibits the expected symmetry in terms of loops C and C_.

The magnetostatic field

In analogy with the electrostatic case, we may attribute the magnetostatic

interaction to a vectorial magnetic field Bstat. I turns out that Bstat can be defined

through

Gauss’s Law and Applications

Though Coulomb’s law is fundamental, one finds it cumbersome to use it to

calculate electric field due to a continuous charge distribution because the

integrals involved can be quite difficult. An alternative but completely equivalent

formulation is Gauss’s Law which is very useful in situations which exhibit

certain symmetry.

Electric Lines of Force :

Electric lines of force (also known as field lines) is a pictorial representation of

the electric field. These consist of directed lines indicating the direction of electric

field at various points in space.

• There is no rule as to how many lines are to be shown. However, it is customary

to draw number of lines proportional to the charge. Thus if N number of lines are

drawn from or into a charge Q, 2N number of lines would be drawn for charge

2Q.

• The electric field at a point is directed along the tangent to the field lines. A

positive charge at this point will move along the tangent in a direction indicated

by the arrow.

• Lines are dense close to a source of the electric field and become sparse as one

moves away.

• Lines originate from a positive charge and end either on a negative charge or

move to infinity.

• Lines of force due to a solitary negative charge is assumed to start at infinity and

end at the negative charge.

• Field lines do not cross each other. ( if they did, the field at the point of crossing

will not be uniquely defined.)

• A neutral point is a point at which field strength is zero. This occurs because of

cancellation of electric field at such a point due to multiple charges.

Exercise : Draw field lines and show the neutral point for a charge +4Q located at

(1, 0) and −Q located at (−1, 0).

Electric Flux

The concept of flux is borrowed from flow of water through a surface. The

amount of water flowing through a surface depends on the velocity of water, the

area of the surface and the orientation of the surface with respect to the direction

of velocity of water.

Though an area is generally considered as a scalar, an element of area may be

considered to be a vector because :

• It has magnitude (measured in m2).

• If the area is infinitisimally small, it can be considered to be in a plane. We can

then associate a direction with it. For instance, if the area element lies in the x-y

plane, it can be considered to be directed along the z–direction.

(Conventionally, the direction of the area is taken to be along the outward

normal.)

We define the flux of the electric field through an area d~S to be given by the

scalar Product

For an arbitrary surface S, the flux is obtainted by integrating over all the surface

Elements

If the electric field is uniform, the angle θ is constant and we have

Thus the flux is equal to the product of magnitude of the electric field and the

projection of area perpendicular to the field.

Unit of flux is N-m2/C. Flux is positive if the field lines come out of the surface

and is negative if they go into it.

Solid Angle :

The concept of solid angle is a natural extension of a plane angle to three

dimensions. Consider an area element dS at a distance r from a point P. Let ˆn be

the unit vector along the outward normal to dS. The element of the solid angle

subtended by the area element at P is defined as

where dS ⊥ is the projection of d~S along a direction perpendicular to ~r. If α is

the angle between ˆr and ˆn, then,

Solid angle is dimensionless. However, for practical reasons it is measured in

terms of a unit called steradian (much like the way a planar angle is measured in

terms of degrees).

The maximum possible value of solid angle is 4π, which is the angle subtended

by an area which encloses the point P completely.

Example 1:

A right circular cone has a semi-vertical angle α. Calculate the solid angle at the

apex P of the cone.

Solution :

The cap on the cone is a part of a sphere of radius R, the slant length of the cone.

Using spherical polar coordinates, an area element on the cap is R2 sin θdθdφ,

where θ is the polar angle and φ is the azimuthal angle. Here, φ goes from 0 to 2π

while θ goes from 0 to α.

Thus the area of the cap is

Example 2

An wedge in the shape of a rectangular box is kept on a horizontal floor. The two

triangular faces and the rectangular face ABFE are in the vertical plane. The

electric field is horizontal, has a magnitude 8 × 104 N/C and enters the wedge

through the face ABFE, as shown. Calculate the flux through each of the faces

and through the entire surface of the wedge.

Solution :

The outward normals to the triangular faces AED, BFC, as well as the normal to

the base are perpendicular to ~E . Hence the flux through each of these faces is

zero.

The vertical rectangular face ABFE has an area 0.06 m2. The outward normal to

this face is perpendicular to the electric field. The flux is entering through this

face and is negative. Thus flux through ABFE is

To find the flux through the slanted face, we need the angle that the normal to this

face makes with the horizontal electric field. Since the electric field is

perpendicular to the side ABFE, this angle is equal to the angle between AE and

AD, which is cos−1(.3/.5). The area of the slanted face ABCD is 0.1 m2. Thus the

flux through ABCD is

GAUSS’S LAW - Integral form

The flux calculation done in Example 4 above is a general result for flux out of

any closed surface, known as Gauss’s law.

Total outward electric flux φ through a closed surface S is equal to 1/ǫ0 times

the charge enclosed by the volume defined by the surface S

Mathematicaly, the surface integral of the electric field over any closed surface is

equal to the net charge enclosed divided by ǫ0

• The law is valid for arbitry shaped surface, real or imaginary.

• Its physical content is the same as that of Coulomb’s law.

• In practice, it allows evaluation of electric field in many practical situations by

forming imagined surfaces which exploit symmetry of the problem. Such surfaces

are called Gaussian surfaces.

GAUSS’S LAW - Differential form

The integral form of Gauss’s law can be converted to a differential form by

using the divergence theorem. If V is the volume enclosed by the surface S,

Direct Calculation of divergence from Coulomb’s Law :

We will use the field expression

to directly evaluate the divergence of the electric field. Since the differentiation

is with respect to ~r while the integration is with respect to ~r′, we can take the

divergence inside the integral,

Divergence of ~r is equal to 3

i.e. ∇・ ~E = 0, which violates Gauss’s Law, that ∇・ ~E = ρ/ǫ0. The problem

arises because the function 1/ | ~r −~r′ | has a singularity at ~r = ~r′. This point has

to be taken care of with care. Except at this point the divergence of the integral is

indeed zero.

Applications of Gauss’s Law

Field due to a uniformly charged sphere of radius R with a charge Q

By symmetry, the field is radial. Gaussian surface is a concentric sphere of radius

r. The outward normals to the Gaussian surface is parallel to the field ~E at every

point. Hence

The field outside the sphere is what it would be if all the charge is concentrated at

the origin of the sphere.

Field due to an infinite line charge of linear charge density λ

Gaussian surface is a cylinder of radius r and length L. By symmetry, the field has

the same magnitude at every point on the curved surface and is directed outwards.

At the end caps, ~E is perpendicular to d~S everywhere and the flux is zero. For

the curved surface, ~E and d~S are parallel,

where ˆρ is a unit vector perpendicular to the line,directed outward for positive

line charge and inward for negative line charge.

UNIT II : Magnetostatics

Biot - Savart Law - Ampere's law - Magnetic vector potential and

magnetic field of a localised current distribution - Magnetic moment,

force and torque on a current distribution in an external field -

Magnetostatic energy - Magnetic induction and magnetic field in

macroscopic media - Boundary conditions - Uniformly magnetised

sphere.

Magnetostatics

Whereas electrostatics deals with static electric charges (electric charges that do

not move), and the interaction between these charges, magnetostatics deals with

static electric currents (electric charges moving with constant speeds), and the

interaction between these currents. Here we shall discuss the theory of

magnetostatics in some detail.

The Biot-Savart Law

In this Section we will discuss the magnetic field produced by a steady current.

A steady current is a flow of charge that has been going on forever, and will be

going on forever. These currents produce magnetic fields that are constant in time.

The magnetic field produced by a steady line current is given by the Biot-Savart

Law:

where   is an element of the wire,   is the vector connecting the element of the

wire and P, and   is the permeability constant which is equal to

The unit of the magnetic field is the Tesla (T). For surface and volume currents

the Biot-Savart law can be rewritten as

and

Example:Problem

Find the magnetic field at point P for each of the steady current configurations

shown in Figure

a) The total magnetic field at P is the vector sum of the magnetic fields produced

by the four segments of the current loop. Along the two straight sections of the

loop,   and   are parallel or opposite, and thus  . Therefore, the magnetic

field produced by these two straight segments is equal to zero. Along the two

circular segments   and   are perpendicular. Using the right-hand rule it is easy

to show that

and

where   is pointing out of the paper. The total magnetic field at P is therefore

equal to

b) The magnetic field at P produced by the circular segment of the current loop is

equal to

where   is pointing out of the paper. The magnetic field produced at P by each of

the two linear segments will also be directed along the negative z axis. The

magnitude of the magnetic field produced by each linear segment is just half of

the field produced by an infinitely long straight wire :

The total field at P is therefore equal to

Example:Problem

Suppose you have two infinite straight-line charges λ, a distance d apart, moving

along at a constant v (see Figure ). How fast would v have to be in order for the

magnetic attraction to balance the electrical repulsion?

Figure Problem

When a line charge moves it looks like a current of magnitude I = λv. The two

parallel currents attract each other, and the attractive force per unit length is

and is attractive. The electric generated by one of the wires can be found using

Gauss' law and is equal to

The electric force per unit length acting on the other wire is equal to

and is repulsive (like charges). The electric and magnetic forces are balanced

when

or

This requires that

This requires that the speed v is equal to the speed of light, and this can therefore

never be achieved. Therefore, at all velocities the electric force will dominate.

Ampère’s law

Experiments on the force interaction between two small loops that carry static

electric currents I and I’ (i.e. the currents I and I’ do not vary in time) have shown

that the loops interact via a mechanical force, much the same way that static

electric charges interact. Let Fms.(x) denote the magnetostatic force on a loop C,

with tangential line vector element dl, located at x and carrying a current I in the

direction of dl, due to the presence of a loop C ‘ , with tangential line element dl’,

located at x’ and carrying a current I’ in the direction of dl’ in otherwise empty

space. This spatial configuration is illustrated in graphical form in figure :

According to Ampère’s law the magnetostatic force in question is given by

Here μ’ is in hennery per metre (Hm-1) is the permeability of free space. From the

definition of ε’ and μ’ (in SI units) we observe that :

which is most useful relation.

At first glance, equation above may appear asymmetric in terms of the loops and

therefore be a force law that does not obey Newton’s third law. However, by

applying the vector triple product ‘bac-cab’ formula )

Ampère’s law postulates how a small loop C, carrying a static electric current I

directed along the line element dl at x, experiences a magnetostatic force Fms (x)

from a small loop C’, carrying

a static electric current I ‘ directed along the line element dl ’ located at x’.

we can rewrite above equation as :

Since the integrand in the first integral over C is an exact differential, this integral

vanishes and we can rewrite the force expression, formula on the preceding page,

in the following symmetric way

Which clearly exhibits the expected interchange symmetry between loops C and

C’

The Vector Potential

The magnetic field generated by a static current distribution is uniquely defined

by the so-called Maxwell equations for magnetostatics:

Similarly, the electric field generated by a static charge distribution is uniquely

defined by the so-called Maxwell equations for electrostatics:

The fact that the divergence of   is equal to zero suggests that there are no point

charges for  . Magnetic field lines therefore do not begin or end anywhere (in

contrast to electric field lines that start on positive point charges and end on

negative point charges). Since a magnetic field is created by moving charges, a

magnetic field can never be present without an electric field being present. In

contrast, only an electric field will exist if the charges do not move.

Maxwell's equations for magnetostatics show that if the current density is known,

both the divergence and the curl of the magnetic field are known. The Helmholtz

theorem indicates that in that case there is a vector potential   such that

However, the vector potential is not uniquely defined. We can add to it the

gradient of any scalar function f without changing its curl:

The divergence of   is equal to

It turns out that we can always find a scalar function f such that the vector

potential   is divergence-less. The main reason for imposing the requirement

that   is that it simplifies many equations involving the vector potential. For

example, Ampere's law rewritten in terms of   is

or

This equation is similar to Poisson's equation for a charge distribution ρ:

Therefore, the vector potential   can be calculated from the current   in a manner

similar to how we obtained V from ρ. Thus

Note: these solutions require that the currents go to zero at infinity (similar to the

requirement that ρ goes to zero at infinity).

Example:Problem

Find the magnetic vector potential of a finite segment of straight wire carrying a

current I. Check that your answer is consistent with eq.

The current at infinity is zero in this problem, and therefore we can use the

expression for   in terms of the line integral of the current I. Consider the wire

located along the z axis between z1 and z2 (see Figure 5.6) and use cylindrical

coordinates. The vector potential at a point P is independent of φ (cylindrical

symmetry) and equal to

Here we have assumed that the origin of the coordinate system is chosen such

that P has z = 0. The magnetic field at P can be obtained from the vector potential

and is equal to

where θ1 and θ2 are defined in Figure. This result is identical to the result of

Example 5 in Griffiths.

Figure . Problem

Example:Problem

If   is uniform, show that  , where   is the vector from the origin to

the point in question. That is check that   and  .

The curl of   is equal to

Since   is uniform it is independent of r, θ, and φ and therefore the second and

third term on the right-hand side of this equation are zero. The first term,

expressed in Cartesian coordinates, is equal to

The fourth term, expressed in Cartesian coordinates, is equal to

Therefore, the curl of   is equal to

The divergence of   is equal to

Example:Problem

Find the vector potential above and below the plane surface current of Example

in Griffiths.

In Example of Griffiths a uniform surface current is flowing in the xy plane,

directed parallel to the x axis:

However, since the surface current extends to infinity, we can not use the surface

integral of   to calculate   and an alternative method must be used to

obtain  . Since Example 8 showed that   is uniform above the plane of the

surface current and   is uniform below the plane of the surface current, we can

use the result of Problem to calculate  :

In the region above the xy plane (z > 0) the magnetic field is equal to

Therefore,

In the region below the xy plane (z < 0) the magnetic field is equal to

Therefore,

We can verify that our solution for   is correct by calculating the curl of   (which

must be equal to the magnetic field). For z > 0:

The vector potential   is however not uniquely defined. For

example,   and   are also possible solutions that generate

the same magnetic field. These solutions also satisfy the requirement that  .

The Three Fundamental Quantities of Magnetostatics

Our discussion of the magnetic fields produced by steady currents has shown that

there are three fundamental quantities of magnetostatics:

1. The current density 

2. The magnetic field 

3. The vector potential 

These three quantities are related and if one of them is known, the other two can

be calculated. The following table summarizes the relations between  ,  , and  :

Magnetic Moment  :

The magnetic moment of a magnet is a quantity that determines the force that the

magnet can exert on electric currents and the torque that a magnetic field will

exert on it. A loop of electric current, a bar magnet, an electron, a molecule, and

a planet all have magnetic moments.

Both the magnetic moment and magnetic field may be considered to

be vectors having a magnitude and direction. The direction of the magnetic

moment points from the south to north pole of a magnet. The magnetic field

produced by a magnet is proportional to its magnetic moment as well. More

precisely, the term magnetic moment normally refers to a system's magnetic

dipole moment, which produces the first term in the multipole expansion of a

general magnetic field. The dipole component of an object's magnetic

Two Defination of moment :

The preferred definition of a magnetic moment has changed over time. Before the

1930s, textbooks defined the moment using magnetic poles. Since then, most have

defined it in terms of Ampèrian currents. field is symmetric about the direction of

its magnetic dipole moment, and decreases as the inverse cube of the distance

from the object.

Magnetic pole definition

The sources of magnetic moments in materials can be represented by poles in

analogy to electrostatics. Consider a bar magnet which has magnetic poles of

equal magnitude but opposite polarity. Each pole is the source of magnetic force

which weakens with distance. Since magnetic poles always come in pairs, their

forces partially cancel each other because while one pole pulls, the other repels.

This cancellation is greatest when the poles are close to each other i.e. when the

bar magnet is short. The magnetic force produced by a bar magnet, at a given

point in space, therefore depends on two factors: the strength p of its poles

(the magnetic pole strength), and the vector ℓ separating them. The moment is

defined as

It points in the direction from South to North pole. The analogy with electric

dipoles should not be taken too far because magnetic dipoles are associated

withangular momentum (see Magnetic moment and angular momentum).

Nevertheless, magnetic poles are very useful for magnetostatic calculations,

particularly in applications to ferromagnets. Practitioners using the magnetic pole

approach generally represent the magnetic field by the irrotational field H, in

analogy to theelectric field E.

An electrostatic analogue for a magnetic moment: two opposing charges separated

by a finite distance.

Current loop definition

Suppose a planar closed loop carries an electric current I and has vector

area S (x, y, and z coordinates of this vector are the areas of projections of the

loop onto the yz, zx, and xy planes). Its magnetic moment m, vector, is defined

as:

By convention, the direction of the vector area is given by the right hand grip

rule (curling the fingers of one's right hand in the direction of the current around

the loop, when the palm of the hand is "touching" the loop's outer edge, and the

straight thumb indicates the direction of the vector area and thus of the magnetic

moment).

If the loop is not planar, the moment is given as

where × is the vector cross product. In the most general case of an arbitrary

current distribution in space, the magnetic moment of such a distribution can be

found from the following equation:

where r is the position vector pointing from the origin to the location of the

volume element, and J is the current density vector at that location.

The above equation can be used for calculating a magnetic moment of any

assembly of moving charges, such as a spinning charged solid, by substituting

where ρ is the electric charge density at a given point and v is the instantaneous

linear velocity of that point.

For example, the magnetic moment produced by an electric charge moving along

a circular path is

,

where r is the position of the charge q relative to the center of the circle and v is

the instantaneous velocity of the charge.

Moment μ of a planar current having magnitude I and

enclosing an area S

Practitioners using the current loop model generally represent the magnetic field

by the solenoidal field B, analogous to the electrostatic field D.

Magnetic moment of a solenoid

3-D image of a solenoid

A generalization of the above current loop is a multi-turn coil, or solenoid. Its

moment is the vector sum of the moments of individual turns. If the solenoid

has N identical turns (single-layer winding) and vector area S

Units of Magnetic Moment :

The unit for magnetic moment is not a base unit in the International System of

Units (SI) and it can be represented in more than one way. For example, in

the current loop definition, the area is measured in square meters and I is

measured in amperes, so the magnetic moment is measured in ampere–square

meters (A m2). In the equation for torque on a moment, the torque is measured

in joules and the magnetic field in tesla, so the moment is measured in Joules per

Tesla (J⋅T−1). These two representations are equivalent:

1 A·m2 = 1 J·T−1.

In the CGS system, there are several different sets of electromagnetism units, of

which the main ones are ESU, Gaussian, and EMU. Among these, there are two

alternative (non-equivalent) units of magnetic dipole moment in CGS:

(ESU CGS) 1 stat A·cm2 = 3.33564095×10−14 (A·m2 or J·T−1)

and (more frequently used) (EMU CGS and Gaussian-CGS) 1 erg/G =

1 abA·cm2 = 10−3 (m2·A or J/T).

The ratio of these two non-equivalent CGS units (EMU/ESU) is equal exactly to

the speed of light in free space, expressed in cm·s−1.

All formulas in this article are correct in SI units, but in other unit systems, the

formulas may need to be changed. For example, in SI units, a loop of current with

current I and area A has magnetic moment I×A (see below), but in Gaussian

units the magnetic moment is I×A/c.

Magnetic moment angular momentum :

The magnetic moment has a close connection with angular momentum called

the gyromagnetic effect. This effect is expressed on a macroscopic scale in

the Einstein-de Haas effect, or "rotation by magnetization," and its inverse,

the Barnett effect, or "magnetization by rotation."In particular, when a magnetic

moment is subject to a torque in a magnetic field that tends to align it with the

applied magnetic field, the moment precesses (rotates about the axis of the

applied field). This is a consequence of the angular momentum associated with

the moment.

Viewing a magnetic dipole as a rotating charged sphere brings out the close

connection between magnetic moment and angular momentum. Both the magnetic

moment and the angular momentum increase with the rate of rotation of the

sphere. The ratio of the two is called the gyromagnetic ratio, usually denoted by

the symbol γ.

For a spinning charged solid with a uniform charge density to mass density ratio,

the gyromagnetic ratio is equal to half the charge-to-mass ratio. This implies that

a more massive assembly of charges spinning with the same angular

momentum will have a proportionately weaker magnetic moment, compared to its

lighter counterpart. Even though atomic particles cannot be accurately described

as spinning charge distributions of uniform charge-to-mass ratio, this general

trend can be observed in the atomic world, where the intrinsic angular momentum

(spin) of each type of particle is a constant: a small half-integer times the

reduced Planck constant ħ. This is the basis for defining the magnetic moment

units of Bohr magneton (assuming charge-to-mass ratio of theelectron)

and nuclear magneton (assuming charge-to-mass ratio of the proton).

Force and Torque on a current distribution in an external

field :

Force on a moment

See also: force between magnets

A magnetic moment in an externally-produced magnetic field has a potential

energy U:

In a case when the external magnetic field is non-uniform, there will be a force,

proportional to the magnetic field gradient, acting on the magnetic moment itself.

There has been some discussion on how to calculate the force acting on a

magnetic dipole. There are two expressions for the force acting on a magnetic

dipole, depending on whether the model used for the dipole is a current loop or

two monopoles (analogous to the electric dipole). The force obtained in the case

of a current loop model is

In the case of a pair of monopoles being used (i.e. electric dipole model)

and one can be put in terms of the other via the relation

In all these expressions m is the dipole and B is the magnetic field at its position.

Note that if there are no currents or time-varying electrical fields ∇ × B = 0 and

the two expressions agree.

An electron, nucleus, or atom placed in a uniform magnetic field will precess with

a frequency known as the Larmor frequency.

Torque on a moment

The magnetic moment can also be defined as a vector relating the

aligning torque on the object from an externally applied magnetic field to the field

vector itself. The relationship is given by 

where τ is the torque acting on the dipole and B is the external magnetic field.

External magnetic field produced by a magnetic dipole moment :

Any system possessing a net magnetic dipole moment m will produce

a dipolar magnetic field (described below) in the space surrounding the system.

While the net magnetic field produced by the system can also have higher-

order multipole components, those will drop off with distance more rapidly, so

that only the dipolar component will dominate the magnetic field of the system at

distances far away from it.

Magnetic field lines around a "magnetostatic dipole" the magnetic dipole itself is

in the center and is seen from the side.

The vector potential of magnetic field produced by magnetic moment m is

and magnetic flux density is

Alternatively one can obtain the scalar potential first from the magnetic pole

perspective,

and hence magnetic field strength is

The magnetic field of an ideal magnetic dipole is depicted on the left.

Internal magnetic field of a dipole

The two models for a dipole (current loop and magnetic poles) give the same

predictions for the magnetic field far from the source. However, inside the source

region they give different predictions. The magnetic field between poles (see

figure for Magnetic pole definition) is in the opposite direction to the magnetic

moment (which points from the negative charge to the positive charge), while

inside a current loop it is in the same direction (see the figure to the right).

Clearly, the limits of these fields must also be different as the sources shrink to

zero size. This distinction only matters if the dipole limit is used to calculate

fields inside a magnetic material

The magnetic field of a current loop

.

If a magnetic dipole is formed by making a current loop smaller and smaller, but

keeping the product of current and area constant, the limiting field is

Unlike the expressions in the previous section, this limit is correct for the internal

field of the dipole.

If a magnetic dipole is formed by taking a "north pole" and a "south pole",

bringing them closer and closer together but keeping the product of magnetic

pole-charge and distance constant, the limiting field is

These fields are related by  , where

is the magnetization.

Forces between two magnetic dipoles

As discussed earlier, the force exerted by a dipole loop with moment m1 on

another with moment m2 is

where B2 is the magnetic field due to moment 2. The result of calculating the

gradient is[8][9]

where   is the unit vector pointing from magnet 1 to magnet 2 and r is the

distance. An equivalent expression is

The force acting on m1 is in opposite direction.

The torque of magnet 2 on magnet 1 is

Examples of Magnetic Moments :

Two kinds of magnetic sources

Fundamentally, contributions to any system's magnetic moment may come from

sources of two kinds: (1) motion of electric charges, such as electric currents, and

(2) the intrinsic magnetism ofelementary particles, such as the electron.

Contributions due to the sources of the first kind can be calculated from knowing

the distribution of all the electric currents (or, alternatively, of all the electric

charges and their velocities) inside the system, by using the formulas below. On

the other hand, the magnitude of each elementary particle's intrinsic magnetic

moment is a fixed number, often measured experimentally to a great precision.

For example, any electron's magnetic moment is measured to be

−9.284764×10−24 J/T.The direction of the magnetic moment of any elementary

particle is entirely determined by the direction of its spin (the minus in front of the

value above indicates that any electron's magnetic moment is antiparallel to its

spin).

The net magnetic moment of any system is a vector sum of contributions from

one or both types of sources. For example, the magnetic moment of an atom

of hydrogen-1 (the lightest hydrogen isotope, consisting of a proton and an

electron) is a vector sum of the following contributions:

1. the intrinsic moment of the electron,

2. the orbital motion of the electron around the proton,

3. the intrinsic moment of the proton.

Similarly, the magnetic moment of a bar magnet is the sum of the intrinsic and

orbital magnetic moments of the unpaired electrons of the magnet's material.

Magnetic moment of an atom

For an atom, individual electron spins are added to get a total spin, and individual

orbital angular momenta are added to get a total orbital angular momentum. These

two then are added usingangular momentum coupling to get a total angular

momentum. The magnitude of the atomic dipole moment is then

where J is the total angular momentum quantum number, gJ is the Landé g-factor,

and μB is the Bohr magneton. The component of this magnetic moment along the

direction of the magnetic field is then

where m is called the magnetic quantum number or the equatorial quantum

number, which can take on any of 2J+1 values:

.

The negative sign occurs because electrons have negative charge.

Due to the angular momentum, the dynamics of a magnetic dipole in a magnetic

field differs from that of an electric dipole in an electric field. The field does exert

a torque on the magnetic dipole tending to align it with the field. However, torque

is proportional to rate of change of angular momentum, so precession occurs: the

direction of spin changes. This behavior is described by theLandau-Lifshitz-

Gilbert equation:

where   is gyromagnetic ratio, m is magnetic moment, λ is damping coefficient

and Heff is effective magnetic field (the external field plus any self-field). The first

term describes precession of the moment about the effective field, while the

second is a damping term related to dissipation of energy caused by interaction

with the surroundings.

Magnetic moment of an electron

Electrons and many elementary particles also have intrinsic magnetic moments,

an explanation of which requires a quantum mechanical treatment and relates to

the intrinsic angular momentumof the particles as discussed in the article electron

magnetic dipole moment. It is these intrinsic magnetic moments that give rise to

the macroscopic effects of magnetism, and other phenomena, such as electron

paramagnetic resonance.

The magnetic moment of the electron is

where μB is the Bohr magneton, S is electron spin, and the g-factor gS is 2

according to Dirac's theory, but due to quantum electrodynamic effects it is

slightly larger in reality: 2.002 319 304 36. The deviation from 2 is known as

the anomalous magnetic dipole moment.

Again it is important to notice that m is a negative constant multiplied by the spin,

so the magnetic moment of the electron is antiparallel to the spin. This can be

understood with the following classical picture: if we imagine that the spin

angular momentum is created by the electron mass spinning around some axis, the

electric current that this rotation creates circulates in the oppositedirection,

because of the negative charge of the electron; such current loops produce a

magnetic moment which is antiparallel to the spin. Hence, for a positron (the anti-

particle of the electron) the magnetic moment is parallel to its spin.

Magnetic moment of a nucleus

The nuclear system is a complex physical system consisting of nucleons,

i.e., protons and neutrons. The quantum mechanical properties of the nucleons

include the spin among others. Since the electromagnetic moments of the nucleus

depend on the spin of the individual nucleons, one can look at these properties

with measurements of nuclear moments, and more specifically the nuclear

magnetic dipole moment.

Most common nuclei exist in their ground state, although nuclei of

some isotopes have long-lived excited states. Each energy state of a nucleus of a

given isotope is characterized by a well-defined magnetic dipole moment, the

magnitude of which is a fixed number, often measured experimentally to a great

precision. This number is very sensitive to the individual contributions from

nucleons, and a measurement or prediction of its value can reveal important

information about the content of the nuclear wave function. There are several

theoretical models that predict the value of the magnetic dipole moment and a

number of experimental techniques aiming to carry out measurements in nuclei

along the nuclear chart.

Magnetic moment of a molecule

Any molecule has a well-defined magnitude of magnetic moment, which may

depend on the molecule's energy state. Typically, the overall magnetic moment of

a molecule is a combination of the following contributions, in the order of their

typical strength:

magnetic moments due to its unpaired electron

spins (paramagnetic contribution), if any

orbital motion of its electrons, which in the ground state is

often proportional to the external magnetic field

(diamagnetic contribution)

the combined magnetic moment of its nuclear spins,

which depends on the nuclear spin configuration.

Examples of molecular magnetism

Oxygen molecule, O2, exhibits strong paramagnetism, due

to unpaired spins of its outermost two electrons.

Carbon dioxide molecule, CO2, mostly

exhibits diamagnetism, a much weaker magnetic moment

of the electron orbitals that is proportional to the external

magnetic field. In the rare instance when a

magnetic isotope, such as 13C or 17O, is present, it will

contribute its nuclear magnetism to the molecule's

magnetic moment.

Hydrogen molecule, H2, in a weak (or zero) magnetic

field exhibits nuclear magnetism, and can be in a para- or

an ortho- nuclear spin configuration.

Magnetostatic Energy :

This is the magnetic potential energy generated when a magnetic body is present

in a magnetic field. For a strongly magnetic body, even in the absence of an

external magnetic field, magnetostatic energy is generated within the magnetic

body by the internal magnetic field generated in the opposite direction from the

magnetization (and called the diamagnetic field). The amount is the amount of

work required for the magnetic poles to exist counter to the internal magnetic

field at both ends of the magnetic body. A strongly magnetic body is given a

magnetic domain structure in order to minimize this magnetostatic energy.

Now we will provide the expression for the contribution of magnetostatic interactions to

the free energy of the system. The derivation of such expression is quite straightforward

if one assumes that the energy density of magnetostatic field is given by:

m

where   is the whole space. In fact, by expressing the magnetostatic field as

Eq. becomes:

m

The first term in Eq.  vanishes owing to the integral orthogonality of the

solenoidal field  m and the conservative field   over the whole space . The

remaining part, remembering that   is nonzero only within the region  , is the

magnetostatic free energy:

m

We observe that magnetostatic energy expresses a nonlocal interaction, since the

magnetostatic field functionally depends, through the boundary value problem ,

on the whole magnetization vector field, as we anticipated in the beginning of the

section. The latter equation has the physical meaning of an interaction energy of

an assigned continuous magnetic moments distribution, namely it can be obtained

by computing the work, made against the magnetic field generated by the

continuous distribution, to bring an elementary magnetic moment   from

infinity to its actual position within the distribution 

Magnetostatic self-energy :

The origin of domains still cannot be explained by the two energy terms above. Another

contribution comes from the magnetostatic self-energy, which originates from the

classical interactions between magnetic dipoles. For a continuous material it is

described by Maxwell's equations

In our magnetostatic problem, we do not have any electric fields   or free

currents  . Thus, there are two remaining equations

The magnetic induction   is given by  . A general solution is

given by

where   is the magnetic scalar potential. Inserting the expressions for   and   

gives :

inside magnetic bodies and

outside in air or vacuum.

These equations have to be solved with the boundary conditions

on the surface of the magnet to obtain   and derive from it  .   is the unit

normal to the magnetic body, taken to be positive in outward direction.

In micromagnetics, the magnetization distribution   is given. With relation

the magnetic scalar potential can be calculated from the magnetization

distribution. The demagnetizing field   is then obtained .

Finally the magnetostatic energy is given by 

Magnetic induction and magnetic field in macroscopic media :

Usually quantum mechanics deals with matter on the scale of atoms and atomic

particles. However, at low temperatures, there are phenomena that are

manifestations of quantum mechanics on a macroscopic scale. The most well-

known effects are superfluidity of helium and superconductivity which both show

spectacular behavior. E.g. in both cases matter can flow with zero flow resistance.

In rotating helium so-called quantum vortices are formed which are all equally

strong and which can organize in beautiful patterns. A similar effect shows up in

superconductors where an applied magnetic field is squeezed in bundles each

containing the same amount of magnetic flux.

Macroscopic quantum effects are among the most elegant phenomena in physics.

Chapter 21 of the Feynman Lectures on Physics on this topic starts with "This

lecture is only for entertainment." In the period from 1996 to 2003 four Nobel

prizes were given for work which was related to macroscopic quantum

phenomena. Macroscopic quantum phenomena can be observed in superfluid

helium and in superconductors, but also in dilute quantum gases and in laser light.

Although these media are very different, their behavior is very similar as they all

show macroscopic quantum behavior.

The phenomena which are called macroscopic quantum

phenomena are macroscopic in two ways:

1. The quantum states are occupied by a large number of particles

(typically Avogadro's number).

2. The quantum states involved are macroscopic in size (up to km size

in superconducting wires).

Maxwell’s macroscopic theory

Under certain conditions, for instance for small magnitudes of the primary field

strengths E and B, we may assume that the response of a substance to the fields

can be approximated by a linear one so that the electric displacement vector D.(t,

x) is only linearly dependent on the electric field E(t, x)., and the magnetising

field H(t, x). is only linearly dependent on the magnetic field B. (t, x). In this

chapter we derive these linearised forms, and then consider a simple, explicit

linear model for a medium from which we derive the expression for the dielectric

permittivity ".ε(t, x), the magnetic susceptibility μ(t, x) , and the refractive index

or index of refraction n. (t, x) of this medium. Using this simple model, we study

certain interesting aspects of the propagation of electromagnetic particles and

waves in the medium

Magnetisation and the magnetising field

An analysis of the properties of magnetic media and the associated currents shows

that three such types of currents exist:

1. In analogy with true charges for the electric case, we may have true currents

jtrue, i.e. a physical transport of true (free) charges.

2. In analogy with the electric polarisation P there may be a form of charge

transport associated with the changes of the polarisation with time. Such currents,

induced by an external field, are called polarisation currents and are identified with

:

3. There may also be intrinsic currents of a microscopic, often atomistic, nature

that are inaccessible to direct observation, but which may produce net effects at

discontinuities and boundaries. These magnetisation currents are denoted jM.

Magnetic monopoles have not yet been unambiguously identified in experiments.

So there is no correspondence in the magnetic case to the electric monopole

moment, formula The lowest order magnetic moment, corresponding to the

electric dipole moment, is the magnetic dipole moment

Analogously to the electric case, one may, for a distribution of magnetic dipole

moments in a volume, describe this volume in terms of its magnetisation, or

magnetic dipole moment per unit volume, M. Via the definition of the vector

potential A one can show that the magnetisation current and the magnetization is

simply related:

In a stationary medium we therefore have a total current which is (approximately)

the sum of the three currents enumerated above:

One might then be led to think that the right-hand side (RHS) of the ∆ × B

Maxwell equation

However, moving the term ∆ × M from the right hand side (RHS) to the left hand

side (LHS) and introducing the magnetising field (magnetic field intensity,

Ampère-turn density) as

and using the definition for D, equation, we find that

Hence, in this simplistic view, we would pick up a term which makes

the equation inconsistent: the divergence of the left hand side vanishes while the

divergence of the right hand side does not! Maxwell realised this and to overcome

this inconsistency he was forced to add his famous displacement current term

which precisely compensates for the last term the RHS expression.

We may, in analogy with the electric case, introduce a magnetic susceptibility for

the medium. Denoting it Xm, we can write

where, approximately

and

is the relative permeability. In the case of anisotropy, Km will be a tensor, but it

is still only a linear approximation

Macroscopic Maxwell equations :

Field equations, expressed in terms of the derived, and therefore in principle

superfluous, field quantities D and H are obtained from the Maxwell-Lorentz

microscopic equations, by replacing the E and B in the two source equations by

using the approximate relations formula,respectively:

This set of differential equations, originally derived by Maxwell himself, are

called Maxwell’s macroscopic equations. Together with the boundary conditions

and the constitutive relations, they describe uniquely (but only approximately) the

properties of the electric and magnetic fields in matter and are convenient to use

in certain simple cases, particularly in engineering applications. However, the

structure of these equations rely on certain linear approximations and

there are many situations where they are not useful or even applicable. Therefore,

these equations, which are the original Maxwell equations (albeit expressed in

their modern vector form as introduced by OLIVER HEAVISIDE), should be

used with some care.

Boundary conditions in general the fields

, B, D and H (vectors) are discontinuous at points where ε, μ and σ also are.

Hence the field vectors will be discontinuous at a boundary between two media

with different constitutive parameters. The integral form of Maxwell’s equations

can be used to determine the relations, called boundary conditions, of the normal

and tangential components of the fields at the interface between two regions with

different constitutive parameters ε, μ and σ where surface density of sources may

exist along the boundary.

The boundary condition for D(vector) can be calculated using a very thin, small

pillbox that crosses the interface of the two media, as shown in Fig. . Applying the

divergence theorem7 to we have

outward flux of D(vector) over them is (Dn1 −Dn2)ds = ( D1 − D2)·ˆnds, where

these Dn are the normal components of D(vector), ds is the area of each base, and

ˆn is the unit normal drawn from medium 2 to medium 1. At the limit, by taking a

shallow enough pillbox, we can disregard the flux over the curved surface,

whereupon the sources of D(vector) reduce to the density of surface free charge

ρs on the interface

Figure : Derivation of boundary conditions at the interface of two media.

Hence the normal component of D(vector) changes discontinously across the

interface by an amount equal to the free charge surface density ρs on the surface

boundary. Similarly the boundary condition for B(vector) can be established

using the Gauss’ law for magnetic fields . Since the magnetic field is solenoidal, it

follows that the normal components of B(vector) are continuous across the

interface between two media

The behavior of the tangential components of E(vector) can be determined using a

infinitesimal rectangular loop at the interface which has sides of lengh dh, normal

to the interface, and sides of lengh dl parallel to it . From the integral form of the

Faraday’s law, and defining ˆt as the unit tangent vector parallel to the direction

of integration on the upper side of the loop, we

Have

In the limit, as dh → 0, the area ds = dldh bounded by the loop approaches zero

and, since B(vector) is finite, the flux of B(vector) vanishes. Hence( E1 − E2)

(vector) · ˆt = 0 and we conclude that the tangential components of E(vector) are

continuous across the interface between two media. In terms of the normal ˆn to

the boundary, this can be written as :

Analogously, using the same infinitesimal rectangular loop, it can be deduced

from the generalized Ampère’s law, (1.2d), that

where, since D(vector) is finite, its flux vanishes. Nevertheless, the flux of the

surface current can have a non-zero value when the integration loop is reduced to

zero, if the conductivity σ of the medium 2, and consequently Js(vector), is

infinite. This requires the surface to be a perfect conductor. Thus

the tangential component of H(vector) is discontinuous by the amount of surface

current density Js(vector). For finite conductivity, the tangential magnetic field is

continuous across the boundary.

A summary of the boundary conditions, given in , are particularized in for the

case when the medium 2 is a perfect conductor (σ2 → ∞).

General boundary conditions :

Boundary Conditions when the mediun 2 is a perfect conductor (σ2 → ∞).

A uniformly magnetized sphere

Consider a sphere of radius  , with a uniform permanent

magnetization  , surrounded by a vacuum region. The simplest way of

solving this problem is in terms of the scalar magnetic potential introduced in Eqs.

, it is clear that  satisfies Laplace's equation,

since there is zero volume magnetic charge density in a vacuum or a uniformly

magnetized magnetic medium. However, according to Eq. , there is a magnetic surface

charge density,

on the surface of the sphere. One of the matching conditions at the surface of the

sphere is that the tangential component of   must be continuous. It follows from

Eq. that the scalar magnetic potential must be continuous at  , so that

Integrating Eq.  over a Gaussian pill-box straddling the surface of the sphere

yields 

In other words, the magnetic charge sheet on the surface of the sphere gives rise

to a iscontinuity in the radial gradient of the magnetic scalar potential at  .

The most general axisymmetric solution to Eq. which satisfies physical boundary

conditions at   and  is

for  , and

for  . The boundary condition yields

for all  . The boundary condition gives

for all  , since  . It follows that

for  , and

Thus,

for  , and

for  . Since there is a uniqueness theorem associated with Poisson's

equation, we can be sure that this axisymmetric potential is the only solution to

the problem which satisfies physical boundary conditions at  and infinity.

In the vacuum region outside the sphere

It is easily demonstrated from Eq. that

where

This, of course, is the magnetic field of a magnetic dipole  . Not surprisingly,

the net dipole moment of the sphere is equal to the integral of the

magnetization   (which is the dipole moment per unit volume) over the volume

of the sphere.

Figure 4: Schematic demagnetization curve for a permanent magnet

Inside the sphere we have   and  , giving

and

Thus, both the   and   fields are uniform inside the sphere. Note that the

magnetic intensity is oppositely directed to the magnetization. In other words,

the   field acts to demagnetize the sphere. How successful it is at achieving this

depends on the shape of the hysteresis curve in the negative   and positive   

quadrant. This curve is sometimes called the demagnetization curve of the

magnetic material which makes up the sphere. Figure 4 shows a schematic

demagnetization curve. The curve is characterized by two quantities: the

retentivity   (i.e., the residual magnetic field strength at zero magnetic

intensity) and the coercivity   (i.e., the negative magnetic intensity required

to demagnetize the material: this quantity is conventionally multiplied by   to

give it the units of magnetic field strength). The operating point (i.e., the values

of   and   inside the sphere) is obtained from the intersection of the

demagnetization curve and the curve  . It is clear from Eqs. and that

for a uniformly magnetized sphere in the absence of external fields. The

magnetization inside the sphere is easily calculated once the operating point has

been determined. In fact,  . It is clear from Fig. 4 that for a

magnetic material to be a good permanent magnet it must possess both a large

retentivity and a large coercivity. A material with a large retentivity but a small

coercivity is unable to retain a significant magnetization in the absence of a strong

external magnetizing field.

UNIT III : Maxwell Equations

Faraday's laws of Induction - Maxwell's displacement current -

Maxwell's equations – free space and linear isotropic media -

Vector and scalar potentials - Gauge invariance - Wave equation

and plane wave solution - Coulomb and Lorentz gauges - Energy

and momentum of the field - Poynting's theorem -Lorentz force -

Conservation laws for a system of charges and electromagnetic

fields.

Faraday’s law of induction :

Any change in the magnetic environment of a coil of wire will cause a voltage

(emf) to be "induced" in the coil. No matter how the change is produced, the

voltage will be generated. The change could be produced by changing the

magnetic field strength, moving a magnet toward or away from the coil, moving

the coil into or out of the magnetic field, rotating the coil relative to the magnet,

etc.

Faraday's experiment showing induction between coils of wire: The liquid

battery (right) provides a current which flows through the small coil (A), creating

a magnetic field. When the coils are stationary, no current is induced. But when

the small coil is moved in or out of the large coil (B), the magnetic flux through

the large coil changes, inducing a current which is detected by the galvanometer

The induced electromotive force in any closed circuit is equal to the negative of

the time rate of change of the magnetic flux through the circuit.

This version of Faraday's law strictly holds only when the closed circuit is a loop

of infinitely thin wire, and is invalid in other circumstances as discussed below. A

different version, the Maxwell–Faraday equation , is valid in all circumstances.

 in physics, a quantitative relationship between a changing magnetic field and the

electric field created by the change, developed on the basis of experimental

observations made in 1831 by the English scientist Michael Faraday.

The phenomenon called electromagnetic induction was first noticed and

investigated by Faraday; the law of induction is its quantitative expression.

Faraday discovered that, whenever the magnetic field about an electromagnet was

made to grow and collapse by closing and opening the electric circuit of which it

was a part, an electric current could be detected in a separate conductor nearby.

Moving a permanent magnet into and out of a coil of wire also induced a current

in the wire while the magnet was in motion. Moving a conductor near a stationary

permanent magnet caused a current to flow in the wire, too, as long as it was

moving.

Faraday visualized a magnetic field as composed of many lines of induction,

along which a small magnetic compass would point. The aggregate of the lines

intersecting a given area is called the magnetic flux. The electrical effects were

thus attributed by Faraday to a changing magnetic flux. Some years later the

Scottish physicist James Clerk Maxwell proposed that the fundamental effect of

changing magnetic flux was the production of an electric field, not only in a

conductor (where it could drive an electric charge) but also in space even in the

absence of electric charges. Maxwell formulated the mathematical expression

relating the change in magnetic flux to the induced electromotive force

(E, or emf). This relationship, known as Faraday’s law of induction (to distinguish

it from his laws of electrolysis), states that the magnitude of the emf induced in a

circuit is proportional to the rate of change of the magnetic flux that cuts across

the circuit. If the rate of change of magnetic flux is expressed in units of webers

per second, the induced emf has units of volts. Faraday’s law is one of the four

Maxwell equations that define electromagnetic theory.

Quantitative

Faraday's law of induction makes use of the magnetic flux ΦB through a

hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be

moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface

integral:

where dA is an element of surface area of the moving surface Σ(t), B is the

magnetic field, and B·dA is a vector dot product (the infinitesimal amount of

magnetic flux). In more visual terms, the magnetic flux through the wire loop is

proportional to the number ofmagnetic flux lines that pass through the loop.

The wire loop (red) forms the boundary of a surface Σ (blue). The black arrows denote

any vector field F(r, t) defined throughout space; in the case of Faraday's law, the relevant

vector field is the magnetic flux density B, and it is integrated over the blue surface. The red

arrow represents the fact that the wire loop may be moving and/or deforming.

The definition of surface integral relies on splitting the surface Σ into small

surface elements. Each element is associated with a vector dA of magnitude

equal to the area of the element and with direction normal to the element and

pointing “outward” (with respect to the orientation of the surface).

When the flux changes—because B changes, or because the wire loop is moved or

deformed, or both—Faraday's law of induction says that the wire loop acquires

an EMF  , defined as the energy available per unit charge that travels once

around the wire loop (the unit of EMF is the volt). Equivalently, it is the voltage

that would be measured by cutting the wire to create an open circuit, and

attaching a voltmeter to the leads. According to the Lorentz force law (in SI

units),

the EMF on a wire loop is:

where E is the electric field, B is the magnetic field (aka magnetic flux density,

magnetic induction), dℓ is an infinitesimal arc length along the wire, and the line

integral is evaluated along the wire (along the curve the conincident with the

shape of the wire).

The EMF is also given by the rate of change of the magnetic flux:

where   is the electromotive force (EMF) in volts and ΦB is the magnetic

flux in webers. The direction of the electromotive force is given by Lenz's law.

For a tightly wound coil of wire, composed of N identical loops, each with the

same ΦB, Faraday's law of induction states that

where N is the number of turns of wire and ΦB is the magnetic flux in webers

through a single loop.

Maxwell–Faraday equation :

The Maxwell–Faraday equation is a generalisation of Faraday's law that states

that a time-varying magnetic field is always accompanied by a spatially-varying,

non-conservative electric field, and vice-versa. The Maxwell–Faraday equation is

(in SI units) where   is the curl operator and again E(r, t) is the electric

field and B(r, t) is the magnetic field. These fields can generally be functions of

position r and time t.

The Maxwell–Faraday equation is one of the four Maxwell's equations, and

therefore plays a fundamental role in the theory of classical electromagnetism. It

can also be written in an integral form by the Kelvin-Stokes theorem:

where, as indicated in the figure:

Σ is a surface bounded by the closed contour ∂Σ,

E is the electric field, B is the magnetic field.

dℓ is an infinitesimal vector element of the contour ∂Σ,

dA is an infinitesimal vector element of surface Σ. If its direction is orthogonal to

that surface patch, the magnitude is the area of an infinitesimal patch of surface.

Both dℓ and dA have a sign ambiguity; to get the correct sign, the right-hand

rule is used, as explained in the article Kelvin-Stokes theorem. For a planar

surface Σ, a positive path element dℓ of curve ∂Σ is defined by the right-hand rule

as one that points with the fingers of the right hand when the thumb points in the

direction of the normal n to the surface Σ.

The integral around ∂Σ is called a path integral or line integral.

Notice that a nonzero path integral for E is different from the behavior of the

electric field generated by charges. A charge-generated E-field can be expressed

as the gradient of a scalar field that is a solution to Poisson's equation, and has a

zero path integral. See gradient theorem.

The integral equation is true for any path ∂Σ through space, and any surface Σ for

which that path is a boundary.

If the path Σ is not changing in time, the equation can be rewritten:

The surface integral at the right-hand side is the explicit expression for

the magnetic flux ΦB through Σ.

Proof of Faraday's law from Maxwell's equations :

The four Maxwell's equations (including the Maxwell–Faraday equation), along

with the Lorentz force law, are a sufficient foundation to

derive everything in classical electromagnetism . Therefore it is possible to

"prove" Faraday's law starting with these equations. Click "show" in the box

below for an outline of this proof. (In an alternative approach, not shown here but

equally valid, Faraday's law could be taken as the starting point and used to

"prove" the Maxwell–Faraday equation and/or other laws.)

Outline of proof of Faraday's law from Maxwell's equations and the

Lorentz force law.

Consider the time-derivative of flux through a possibly moving loop, with

area  :

The integral can change over time for two reasons: The integrand can change,

or the integration region can change. These add linearly, therefore:

where t0 is any given fixed time. We will show that the first term on the right-

hand side corresponds to transformer EMF, the second to motional EMF (see

above). The first term on the right-hand side can be rewritten using the integral

form of the Maxwell–Faraday equation:

Area swept out by vector element dℓ of curve ∂Σ in timedt when

moving with velocity v.

Next, we analyze the second term on the right-hand side:

This is the most difficult part of the proof; more details and alternate

approaches can be found in references. As the loop moves and/or deforms, it

sweeps out a surface (see figure on right). The magnetic flux through this

swept-out surface corresponds to the magnetic flux that is either entering or

exiting the loop, and therefore this is the magnetic flux that contributes to the

time-derivative. (This step implicitly uses Gauss's law for magnetism: Since

the flux lines have no beginning or end, they can only get into the loop by

getting cut through by the wire.) As a small part of the loop   moves with

velocity v for a short time  , it sweeps out a vector area

vector  . Therefore, the change in magnetic flux through the

loop here is

Therefore:

where v is the velocity of a point on the loop  .

Putting these together,

Meanwhile, EMF is defined as the energy available per unit charge that travels

once around the wire loop. Therefore, by the Lorentz force law,

Combining these, 

Examples of Faraday’s Law :

Faraday's disc electric generator. The disc rotates with angular rate ω, sweeping

the conducting radius circularly in the static magnetic field B. The magnetic

Lorentz force v × Bdrives the current along the conducting radius to the

conducting rim, and from there the circuit completes through the lower brush and

the axle supporting the disc. Thus, current is generated from mechanical motion.

A counterexample to Faraday's Law when over-broadly interpreted. A wire (solid

red lines) connects to two touching metal plates (silver) to form a circuit. The

whole system sits in a uniform magnetic field, normal to the page. If the word

"circuit" is interpreted as "primary path of current flow" (marked in red), then the

magnetic flux through the "circuit" changes dramatically as the plates are rotated,

yet the EMF is almost zero, which contradicts Faraday's Law.

Although Faraday's law is always true for loops of thin wire, it can give the

wrong result if naively extrapolated to other contexts.  One example is

the homopolar generator (above left): A spinning circular metal disc in a

homogeneous magnetic field generates a DC (constant in time) EMF. In

Faraday's law, EMF is the time-derivative of flux, so a DC EMF is only possible

if the magnetic flux is getting uniformly larger and larger perpetually. But in the

generator, the magnetic field is constant and the disc stays in the same position,

so no magnetic fluxes are growing larger and larger. So this example cannot be

analyzed directly with Faraday's law.

Another example, due to Feynman, has a dramatic change in flux through a

circuit, even though the EMF is arbitrarily small. See figure and caption above

right.

In both these examples, the changes in the current path are different from the

motion of the material making up the circuit. The electrons in a material tend to

follow the motion of the atoms that make up the material, due to scattering in the

bulk and work function confinement at the edges. Therefore, motional EMF is

generated when a material's atoms are moving through a magnetic field, dragging

the electrons with them, thus subjecting the electrons to the Lorentz force. In the

homopolar generator, the material's atoms are moving, even though the overall

geometry of the circuit is staying the same. In the second example, the material's

atoms are almost stationary, even though the overall geometry of the circuit is

changing dramatically. On the other hand, Faraday's law always holds for thin

wires, because there the geometry of the circuit always changes in a direct

relationship to the motion of the material's atoms.

Although Faraday's law does not apply to all situations, the Maxwell–Faraday

equation and Lorentz force law are always correct and can always be used

directly.

Both of the above examples can be correctly worked by choosing the appropriate

path of integration for Faraday's Law. Outside of context of thin wires, the path

must never be chosen to go through the conductor in the shortest direct path. 

Applications of faraday’s law :

The principles of electromagnetic induction are applied in many devices and

systems, including:

Current clamp

Electrical generators

Electromagnetic forming

Graphics tablet

Hall effect meters

Induction cookers

Induction motors

Induction sealing

Induction welding

Inductive charging

Inductors

Magnetic flow meters

Mechanically powered flashlight

Pickups

Rowland ring

Transcranial magnetic stimulation

Transformers

Wireless energy transfer

The Maxwell Displacement Current

Maxwell's Equations (ME) consist of two inhomogeneous partial differential

equations and two homogeneous partial differential equations. At this point you

should be familiar at least with the ``static'' versions of these equations by name

and function: 

in SI units, where   and  .

The astute reader will immediately notice two things. One is that these equations

are not all, strictly speaking, static - Faraday's law contains a time derivative, and

Ampere's law involves moving charges in the form of a current. The second is

that they are almost symmetric. There is a divergence equation and a curl equation

for each kind of field. The inhomogenous equations (which are connected

to sources in the form of electric charge) involve the electric displacement and

magnetic field, where the homogeneous equations suggest that there is no

magnetic charge and consequently no screening of the magnetic induction or

electric field due to magnetic charge. One asymmetry is therefore the

presence/existence of electric charge in contrast with the absence/nonexistence of

magnetic charge.

The other asymmetry is that Faraday's law connects the curl of the   field to the

time derivative of the   field, but its apparent partner, Ampere's Law,

does not connect the curl of   to the time deriviative of   as one might

expect from symmetry alone.

If one examines Ampere's law in its integral form, however: 

one quickly concludes that the current through the open surface   bounded by

the closed curve   is not invariant as one chooses different surfaces. Let us

analyze this and deduce an invariant form for the current (density), two ways.

Figure : Current flowing through a closed curve   bounded by two surfaces,   

and  .

Consider a closed curve   that bounds two distinct open surfaces   and   

that together form a closed surface  . Now consider a current

(density) ``through'' the curve  , moving from left to right. Suppose that some of

this current accumulates inside the volume   bounded by  . The law of charge

conservation states that the flux of the current density out of the closed surface   

is equal to the rate that the total charge inside decreases. Expressed as an integral: 

With this in mind, examine the figure above. If we rearrange the integrals on the

left and right so that the normal   points in to the volume (so we can compute

the current through the surface   moving from left to right) we can easily see

that charge conservation tells us that the current in through   minus the current

out through   must equal the rate at which the total charge inside this volume

increases. If we express this as integrals: 

 

 

In this expression and figure, note well that   and   point through the loop in

the same sense (e.g. left to right) and note that the volume integral is over the

volume   bounded by the closed surface formed by  and   together.

Using Gauss's Law for the electric field, we can easily connect this volume

integral of the charge to the flux of the electric field integrated over these two

surfaces with outward directed normals: 

 

 

Combining these two expressions, we get: 

 

   

 

   

From this we see that the flux of the ``current density'' inside the brackets

is invariant as we choose different surfaces bounded by the closed curve  .

In the original formulation of Ampere's Law we can clearly get a different answer

on the right for the current ``through'' the closed curve depending on which

surface we choose. This is clearly impossible. We therefore modify Ampere's

Law to use the invariant current density: 

where the flux of the second term is called the Maxwell displacement

current (MDC). Ampere's Law becomes: 

 

 

or 

in terms of the magnetic field   and electric displacement  . The origin of the

term ``displacement current'' is obviously clear in this formulation.

Using vector calculus on our old form of Ampere's Law allows us to arrive at this

same conclusion much more simply. If we take the divergence of Ampere's Law

we get: 

If we apply the divergence theorem to the law of charge conservation expressed

as a flux integral above, we get its differential form: 

and conclude that in general we can not conclude that the divergence of   

vanishes in general as this expression requires, as there is no guarantee that   

vanishes everywhere in space. It only vanishes for ``steady state currents'' on a

background of uniform charge density, justifying our calling this form of

Ampere's law a magnetostatic version.

If we substitute in   (Gauss's Law) for  , we can see that it is true

that: 

as an identity. A sufficient (but not necessary!) condition for this to be true is: 

or 

This expression is identical to the magnetostatic form in the cases where   is

constant in time but respects charge conservation when the associated

(displacement) field is changing.

We can now write the complete set of Maxwell's equations, including the

Maxwell displacement current discovered by requiring formal invariance of the

current and using charge conservation to deduce its form. Keep the latter in mind;

it should not be surprising to us later when the law of charge conservation

pops out of Maxwell's equations when we investigate their formal properties we

can see that we deliberately encoded it into Ampere's Law as the MDC.

Anyway, here they are. Learn them. They need to be second nature as we will

spend a considerable amount of time using them repeatedly in many, many

contexts as we investigate electromagnetic radiation. 

(where I introduce and obvious and permanent abbreviations for each equation by

name as used throughout the rest of this text).

Aren't they pretty! The no-monopoles asymmetry is still present, but we now

have two symmetric dynamic equations coupling the electric and magnetic fields

and are ready to start studying electrodynamics instead of electrostatics.

Note well that the two inhomogeneous equations use the in-media forms of the

electric and magnetic field. These forms are already coarse-grain averaged over

the microscopic distribution of point charges that make up bulk matter. In a truly

microscopic description, where we consider only bare charges wandering around

in free space, we should use the free space versions: 

It is time to make these equations jump through some hoops.

Maxwell's Equations

Maxwell's equations represent one of the most elegant and concise ways to state

the fundamentals of electricity and magnetism. From them one can develop most

of the working relationships in the field. Because of their concise statement, they

embody a high level of mathematical sophistication and are therefore not

generally introduced in an introductory treatment of the subject, except perhaps as

summary relationships

Maxwell's equations are a set of partial differential equations that, together with

the Lorentz force law, form the foundation of classical electrodynamics,

classical optics, and electric circuits. These fields in turn underlie modern

electrical and communications technologies. Maxwell's equations describe

howelectric and magnetic fields are generated and altered by each other and

by charges and currents. They are named after the Scottish physicist and

mathematician James Clerk Maxwell who published an early form of those

equations between 1861 and 1862.

The equations have two major variants. The "microscopic" set of Maxwell's

equations uses total charge and total current, including the complicated charges

and currents in materials at the atomic scale; it has universal applicability, but

may be unfeasible to calculate. The "macroscopic" set of Maxwell's equations

defines two new auxiliary fields that describe large-scale behavior without having

to consider these atomic scale details, but it requires the use of parameters

characterizing the electromagnetic properties of the relevant materials.

The term "Maxwell's equations" is often used for other forms of Maxwell's

equations. For example, space-time formulations are commonly used in high

energy and gravitational physics. These formulations defined on space-time,

rather than space and time separately are manifestly[1] compatible

with specialand general relativity. In quantum mechanics, versions of Maxwell's

equations based on the electric and magnetic potentials are preferred.

Since the mid-20th century, it has been understood that Maxwell's equations are

not exact laws of the universe, but are a classical approximation to the more

accurate and fundamental theory of quantum electrodynamics. In most cases,

though, quantum deviations from Maxwell's equations are immeasurably small.

Exceptions occur when the particle nature of light is important or for very strong

electric fields.

Conceptually, Maxwell's equations describe how electric charges and electric

currents act as sources for the electric and magnetic fields and how they affect

each other. (Mathematical descriptions are below). Using vector calculus there are

four equations. Two describe how the fields vary in space due to sources (if any);

electric fields emanating from electric charges in Gauss's law, and magnetic fields

as closed field lines (not due to magnetic monopoles) in Gauss's law for

magnetism. The other two describe how the fields 'circulate' around their

respective sources; the magnetic field 'circulates' around electric currents and time

varying electric fields in Ampère's law with Maxwell's correction, while the

electric field 'circulates' around time varying magnetic fields inFaraday's law.

Gauss's law

Gauss's law describes the relationship between a static electric field and

the electric charges that cause it: The static electric field points away from

positive charges and towards negative charges. In the field line description,

electric field lines begin only at positive electric charges and end only at negative

electric charges. 'Counting' the number of field lines passing though a closed

surface, therefore, yields the total charge (including bound charge due to

polarization of material) enclosed by that surface divided by dielectricity of free

space (the vacuum permittivity). More technically, it relates the electric

flux through any hypothetical closed "Gaussian surface" to the enclosed electric

charge.

Gauss's law for magnetism: magnetic field lines never begin nor end but form

loops or extend to infinity as shown here with the magnetic field due to a ring of

current.

Gauss's law for magnetism

Gauss's law for magnetism states that there are no "magnetic charges" (also

called magnetic monopoles), analogous to electric charges. Instead, the magnetic

field due to materials is generated by a configuration called a dipole. Magnetic

dipoles are best represented as loops of current but resemble positive and negative

'magnetic charges', inseparably bound together, having no net 'magnetic charge'.

In terms of field lines, this equation states that magnetic field lines neither begin

nor end but make loops or extend to infinity and back. In other words, any

magnetic field line that enters a given volume must omewhere exit that volume.

Equivalent technical statements are that the sum total magnetic flux through any

Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

Faraday's law

In a geomagnetic storm, a surge in the flux of charged particles temporarily alters

Earth's magnetic field, which induces electric fields in Earth's atmosphere, thus

causing surges in electrical power grids. Artist's rendition; sizes are not to scale.

Faraday's law describes how a time varying magnetic field creates ("induces")

an electric field. This dynamically induced electric field has closed field lined just

as the magnetic field, if NOT superposed by a static (charge induced) electric

field. This aspect of electromagnetic induction is the operating principle behind

many electric generators: for example, a rotating bar magnet creates a changing

magnetic field, which in turn generates an electric field in a nearby wire. (Note:

there are two closely related equations which are called Faraday's law. The form

used in Maxwell's equations is always valid but more restrictive than that

originally formulated by Michael Faraday.)

Ampère's law with Maxwell's correction

An Wang's magnetic core memory (1954) is an application of Ampère's law.

Each corestores one bit of data.

Ampère's law with Maxwell's correction states that magnetic fields can be

generated in two ways: byelectrical current (this was the original "Ampère's law")

and by changing electric fields (this was "Maxwell's correction").

Maxwell's correction to Ampère's law is particularly important: it shows that not

only does a changing magnetic field induce an electric field, but also a changing

electric field induces a magnetic field. Therefore, these equations allow self-

sustaining "electromagnetic waves" to travel through empty space

(see electromagnetic wave equation).

The speed calculated for electromagnetic waves, which could be predicted from

experiments on charges and currents, exactly matches the speed of light;

indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves,

and others). Maxwell understood the connection between electromagnetic waves

and light in 1861, thereby unifying the theories of electromagnetism and optics.

Maxwell's Equations and Electromagnetic Waves

 The Equations

Maxwell’s four equations describe the electric and magnetic fields arising from

varying distributions of electric charges and currents, and how those fields change

in time.  The equations were the mathematical distillation of decades of

experimental observations of the electric and magnetic effects of charges and

currents.  Maxwell’s own contribution is just the last term of the last equation

but realizing the necessity of that term had dramatic consequences.  It made

evident for the first time that varying electric and magnetic fields could feed off

each other these fields could propagate indefinitely through space, far from the

varying charges and currents where they originated.  Previously the fields had

been envisioned as tethered to the charges and currents giving rise to them.  

Maxwell’s new term (he called it the displacement current) freed them to move

through space in a self-sustaining fashion, and even predicted their velocity it

was the velocity of light!

Here are the equations:

1. Gauss’ Law for electric fields:    (The integral of the

outgoing electric field over an area enclosing a volume equals the total

charge inside, in appropriate units.)

2. The corresponding formula for magnetic fields:   (No magnetic

charge exists: no “monopoles”.)

3. Faraday’s Law of Magnetic Induction:    The

first term is integrated round a closed line, usually a wire, and gives the

total voltage change around the circuit, which is generated by a varying

magnetic field threading through the circuit.

4. Ampere’s Law plus Maxwell’s displacement

current:    This gives the total magnetic

force around a circuit in terms of the current through the circuit, plus any

varying electric field through the circuit (that’s the “displacement

current”).

The purpose of this lecture is to review the first three equations and the

original Ampere’s law fairly briefly, as they were already covered earlier in

the course, then to demonstrate why the displacement current term must be

added for consistency, and finally to show, without using differential

equations, how measured values of static electrical and magnetic attraction are

sufficient to determine the speed of light.

 

Preliminaries: Definitions of µ0 and ε0, the Ampere and the Coulomb

Ampere discovered that two long parallel wires carrying electric currents in the

same direction attract each other magnetically, the force per unit length being

proportional to the product of the currents (so oppositely directed currents repel)

and decaying with distance as 1/r.  In modern (SI) notation, his discovery is

written (F in Newtons)

The modern convention is that the constant   appearing here is exactly 10-7,

this defines our present unit of current, the ampere.  To repeat:   is not

something to measure experimentally, it's just a funny way of writing the number

10-7!  That's not quite fair it has dimensions to ensure that both sides of the above

equation have the same dimensionality. (Of course, there's a historical reason for

this strange convention, as we shall see later).  Anyway, if we bear in mind that

dimensions have been taken care of, and just write the equation 

 

it's clear that this defines the unit current one ampere as that current in a long

straight wire which exerts a magnetic force of   newtons per meter of wire

on a parallel wire one meter away carrying the same current.

However, after we have established our unit of current the ampere we have

also thereby defined our unit of charge, since current is a flow of charge, and the

unit of charge must be the amount carried past a fixed point in unit time by unit

current.  Therefore, our unit of charge the coulomb is defined by stating

that a one amp current in a wire carries one coulomb per second past a fixed

point.

 To be consistent, we must do electrostatics using this same unit of charge. Now,

the electrostatic force between two charges is   The constant

appearing here, now written  , must be experimentally measured its value

turns out to be  .

 To summarize:  to find the value of  , two experiments have to be

performed. We must first establish the unit of charge from the unit of current by

measuring the magnetic force between two current-carrying parallel wires.

Second, we must find the electrostatic force between measured charges. (We

could, alternatively, have defined some other unit of current from the start, then

we would have had to find both   and   by experiments on magnetic and

electrostatic attraction.  In fact, the ampere was originally defined as the current

that deposited a definite weight of silver per hour in an electrolytic cell). 

 

Maxwell's Equations

We established earlier in the course that the total flux of electric field out of a

closed surface is just the total enclosed charge multiplied by  , 

This is Maxwell’s first equation.  It represents completely covering the surface

with a large number of tiny patches having areas  .  (The little areas are small

enough to be regarded as flat, the vector magnitude dA is just the value of the

area, the direction of the vector is perpendicular to the area element, pointing

outwards away from the enclosed volume.)  Hence the dot product with the

electric field selects the component of that field pointing perpendicularly

outwards (it would count negatively if the field were pointing inwards) this is

the only component of the field that contributes to actual electric flux across the

surface.  (Remember flux just means flow the picture of the electric field in

this context is like a fluid flowing out from the charges, the field vector

representing the direction and velocity of the flowing fluid.)

 The second Maxwell equation is the analogous one for the magnetic field, which

has no sources or sinks (no magnetic monopoles, the field lines just flow around

in closed curves).  Again thinking of the force lines as representing a kind of fluid

flow, the so-called "magnetic flux", we see that for a closed surface, as much

magnetic flux flows into the surface as flows out since there are no sources.

This can perhaps be visualized most clearly by taking a group of neighboring

lines of force forming a slender tube the "fluid" inside this tube flows round

and round, so as the tube goes into the closed surface then comes out again

(maybe more than once) it is easy to see that what flows into the closed surface at

one place flows out at another. Therefore the net flux out of the enclosed volume

is zero,  Maxwell’s second equation:

The first two Maxwell's equations, given above, are for integrals of the electric

and magnetic fields over closed surfaces.  Maxwell's other two equations,

discussed below, are for integrals of electric and magnetic fields around closed

curves (taking the component of the field pointing along the curve). These

represent the work that would be needed to take a charge around a closed curve in

an electric field, and a magnetic monopole (if one existed!) around a closed curve

in a magnetic field.

 The simplest version of Maxwell's third equation is for the

special electrostatic case: 

The path integral   for electrostatics.

However, we know that this is only part of the truth, because from Faraday's Law

of Induction, if a closed circuit has a changing magnetic flux through it, a

circulating current will arise, which means there is a nonzero voltage around the

circuit.

 The complete Maxwell's third equation is: 

where the area integrated over on the right hand side spans the path (or circuit) on

the left hand side, like a soap film on a loop of wire.  (The best way to figure out

the sign is to use Lenz’ law: the induced current will generate a magnetic field

opposing the changing of the external  field, so if an external upward field is

decreasing, the current thereby generated around the loop will give an upward

pointing field.)

 It may seem that the integral on the right hand side is not very clearly defined,

because if the path or circuit lies in a plane, the natural choice of spanning surface

(the "soap film") is flat, but how do you decide what surface to choose to do the

integral over for a wire bent into a circuit that doesn’t lie in a plane?   The answer

is that it doesn’t matter what surface you choose, as long as the wire forms its

boundary.  Consider two different surfaces both having the wire as a boundary

(just as both the northern hemisphere of the earth’s surface and the southern

hemisphere have the equator as a boundary).  If you add these two surfaces

together, they form a single closed surface, and we know that for a closed

surface  .  This implies that   for one of the two surfaces

bounded by the path is equal to   for the other one, so that the two will

add to zero for the whole closed surface.  But don’t forget these integrals for the

whole closed surface are defined with the little area vectors pointing outwards

from the enclosed volume. By imagining two surfaces spanning the wire that are

actually close to each other, it is clear that the integral over one of them is equal to

the integral over the other if we take the   vectors to point in the same direction

for both of them, which in terms of the enclosed volume would be outwards for

one surface, inwards for the other one. The bottom line of all this is that the

surface integral   is the same for any surface spanning the path, so it

doesn’t matter which we choose. 

The equation analogous to the electrostatic version of the third equation given

above, but for the magnetic field, is Ampere's law, 

 for

the magnetostatic case,

 where the currents counted are those threading through the path we're integrating

around, so if there is a soap film spanning the path, these are the currents that

punch through the film (of course, we have to agree on a direction, and subtract

currents flowing in the opposite direction).

 We must now consider whether this equation, like the electrostatic one, has

limited validity. In fact, it was not questioned for a generation after Ampere wrote

it down: Maxwell's great contribution, in the 1860's, was to realize that it

was not always valid.

 

When Does Ampere's Law Go Wrong?

 A simple example to see that something must be wrong with Ampere's Law in

the general case is given by Feynman in his Lectures in Physics (II, 18-3).

Suppose we use a hypodermic needle to insert a spherically symmetric blob of

charge in the middle of a large vat of solidified jello (which we assume conducts

electricity).  Because of electrostatic repulsion, the charge will dissipate, currents

will flow outwards in a spherically symmetric way.  Question: does this outward-

flowing current distribution generate a magnetic field?  The answer must be no ,

because since we have a completely spherically symmetric situation, it could only

generate a spherically symmetric magnetic field.  But the only possible such fields

are one pointing outwards everywhere and one pointing inwards everywhere, both

corresponding to non-existent monopoles.  So, there can be no magnetic field.

 However, imagine we now consider checking Ampere's law by taking as a path a

horizontal circle with its center above the point where we injected the charge

(think of a halo above someone’s head.)  Obviously, the left hand side of

Ampere's equation is zero, since there can be no magnetic field.  On the other

hand, the right hand side is most definitely not zero, since some of the outward

flowing current is going to go through our circle.  So the equation must be wrong.

 Ampere's law was established as the result of large numbers of careful

experiments on all kinds of current distributions.  So how could it be that

something of the kind we describe above was overlooked?  The reason is really

similar to why electromagnetic induction was missed for so long.  No-one thought

about looking at changing fields, all the experiments were done on steady

situations.  With our ball of charge spreading outward in the jello, there is

obviously a changing electric field.  Imagine yourself in the jello near where the

charge was injected: at first, you would feel a strong field from the nearby

concentrated charge, but as the charge spreads out spherically, some of it going

past you, the field will decrease with time.

 

Maxwell's Example

Maxwell himself gave a more practical example:  consider Ampere's law for the

usual infinitely long wire carrying a steady current I , but now break the wire at

some point and put in two large circular metal plates, a capacitor, maintaining the

steady current I in the wire everywhere else, so that charge is simply piling up on

one of the plates and draining off the other.

 Looking now at the wire some distance away from the plates, the situation

appears normal, and if we put the usual circular path around the wire, application

of Ampere's law tells us that the magnetic field at distancer , from 

is just

(Reminder on field direction:  the right hand rule if you curl the fingers of your

right hand around an imaginary wire, a current flowing in the direction indicated

by your thumb will generate circular magnetic field lines in the direction indicated

by your fingers.)

Recall, however, that we defined the current threading the path in terms of current

punching through a soap film spanning the path, and said this was independent of

whether the soap film was flat, bulging out on one side, or whatever. With a

single infinite wire, there was no escape no contortions of this covering surface

could wriggle free of the wire going through it (actually, if you distort the surface

enough, the wire could penetrate it several times, but you have to count the net

flow across the surface, and the new penetrations would come in pairs with the

current crossing the surface in opposite directions, so they would cancel).

 Once we bring in Maxwell's parallel plate capacitor, however, there is a way to

distort the surface so that no current penetrates it at all: we can run it between the

plates!

The question then arises: can we rescue Ampere's law by adding another term just

as the electrostatic version of the third equation was rescued by adding Faraday's

induction term? The answer is of course yes:  although there is no current crossing

the surface if we put it between the capacitor plates, there is certainly a changing

electric field , because the capacitor is charging up as the current I flows in.

Assuming the plates are close together, we can take all the electric field lines from

the charge q on one plate to flow across to the other plate, so the total electric flux

across the surface between the plates, 

Now, the current in the wire, I , is just the rate of change of charge on the plate, 

Putting the above two equations together, we see that 

Ampere's law can now be written in a way that is correct no matter where we put

the surface spanning the path we integrate the magnetic field around: 

This is Maxwell’s fourth equation.

Notice that in the case of the wire, either the current in the wire, or the increasing

electric field, contribute on the right hand side, depending on whether we have the

surface simply cutting through the wire, or positioned between the plates.

(Actually, more complicated situations are possible we could imaging the

surface partly between the plates, then cutting through the plates to get out!  In

this case, we would have to figure out the current actually in the plate to get the

right hand side, but the equation would still apply).

 

"Displacement Current"

 Maxwell referred to the second term on the right hand side, the changing electric

field term, as the "displacement current".  This was an analogy with a dielectric

material.  If a dielectric material is placed in an electric field, the molecules are

distorted, their positive charges moving slightly to the right, say, the negative

charges slightly to the left.  Now consider what happens to a dielectric in

an increasing electric field.  The positive charges will be displaced to the right by

a continuously increasing distance, so, as long as the electric field is increasing in

strength, these charges are moving: there is actually a displacement current .

(Meanwhile, the negative charges are moving the other way, but that is a current

in the same direction, so adds to the effect of the positive charges' motion.)

Maxwell's picture of the vacuum, the aether, was that it too had dielectric

properties somehow, so he pictured a similar motion of charge in the vacuum to

that we have just described in the dielectric.  This is why the changing electric

field term is often called the "displacement current", and in Ampere's law

(generalized) is just added to the real current, to give Maxwell's fourth and final

equation. 

 

Another Angle on the Fourth Equation: the Link to Charge Conservation

Going back for a moment to Ampere's law, we stated it as:          

 for magnetostatics

where the currents counted are those threading through the path we’re integrating

around, so if there is a soap film spanning the path, these are the currents that

punch through the film. Our mental picture here is usually of a few thin wires,

maybe twisted in various ways, carrying currents. More generally, thinking of

electrolytes, or even of fat wires, we should be envisioning a current density

varying from point to point in space. In other words, we have a flux of current and

the natural expression for the current threading our path is (analogous to the

magnetic flux in the third equation) to write a surface integral of the current

density   over a surface spanning the path, giving for magnetostatics 

path integral  ,  (surface integral, over surface spanning path)

The question then arises as to whether the surface integral we have written on the

right hand side above depends on which surface we choose spanning the path.

From an argument exactly parallel to that for the magnetic flux in the third

equation (see above), this will be true if and only if    for

a closed surface (with the path lying in the surface this closed surface is made up

by combining two different surfaces spanning the path).

 Now,   taken over a closed surface is just the net current flow out of the

enclosed volume. Obviously, in a situation with steady currents flowing along

wires or through conductors, with no charge piling up or draining away from

anywhere, this is zero. However, if the total electric charge q , say, enclosed by

the closed surface is changing as time goes on, then evidently 

where we put in a minus sign because, with our convention,   is a little vector

pointing outwards, so the integral represents net flow of charge out from the

surface, equal to the rate of decrease of the enclosed total charge.

 To summarize: if the local charge densities are changing in time, that is, if charge

is piling up in or leaving some region, then   over a closed surface

around that region. That implies that   over one surface spanning the wire

will be different from   over another surface spanning the wire if these two

surfaces together make up a closed surface enclosing a region containing a

changing amount of charge.

 The key to fixing this up is to realize that

although 

 it can be written as another surface integral over the same surface, using

the first Maxwell equation, that is, the integral over a closed surface

where q is the total charge in the volume enclosed by the surface. 

By taking the time rate of change of both sides, we find 

Putting this together with   gives: 

for any closed surface, and consequently this is a surface integral that must be the

same for any surface spanning the path or circuit!  (Because two different surfaces

spanning the same circuit add up to a closed surface. We’ll ignore the technically

trickier case where the two surfaces intersect each other, creating multiple

volumes there one must treat each created volume separately to get the signs

right.) 

Therefore, this is the way to generalize Ampere's law from the magnetostatic

situation to the case where charge densities are varying with time, that is to

say the path integral

and this gives the same result for any surface spanning the path. 

 

A Sheet of Current: A Simple Magnetic Field

As a preliminary to looking at electromagnetic waves, we consider the magnetic

field configuration from a sheet of uniform current of large extent.  Think of the

sheet as perpendicular to this sheet of paper, the current running vertically down

into the paper.   It might be helpful to visualize the sheet as many equal parallel

fine wires uniformly spaced close together, carrying equal (small) currents.

 

The magnetic field from this current sheet can be found using Ampere's law

applied to a rectangular contour in the plane of the paper, with the current sheet

itself bisecting the rectangle, so the rectangle's top and bottom are equidistant

from the current sheet in opposite directions.

 Applying Ampere’s law to the above rectangular contour, there are contributions

to   (taken clockwise) only from the top and bottom, and they add to

give 2BL if the rectangle has side L. The total current enclosed by the rectangle

is IL, taking the current density of the sheet to be I amperes per meter (how many

little wires per meter multiplied by the current in each wire). 

Thus,   immediately gives: 

B = µ0I/2 

a magnetic field strength independent of distance d from the sheet. (This is the

magnetostatic analog of the electrostatic result that the electric field from an

infinite sheet of charge is independent of distance from the sheet.) In real life,

where there are no infinite sheets of anything, these results are good

approximations for distances from the sheet small compared with the extent of the

sheet. 

 

Switching on the Sheet: How Fast Does the Field Build Up? 

Consider now how the magnetic field develops if the current in the sheet is

suddenly switched on at time t = 0. We will assume that sufficiently close to the

sheet, the magnetic field pattern found above using Ampere's law is rather rapidly

established.  

In fact, we will assume further that the magnetic field spreads out from the sheet

like a tidal wave, moving in both directions at some speed v , so that after

time t the field within distance vt of the sheet is the same as that found above for

the magnetostatic case, but beyond vt there is at that instant no magnetic field

present.

Let us now apply Maxwell's equations to this guess to see if it can make sense.

Certainly Ampere's law doesn't work by itself, because if we take a rectangular

path as we did in the previous section, for d < vteverything works as before, but

for a rectangle extending beyond the spreading magnetic field, d > vt , there will

be no magnetic field contribution from the top and bottom of the rectangle, and

hence 

 

but there is definitely enclosed current!  

We are forced to conclude that for Maxwell's fourth equation to be correct, there

must also be a changing electric field through the rectangular contour. 

Let us now try to nail down what this electric field through the contour must look

like.  First, it must be through the contour, that is, have a component

perpendicular to the plane of the contour, in other words, perpendicular to the

magnetic field.  In fact, electric field components in other directions won't affect

the fourth equation we are trying to satisfy, so we shall ignore them.  Notice first

that for a rectangular contour with d < vt,  Ampere's law works, so we don't want

a changing electric field through such a contour (but a constant electric field

would be ok). 

Now apply Maxwell's fourth equation to a rectangular contour with d > vt, 

It is:  path integral   (over surface spanning path).   

For the rectangle shown above, the integral on the left hand side is zero because   

is perpendicular to   along the sides, so the dot product is zero, and   is zero at

the top and bottom, because the outward moving "wave" of magnetic field hasn’t

gotten there yet. Therefore, the right hand side of the equation must also be zero. 

We know  , so we must have:    

 

Finding the Speed of the Outgoing Field Front: the Connection with Light

So, as long as the outward moving front of magnetic field, travelling at v , hasn't

reached the top and bottom of the rectangular contour, the electric field through

the contour increases linearly with time, but the increase drops to zero (because

Ampere's law is satisfied) the moment the front reaches the top and bottom of the

rectangle. The simplest way to get this behavior is to have an electric field of

strength E, perpendicular to the magnetic field, everywhere there is a magnetic

field, so the electric field also spreads outwards at speed v.  (Note that, unlike the

magnetic field, the electric field must point the same way on both sides of the

current sheet, otherwise its net flux through the rectangle would be zero.) 

After time t , then, the electric field flux through the rectangular contour   

(in the yz-plane in the diagram above) will be just field x area = E.2.vtL , and the

rate of change will be 2EvL . (It's spreading both ways, hence the 2). 

Therefore ε0E.2.vL = -LI , the electric field is downwards and of

strength E = I/(2ε0v ). 

Since B = µ0I/2, this implies: 

B = µ0ε0vE.

But we have another equation linking the field strengths of the electric and

magnetic fields, Maxwell's third equation: 

We can apply this equation to a rectangular contour with sides parallel to

the E field, one side being within vt of the current sheet, the other more distant, so

the only contribution to the integral is EL from the first side, which we take to

have length L.  (This contour is all on one side of the current sheet.) The area of

the rectangle the magnetic flux is passing through will be increasing at a

rate Lv (square meters per second) as the magnetic field spreads outwards. 

It follows that 

E = vB.

Putting this together with the result of the fourth equation, 

B = µ0ε0vE, 

we deduce

v2 = 1/µ0ε0 

Substituting the defined value of µ0, and the experimentally measured value of ε0,

we find that the electric and magnetic fields spread outwards from the switched-

on current sheet at a speed of 3 x 108 meters per second. 

To understand how this relates to wave propagation, imagine now that shortly

after the current is switched on, the value of the current is suddenly doubled. 

Repeating the argument above for this more complicated situation, we find the

following scenario:

We could have ramped up the field in a series of steps and the profile of the

magnetic and electric fields would, effectively, be a graph of how the current built

up over time.

The next step is to imagine an electric current in the sheet that’s oscillating like a sine

wave as a function of time: the magnetic and electric fields will evidently be sine waves

too! In fact, this is how electromagnetic waves are generated. Of course, there’s no

such thing as an infinite current sheet, an antenna has an oscillating current going up

and down a wire. But the mechanism is essentially the same: the only difference is that

the geometry of the waves is complicated. Far away, they’ll look like expanding

spheres, a three-dimensional version of the ripples on a pond when a stone falls in,

instead of propagating planes. But at large distances a small fraction of these expanding

spheres w, and that looks like a series of planes. The picture above of how the electric

and magnetic fields relate to each other and to the direction of propagation of the wave

is correct.

Formu

lationName

"Microscopic" equations

(material contributes to

charge and current!)

"Macroscopic" equations

(easy free charge and

current in material)

Integr

al

Gauss

's law

Gauss

's law

for

magn

etism

Same as microscopic

Max

well–

Farad

ay

equat

ion

(Fara

day's

law of

induc

tion)

Same as

microscopic; 

Ampè

re's

circui

tal

law (

with

Max

well's

corre

ction)

Differe

ntial

Gauss

's law

Gauss

's law

for

magn

etism

Same as microscopic

Max

well–

Farad

ay

equat

ion

(Fara

day's

law of

induc

tion)

Same as microscopic

Ampè

re's

circui

tal

law

(with

Max

well's

corre

ction)

This is how Maxwell discovered a speed equal to the speed of light from a purely

theoretical argument based on experimental determinations of forces between

currents in wires and forces between electrostatic charges. This of course led to

the realization that light is an electromagnetic wave, and that there must be other

such waves with different wavelengths.  Hertz detected other waves, of much

longer wavelengths, experimentally, and this led directly to radio, t v, radar, cell

phones,  etc. 

Maxwell's equations – free space and linear isotropic media :

(i)  In Free Space: In absence of any free charges or currents or in the empty

space i.e.,when q or p and i or J are zero. Then Maxwell’s equations take the form

(ii) In Isotropic Medium:

In an isotropic dielectric (or non-conducting isotropic medium)

 

D =ε E , µ H ,J =σ E =0 and  p=0

The Magnetic Vector Potential :

Electric fields generated by stationary charges obey 

This immediately allows us to write 

since the curl of a gradient is automatically zero. In fact, whenever we come

across an irrotational vector field in physics we can always write it as the gradient

of some scalar field. This is clearly a useful thing to do, since it enables us to

replace a vector field by a much simpler scalar field. The quantity   in the above

equation is known as the electric scalar potential.

Magnetic fields generated by steady currents (and unsteady currents, for that

matter) satisfy 

This immediately allows us to write 

since the divergence of a curl is automatically zero. In fact, whenever we come

across a solenoidal vector field in physics we can always write it as the curl of

some other vector field. This is not an obviously useful thing to do, however,

since it only allows us to replace one vector field by another. Nevertheless, Eq.  is

one of the most useful equations we shall come across in this lecture course. The

quantity   is known as the magnetic vector potential.

We know from Helmholtz's theorem that a vector field is fully specified by its

divergence and its curl. The curl of the vector potential gives us the magnetic field

via Eq. . However, the divergence of   has no physical significance. In fact, we

are completely free to choose   to be whatever we like. Note that,

according to Eq. , the magnetic field is invariant under the transformation 

In other words, the vector potential is undetermined to the gradient of a scalar

field. This is just another way of saying that we are free to choose  . Recall

that the electric scalar potential is undetermined to an arbitrary additive constant,

since the transformation 

leaves the electric field invariant in Eq. . The transformations are examples of

what mathematicians call gauge transformations. The choice of a particular

function  or a particular constant   is referred to as a choice of the gauge. We

are free to fix the gauge to be whatever we like. The most sensible choice is the

one which makes our equations as simple as possible. The usual gauge for the

scalar potential   is such that   at infinity. The usual gauge for   is such

that 

This particular choice is known as the Coulomb gauge.

It is obvious that we can always add a constant to   so as to make it zero at

infinity. But it is not at all obvious that we can always perform a gauge

transformation such as to make  zero. Suppose that we have found some

vector field   whose curl gives the magnetic field but whose divergence in non-

zero. Let 

The question is, can we find a scalar field   such that after we perform the gauge

transformation we are left with  . Taking the divergence of Eq.  it is

clear that we need to find a function   which satisfies 

But this is just Poisson's equation. We know that we can always find a unique

solution of this equation This proves that, in practice, we can always set the

divergence of   equal to zero.

Let us again consider an infinite straight wire directed along the  -axis and

carrying a current  . The magnetic field generated by such a wire is written 

We wish to find a vector potential   whose curl is equal to the above magnetic

field, and whose divergence is zero. It is not difficult to see that 

fits the bill. Note that the vector potential is parallel to the direction of the current.

This would seem to suggest that there is a more direct relationship between the

vector potential and the current than there is between the magnetic field and the

current. The potential is not very well-behaved on the  -axis, but this is just

because we are dealing with an infinitely thin current.

Let us take the curl of Eq.  We find that 

where use has been made of the Coulomb gauge condition . We can combine the

above relation with the field equation to give 

Writing this in component form, we obtain 

But, this is just Poisson's equation three times over. We can immediately write the

unique solutions to the above equations: 

   

   

   

These solutions can be recombined to form a single vector solution 

Of course, we have seen a equation like this before: 

Equations are the unique solutions (given the arbitrary choice of gauge) to the

field equations : they specify the magnetic vector and electric scalar potentials

generated by a set of stationary charges, of charge density  , and a set of

steady currents, of current density  . Incidentally, we can prove that above

Eq. satisfies the gauge condition   by repeating the analysis of Eqs.

(with   and  ), and using the fact that   for steady

currents.

Scalar potential

Scalar potential, simply stated, describes the situation where the difference in the

potential energies of an object in two different positions depends only on the

positions, not upon the path taken by the object in traveling from one position to

the other. It is a scalar field in three-space: a directionless value (scalar) that

depends only on its location. A familiar example is potential energy due to

gravity.

gravitational potential well of an increasing mass where 

A scalar potential is a fundamental concept in vector analysis and physics (the

adjective scalar is frequently omitted if there is no danger of confusion

with vector potential). The scalar potential is an example of a scalar field. Given

a vector field F, the scalar potential P is defined such that:

where ∇P is the gradient of P and the second part of the equation is minus the

gradient for a function of the Cartesian coordinates x,y,z. In some cases,

mathematicians may use a positive sign in front of the gradient to define the

potential. Because of this definition of P in terms of the gradient, the direction

of F at any point is the direction of the steepest decrease of P at that point, its

magnitude is the rate of that decrease per unit length.

In order for F to be described in terms of a scalar potential only, the following

have to be true:

1. , where the integration is over

a Jordan arc passing from location a to location b and P(b) is P

evaluated at location b .

2. , where the integral is over any simple closed path,

otherwise known as a Jordan curve.

3.

The first of these conditions represents the fundamental theorem of the

gradient and is true for any vector field that is a gradient of

a differentiable single valued scalar field P. The second condition is a

requirement of F so that it can be expressed as the gradient of a scalar

function. The third condition re-expresses the second condition in terms of

the curl of F using the fundamental theorem of the curl. A vector field F that

satisfies these conditions is said to be irrotational (Conservative).

Scalar potentials play a prominent role in many areas of physics and

engineering. The gravity potential is the scalar potential associated with the

gravity per unit mass, i.e., the acceleration due to the field, as a function of

position. The gravity potential is the gravitational potential energy per unit

mass. In electrostatics the electric potential is the scalar potential associated

with the electric field, i.e., with the electrostatic force per unit charge. The

electric potential is in this case the electrostatic potential energy per unit

charge. In fluid dynamics, irrotational lamellar fields have a scalar potential

only in the special case when it is a Laplacian field. Certain aspects of

the nuclear force can be described by a Yukawa potential. The potential play a

prominent role in theLagrangian and Hamiltonian formulations of classical

mechanics. Further, the scalar potential is the fundamental quantity

in quantum mechanics.

Not every vector field has a scalar potential. Those that do are

called conservative, corresponding to the notion of conservative force in

physics. Examples of non-conservative forces include frictional forces,

magnetic forces, and in fluid mechanics a solenoidal field velocity field. By

the Helmholtz decomposition theorem however, all vector fields can be

describable in terms of a scalar potential and corresponding vector potential.

In electrodynamics the electromagnetic scalar and vector potentials are known

together as the electromagnetic four-potential.

Integrablity Condition :

If F is a conservative vector field (also called irrotational, curl-free, or potential),

and its components have continuous partial derivatives, the potential of F with

respect to a reference point   is defined in terms of the line integral:

where C is a parametrized path from   to 

The fact that the line integral depends on the path C only through its terminal

points   and   is, in essence, the path independence property of a conservative

vector field. The fundamental theorem of calculus for line integrals implies that

if V is defined in this way, then   so that V is a scalar potential of the

conservative vector field F. Scalar potential is not determined by the vector field

alone: indeed, the gradient of a function is unaffected if a constant is added to it.

If V is defined in terms of the line integral, the ambiguity of V reflects the

freedom in the choice of the reference point 

The Vector and Scalar Potentials

Problem:

Given Maxwell's four equations, demonstrate the existence of a vector magnetic

potential and a scalar electric potential. Derive field equations for these potentials.

Solution:

Maxwell's four equations read:

 x H =  0  E/ t + j 1.

 x E = -  0  H/ t 2.

 . H = 0 3.

 . E =  / 0 4.

where H is the magnetic field (A/m), E is the electric field (V/m), j is the vector

current density (A/m2),  0 = 8.8542 x 10-12 F/m is the permittivity of free

space,  0 = 4  x 10-7H/m is the permeability of free space, and   is the scalar

charge density (C/m3).

 

Existence Of The Vector And Scalar Potentials

Let us begin with eq. 3. By a theorem of vector calculus,  .H = 0 if and only if

there exists a vector field A such that

H =   x A 5.

Call A the vector magnetic potential.

Let us now take eq. 2. We will find a characteristic solution Ec and a particular

solution Ep. The total electric field will then be the sum Ec + Ep.

a. Characteristic solution:   x Ec = 0. By another theorem of vector calculus, this

expression can be true if and only if there exists a scalar field   such that Ec =

-  . Call   the scalar electric potential.

b. Particular solution:   x Ep = -  0  H/ t, and H =   x A allows us to write

 x Ep = -  0  (  x A)/ t =   x (-  0 A/ t)

We may obtain a particular solution Ep by simply choosing Ep = -  0  A/ t.

c. Total electric field:

E = Ec + Ep = -   -  0  A/ t 6.

Relationships Between The Vector And Scalar Potentials

Let us now substitute the expressions derived above intoeqs. 4 and 1. From eq. 4,

we obtain

. (-   -  0  A/ t) =  / 0 7.

and from eq. 1, we obtain

 x (  x A) =  0  (-   -  0  A/ t)/ t + j 8.

By still another theorem of vector calculus, we have the identity

 x (  x A) =  (  . A) -  2A

so that eq. 8 becomes

(  . A) -  2A =  0  (-   -  0  A/ t)/ t + j 8a.

or, after some simplification

(  . A) -  2A = -  ( 0  / t) -  0  0 2A/ t2+ j 9.

Since   x grad = 0, eq. 9 may be manipulated by taking the curl of both sides.

The terms involving the gradient then vanish leaving us with the identity

 x ( 2A) =   x ( 0  0 2A/ t2 + j)

from which we may infer, without loss of generality, that

2A =  0  0  2A/ t2 + j 10.

or 2A -  0  0  2A/ t2 = j 10a.

Eq. 10a is one of the field equations we sought. We may simplify eq.

10a somewhat if we recognize that  0  0 = 1/c2 where c is the speed of light. We

now introduce a new operator  2 defined by

2 =  2 - (1/c2)  2 / t2

so that eq. 10a becomes

2 A = j 11.

Using eq. 10 in eq. 9, we obtain

(  . A) = -  ( 0  / t)

from which we may infer, again without loss of generality, that

 . A = -  0  / t 12.

We must next rewrite eq. 7 by distributing the operator   .

 . (-   -  0  A/ t) =  / 0 7.

becomes -  2  -  0  (  .A)/ t =  / 0 7a.

Substituting eq. 12 into eq. 7a gives

-  2  -  0  (-  0 / t)/ t =  / 0

which may be simplified:

2  -  0  0  2 / t2 = -  / 0 13.

or, recalling that  0  0 = 1/c2, and using the operator  2

2   = -  / 0 13a.

Eq. 13a is the other field equation that we sought.

To summarize:

1. The existence of a vector magnetic potential A and a scalar electric

potential   was demonstrated. The respective field equations for A

and   were found to be  2 A = j

and  2   = -  / 0

1. where j is the vector current density (A/m2),   is the scalar charge

potential (C/m3), and  0 = 8.8542 x 10-12 F/m is the permittivity of free

space.

Dimensional analysis on the above equations shows that A has units of electric

current (amperes) and  has units of electric potential (volts).

The Vector and Scalar Potentials

Problem:

Given Maxwell's four equations, demonstrate the existence of a vector magnetic

potential and a scalar electric potential. Derive field equations for these

potentials.

Solution:

Maxwell's four equations read:

 x H =  0  E/ t + j 1.

 x E = -  0  H/ t 2.

 . H = 0 3.

 . E =  / 0 4.

where H is the magnetic field (A/m), E is the electric field (V/m), j is the vector

current density (A/m2),  0 = 8.8542 x 10-12 F/m is the permittivity of free

space,  0 = 4  x 10-7H/m is the permeability of free space, and   is the scalar

charge density (C/m3).

 

Existence Of The Vector And Scalar Potentials

Let us begin with eq. 3. By a theorem of vector calculus,  .H = 0 if and only if

there exists a vector field A such that

H =   x A 5.

Call A the vector magnetic potential.

Let us now take eq. 2. We will find a characteristic solution Ec and a particular

solution Ep. The total electric field will then be the sum Ec + Ep.

a. Characteristic solution:   x Ec = 0. By another theorem of vector calculus,

this expression can be true if and only if there exists a scalar field   such that

Ec = -  . Call   the scalar electric potential.

b. Particular solution:   x Ep = -  0  H/ t, and H =   x A allows us to write

 x Ep = -  0  (  x A)/ t =   x (-  0 A/ t)

We may obtain a particular solution Ep by simply choosing Ep = -  0  A/ t.

c. Total electric field:

E = Ec + Ep = -   -  0  A/ t 6.

Relationships Between The Vector And Scalar Potentials

Let us now substitute the expressions derived above intoeqs. 4 and 1. From eq. 4,

we obtain

. (-   -  0  A/ t) =  / 0 7.

and from eq. 1, we obtain

 x (  x A) =  0  (-   -  0  A/ t)/ t + j 8.

By still another theorem of vector calculus, we have the identity

 x (  x A) =  (  . A) -  2A

so that eq. 8 becomes

(  . A) -  2A =  0  (-   -  0  A/ t)/ t + j 8a.

or, after some simplification

(  . A) -  2A = -  ( 0  / t) -  0  0 2A/ t2+ j 9.

Since   x grad = 0, eq. 9 may be manipulated by taking the curl of both sides.

The terms involving the gradient then vanish leaving us with the identity

 x ( 2A) =   x ( 0  0 2A/ t2 + j)

from which we may infer, without loss of generality, that

2A =  0  0  2A/ t2 + j 10.

or 2A -  0  0  2A/ t2 = j 10a.

Eq. 10a is one of the field equations we sought. We may simplify eq.

10a somewhat if we recognize that  0  0 = 1/c2 where c is the speed of light.

We now introduce a new operator  2 defined by

2 =  2 - (1/c2)  2 / t2

so that eq. 10a becomes

2 A = j 11.

Using eq. 10 in eq. 9, we obtain

(  . A) = -  ( 0  / t)

from which we may infer, again without loss of generality, that

 . A = -  0  / t 12.

We must next rewrite eq. 7 by distributing the operator   .

 . (-   -  0  A/ t) =  / 0 7.

becomes -  2  -  0  (  .A)/ t =  / 0 7a.

Substituting eq. 12 into eq. 7a gives

-  2  -  0  (-  0 / t)/ t =  / 0

which may be simplified:

2  -  0  0  2 / t2 = -  / 0 13.

or, recalling that  0  0 = 1/c2, and using the operator  2

2   = -  / 0 13a.

Eq. 13a is the other field equation that we sought.

To summarize:

2. The existence of a vector magnetic potential A and a scalar electric

potential   was demonstrated. The respective field equations for A

and   were found to be  2 A = j

and  2   = -  / 0

2. where j is the vector current density (A/m2),   is the scalar charge

potential (C/m3), and  0 = 8.8542 x 10-12 F/m is the permittivity of free

space.

3. Dimensional analysis on the above equations shows that A has units of

electric current (amperes) and  has units of electric potential (volts).

Gauge Theory

In physics, a gauge theory is a type of field theory in which

the Lagrangian is invariant under a continuous group of local transformations.

The term gauge refers to redundant degrees of freedom in the Lagrangian. The

transformations between possible gauges, called gauge transformations, form

a Lie group which is referred to as thesymmetry group or the gauge group of the

theory. Associated with any Lie group is the Lie algebra of group generators. For

each group generator there necessarily arises a corresponding vector field called

the gauge field. Gauge fields are included in the Lagrangian to ensure its

invariance under the local group transformations (called gauge invariance). When

such a theory is quantized, the quanta of the gauge fields are called gauge bosons.

If the symmetry group is non-commutative, the gauge theory is referred to as non-

abelian, the usual example being the Yang–Mills theory.

Many powerful theories in physics are described by Lagrangians which

are invariant under some symmetry transformation groups. When they are

invariant under a transformation identically performed at every point in the space

in which the physical processes occur, they are said to have a global symmetry.

The requirement of local symmetry, the cornerstone of gauge theories, is a stricter

constraint. In fact, a global symmetry is just a local symmetry whose group's

parameters are fixed in space-time.

Gauge theories are important as the successful field theories explaining the

dynamics of elementary particles. Quantum electrodynamics is an abelian gauge

theory with the symmetry group U(1)and has one gauge field, the electromagnetic

four-potential, with the photon being the gauge boson. The Standard Model is a

non-abelian gauge theory with the symmetry group U(1)×SU(2)×SU(3)and has a

total of twelve gauge bosons: the photon, three weak bosons and eight gluons.

Gauge theories are also important in explaining gravitation in the theory

of general relativity. Its case is somewhat unique in that the gauge field is a

tensor, the Lanczos tensor. Theories ofquantum gravity, beginning with gauge

gravitation theory, also postulate the existence of a gauge boson known as

the graviton. Gauge symmetries can be viewed as analogues of the principle of

general covariance of general relativity in which the coordinate system can be

chosen freely under arbitrary diffeomorphisms of spacetime. Both gauge

invariance and diffeomorphism invariance reflect a redundancy in the description

of the system. An alternative theory of gravitation, gauge theory gravity, replaces

the principle of general covariance with a true gauge principle with new gauge

fields.

Historically, these ideas were first stated in the context of classical

electromagnetism and later in general relativity. However, the modern importance

of gauge symmetries appeared first in therelativistic quantum

mechanics of electrons – quantum electrodynamics, elaborated on below. Today,

gauge theories are useful in condensed matter, nuclear and high energy

physics among other subfields.

Gauge fields

The "gauge covariant" version of a gauge theory accounts for this effect by

introducing a gauge field (in mathematical language, an Ehresmann connection)

and formulating all rates of change in terms of the covariant derivative with

respect to this connection. The gauge field becomes an essential part of the

description of a mathematical configuration. A configuration in which the gauge

field can be eliminated by a gauge transformation has the property that its field

strength (in mathematical language, its curvature) is zero everywhere; a gauge

theory is not limited to these configurations. In other words, the distinguishing

characteristic of a gauge theory is that the gauge field does not merely

compensate for a poor choice of coordinate system; there is generally no gauge

transformation that makes the gauge field vanish.

When analyzing the dynamics of a gauge theory, the gauge field must be treated

as a dynamical variable, similarly to other objects in the description of a physical

situation. In addition to itsinteraction with other objects via the covariant

derivative, the gauge field typically contributes energy in the form of a "self-

energy" term. One can obtain the equations for the gauge theory by:

starting from a naïve ansatz without the gauge field (in which the

derivatives appear in a "bare" form);

listing those global symmetries of the theory that can be characterized by a

continuous parameter (generally an abstract equivalent of a rotation angle);

computing the correction terms that result from allowing the symmetry

parameter to vary from place to place; and

reinterpreting these correction terms as couplings to one or more gauge

fields, and giving these fields appropriate self-energy terms and dynamical

behavior.

This is the sense in which a gauge theory "extends" a global symmetry to a local

symmetry, and closely resembles the historical development of the gauge theory

of gravity known as general relativity.

Physical experiments

Gauge theories are used to model the results of physical experiments, essentially

by:

limiting the universe of possible configurations to those consistent with

the information used to set up the experiment, and then

computing the probability distribution of the possible outcomes that the

experiment is designed to measure.

The mathematical descriptions of the "setup information" and the "possible

measurement outcomes" (loosely speaking, the "boundary conditions" of the

experiment) are generally not expressible without reference to a particular

coordinate system, including a choice of gauge. (If nothing else, one assumes that

the experiment has been adequately isolated from "external" influence, which is

itself a gauge-dependent statement.) Mishandling gauge dependence in boundary

conditions is a frequent source of anomalies in gauge theory calculations, and

gauge theories can be broadly classified by their approaches to anomaly

avoidance.

Continuum theories

The two gauge theories mentioned above (continuum electrodynamics and

general relativity) are examples of continuum field theories. The techniques of

calculation in a continuum theory implicitly assume that:

given a completely fixed choice of gauge, the boundary conditions of an

individual configuration can in principle be completely described;

given a completely fixed gauge and a complete set of boundary conditions,

the principle of least action determines a unique mathematical configuration

(and therefore a unique physical situation) consistent with these bounds;

the likelihood of possible measurement outcomes can be determined by:

establishing a probability distribution over all physical situations

determined by boundary conditions that are consistent with the setup

information,

establishing a probability distribution of measurement outcomes

for each possible physical situation, and

convolving these two probability distributions to get a distribution

of possible measurement outcomes consistent with the setup information;

and

fixing the gauge introduces no anomalies in the calculation, due either to

gauge dependence in describing partial information about boundary

conditions or to incompleteness of the theory.

These assumptions are close enough to be valid across a wide range of energy

scales and experimental conditions, to allow these theories to make accurate

predictions about almost all of the phenomena encountered in daily life, from

light, heat, and electricity to eclipses and spaceflight. They fail only at the

smallest and largest scales (due to omissions in the theories themselves) and when

the mathematical techniques themselves break down (most notably in the case

of turbulence and other chaotic phenomena).

Gauge invariance

The electric and magnetic fields are obtained from the vector and scalar potentials

according to the prescription

These fields are important because they determine the electromagnetic forces

exerted on charged particles. Note that the above prescription does not uniquely

determine the two potentials. It is possible to make the following transformation,

known as a gauge transformation, which leaves the fields unaltered:

where   is a general scalar field. It is necessary to adopt some form of

convention, generally known as a gauge condition, to fully specify the two

potentials. In fact, there is only one gauge condition which is consistent with Eqs. 

This is the Lorentz gauge condition,

Note that this condition can be written in the Lorentz invariant form

This implies that if the Lorentz gauge holds in one particular inertial frame then it

automatically holds in all other inertial frames. A general gauge transformation can be

written

Note that even after the Lorentz gauge has been adopted the potentials are

undetermined to a gauge transformation using a scalar field  which satisfies the

sourceless wave equation

However, if we adopt ``sensible'' boundary conditions in both space and time then

the only solution to the above equation is 

Plane Wave Solution

D = 2, Supersymmetric Yang-Mills Quantum

Mechanics

Supersymmetric Yang-Mills quantum mechanics can be obtained by a dimen-

sional reduction of a supersymmetric, D = d + 1 dimensional Yang-Mills quan-

tum field theory to one point in d-dimensional space. The initial local gauge

symmetry of field theory is thus reduced to a global symmetry of the quantum

mechanical system. The simplest ones among such systems are those obtained by

reduction of the N = 1 Yang-Mills gauge theory in two dimensions They are

described by a scalar field φA and a complex fermion λA, where A labels the

generators of the gauge group. The Hamiltonian is1

with the algebra of operators given by

The generator of the gauge transformations is

The supersymmetry charges are

And

The fermionic term of the Hamiltonian is proportional to the generator of the

gauge symmetry and hence it is supposed to vanish on any physical (gauge

invariant) state due to the Gauss law. Hence, the Hamiltonian is simply

Subsequently, we introduce bosonic creation and annihilation operators,

and express them, as well as the fermionic creation and annihilation operators, in

the matrix notation,

where T Aij are the generators of the SU(N) group in the fundamental represen-

tation. Hence, all operators become operator valued N × N matrices. Such a

notation has a very practical feature, namely, any gauge invariant operator can be

simply written as a trace of a product of appropriate operator valued matri- ces .

In situations when it is self-evident we will use a simplified notation for the trace

of any such matrix, namely, tr(O) ≡ (O). We get

and

Therefore, in the case of the SU(3) group we have,

Obviously, the Hamiltonian eq.(11) conserves the fermionic occupation number.

Hence, we can analyze its spectrum separately in each subspace of the Hilbert

space with fixed fermionic occupation number. There are 9 such fermionic sec-

tors because of the Pauli exclusion principle. This Hamiltonian possess also

another symmetry, namely, the particle-hole symmetry , which can be thought of

as a quantum mechanical precursor of the charge conjugation sym- metry in

quantum field theory. This symmetry can be observed as a perfect matching of the

energy eigenvalues in the sectors with nF and 8−nF fermionic quanta2. Thus, it is

sufficient to consider only the sectors with nF = 0, . . . , 4 fermionic quanta.

The cut Fock basis method

The cut Fock space approach turned out to be very useful in the studies of gauge

systems for several reasons. First of all, it is a fully non-perturbative tool. Second,

it treats bosons and fermions on an equal footing, therefore calculations can be

extended to all fermionic sectors without difficulties. Finally, it can be

generalized, to handle SU(N) gauge groups with N ≥ 2, as well as systems defined

in spaces of various dimensionality. The analytic approach used in this work is

inspired by this numerical treat- ment. Hence, we now briefly summarize the

numerical approach in order to introduce the basic notions needed in the

following parts. For a more extensive discussion see The basic ingredient of the

numerical approach is a systematic construction of the Fock basis using the

concepts of elementary bosonic bricks and composite

fermionic bricks (see tables 1 and 2). Elementary bricks are linearly independent

single trace operators, composed uniquely of creation operators, which cannot be

reduced by the Cayley-Hamilton theorem. The set of states obtained by acting

with all possible products of powers of bosonic elementary bricks on the Fock

vacuum spans the bosonic sector of the Hilbert space . As far as the fermionic

sectors are concerned, apart of the bosonic bricks one has to use fermionic bricks

which need not to be single traces operators . The spectra of the Hamiltonian are

obtained by diagonalizing numerically the Hamiltonian matrix evaluated in the

cut basis. Therefore, one introduces a cut-off, Ncut, which limits the maximal

number of quanta contained in the basis

states. Once the cut-off is set the Hamiltonian matrix becomes finite and it is

possible to evaluate all its elements. Subsequently, we can diagonalize numeri-

cally such matrix, as well as any other matrices corresponding to other physical

observables.

The physically reliable results are those obtained in the infinite cut-off limit,

called also the continuum limit . The procedure requires to perform several

calculations with increasing cut-off and to extrapolate the re- sults to the

continuum limit. It was shown that the eigenenergies corresponding to localized

and nonlocalized states behave differently . The former ones converge rapidly to

their physical limit, whereas the latter ones slowly fall to zero.

SU(3) fermionic bricks in the sectors with nF = 1, . . . , 4 fermions.

Our analytic approach uses the cut Fock basis since it provides a system-atic

control of the Hilbert space. However, instead of evaluating the matrix elements

of the Hamiltonian operator, one transforms the eigenvalue problem into a

problem of finding a solution of some recursion relation. The analytic formulae

derived in this article are particularly useful since they describe both the

approximate numerical results and the exact continuum solutions. This is possible

because they are parameterized by the cut-off Ncut. In the limit of infinite Ncut

our formulae give the exact solutions, whereas for any finite cut-off they can be

directly compared with numerical results.

Gauge fixing :  

In the physics of gauge theories, gauge fixing (also called choosing a gauge)

denotes a mathematical procedure for coping with redundant degrees of

freedom in field variables. By definition, a gauge theory represents each

physically distinct configuration of the system as an equivalence class of

detailedlocal field configurations. Any two detailed configurations in the same

equivalence class are related by a gauge transformation, equivalent to

a shear along unphysical axes in configuration space. Most of the quantitative

physical predictions of a gauge theory can only be obtained under a coherent

prescription for suppressing or ignoring these unphysical degrees of freedom.

Although the unphysical axes in the space of detailed configurations are a

fundamental property of the physical model, there is no special set of directions

"perpendicular" to them. Hence there is an enormous amount of freedom involved

in taking a "cross section" representing each physical configuration by

aparticular detailed configuration (or even a weighted distribution of them).

Judicious gauge fixing can simplify calculations immensely, but becomes

progressively harder as the physical model becomes m

Coulomb gauge  :

The Coulomb gauge (also known as the transverse gauge) is much used

in quantum chemistry and condensed matter physics and is defined by the gauge

condition (more precisely, gauge fixing condition)

It is particularly useful for "semi-classical" calculations in quantum mechanics, in

which the vector potential is quantized but the Coulomb interaction is not.

The Coulomb gauge has a number of properties:

1. The potentials can be expressed in terms of instantaneous values of

the fields and densities (in SI units)

where ρ(r, t) is the electric charge density, R = |r - r'| (where r is any position

vector in space and r' a point in the charge or current distribution), the del

operates on r and d3r is the volume element at r'. The instantaneous nature of

these potentials appears, at first sight, to violate causality, since motions of

electric charge or magnetic field appear everywhere instantaneously as changes to

the potentials. This is justified by noting that the scalar and vector potentials

themselves do not affect the motions of charges, only the combinations of their

derivatives that form the electromagnetic field strength. Although one can

compute the field strengths explicitly in the Coulomb gauge and demonstrate that

changes in them propagate at the speed of light, it is much simpler to observe that

the field strengths are unchanged under gauge transformations and to demonstrate

causality in the manifestly Lorentz covariant Lorenz gauge described below.

Another expression for the vector potential, in terms of the time-retarded electric

current density J(r, t), has been obtained to be:[1]

.

2. Further gauge transformations that retain the Coulomb gauge

condition might be made with gauge functions that satisfy ∇2 ψ = 0,

but as the only solution to this equation that vanishes at infinity

(where all fields are required to vanish) is ψ(r, t) = 0, no gauge

arbitrariness remains. Because of this, the Coulomb gauge is said to

be a complete gauge, in contrast to gauges where some gauge

arbitrariness remains, like the Lorenz gauge below.

3. The Coulomb gauge is a minimal gauge in the sense that the integral

of A2 over all space is minimal for this gauge: all other gauges give a

larger integral.[2] The minimum value given by the Coulomb gauge is

.

4. In regions far from electric charge the scalar potential becomes zero.

This is known as the radiation gauge. Electromagnetic radiation was

first quantized in this gauge.

5. The Coulomb gauge is not Lorentz covariant. If a Lorentz

transformation to a new inertial frame is carried out, a further gauge

transformation has to be made to retain the Coulomb gauge condition.

Because of this, the Coulomb gauge is not used in covariant

perturbation theory, which has become standard for the treatment of

relativistic quantum field theories such as quantum electrodynamics.

Lorentz covariant gauges such as the Lorenz gauge are used in these

theories.

6.

For a uniform and constant magnetic field B the vector potential in the

Coulomb gauge is

which can be confirmed by calculating the div and curl of A. The

divergence of A at infinity is a consequence of the unphysical

assumption that the magnetic field is uniform throughout the whole of

space. Although this vector potential is unrealistic in general it can

provide a good approximation to the potential in a finite volume of

space in which the magnetic field is uniform.

7. As a consequence of the considerations above, the electromagnetic

potentials may be expressed in their most general forms in terms of the

electromagnetic fields as

where ψ(r, t) is an arbitrary scalar field called the gauge function. The fields that

are the derivatives of the gauge function are known as pure gauge fields and the

arbitrariness associated with the gauge function is known as gauge freedom. In a

calculation that is carried out correctly the pure gauge terms have no effect on any

physical observable. A quantity or expression that does not depend on the gauge

function is said to be gauge invariant: all physical observables are required to be

gauge invariant. A gauge transformation from the Coulomb gauge to another

gauge is made by taking the gauge function to be the sum of a specific function

which will give the desired gauge transformation and the arbitrary function. If the

arbitrary function is then set to zero, the gauge is said to be fixed. Calculations

may be carried out in a fixed gauge but must be done in a way that is gauge

invariant.

Lorenz Gauge :

The Lorenz gauge is given, in SI units, by:

and in Gaussian units by:

This may be rewritten as:

where Aμ = (φ/c, −A) is the electromagnetic four-potential, ∂μ the 4-

gradient [using the metric signature (+−−−)].

It is unique among the constraint gauges in retaining manifest Lorentz invariance.

Note, however, that this gauge was originally named after the Danish

physicist Ludvig Lorenz and not afterHendrik Lorentz; it is often misspelled

"Lorentz gauge". (Neither was the first to use it in calculations; it was introduced

in 1888 by George F. FitzGerald.)

The Lorenz gauge leads to the following inhomogeneous wave equations for the

potentials:

It can be seen from these equations that, in the absence of current and charge, the

solutions are potentials which propagate at the speed of light.

The Lorenz gauge is incomplete in the sense that there remains a subspace of

gauge transformations which preserve the constraint. These remaining degrees of

freedom correspond to gauge functions which satisfy the wave equation

These remaining gauge degrees of freedom propagate at the speed of light. To

obtain a fully fixed gauge, one must add boundary conditions along the light

cone of the experimental region.

Maxwell's equations in the Lorenz gauge simplify to

where jν = (ρc, −j) is the four-current.

Two solutions of these equations for the same current configuration differ by a

solution of the vacuum wave equation

.

In this form it is clear that the components of the potential separately satisfy

the Klein-Gordon equation, and hence that the Lorenz gauge condition allows

transversely, longitudinally, and "time-like" polarized waves in the four-potential.

The transverse polarizations correspond to classical radiation, i. e., transversely

polarized waves in the field strength. To suppress the "unphysical" longitudinal

and time-like polarization states, which are not observed in experiments at

classical distance scales, one must also employ auxiliary constraints known

as Ward identities. Classically, these identities are equivalent to the continuity

equation

.

Many of the differences between classical and quantum electrodynamics can be

accounted for by the role that the longitudinal and time-like polarizations play in

interactions between charged particles at microscopic distances.

Gauge transformations

Electric and magnetic fields can be written in terms of scalar and vector

potentials, as follows: 

However, this prescription is not unique. There are many different potentials

which can generate the same fields. We have come across this problem before. It

is called gauge invariance. The most general transformation which leaves the   

and   fields unchanged in Eqs.

This is clearly a generalization of the gauge transformation which we found

earlier for static fields: 

where   is a constant. In fact, if   then Eqs.  becomes

We are free to choose the gauge so as to make our equations as simple as

possible. As before, the most sensible gauge for the scalar potential is to make it

go to zero at infinity: 

For steady fields, we found that the optimum gauge for the vector potential was

the so-called Coulomb gauge: 

We can still use this gauge for non-steady fields , that it is always possible to

transform away the divergence of a vector potential, remains valid. One of the

nice features of the Coulomb gauge is that when we write the electric field, 

we find that the part which is generated by charges (i.e., the first term on the

right-hand side) is conservative, and the part induced by magnetic fields (i.e., the

second term on the right-hand side) is purely solenoidal. Earlier on, we proved

mathematically that a general vector field can be written as the sum of a

conservative field and a solenoidal field . Now we are finding that when we split

up the electric field in this manner the two fields have different physical origins:

the conservative part of the field emanates from electric charges, whereas the

solenoidal part is induced by magnetic fields.

Equation  can be combined with the field equation 

(which remains valid for non-steady fields) to give 

With the Coulomb gauge condition,  , the above expression reduces

to 

which is just Poisson's equation. Thus, we can immediately write down an

expression for the scalar potential generated by non-steady fields. It is exactly the

same as our previous expression for the scalar potential generated by steady

fields, namely 

However, this apparently simple result is extremely deceptive. Equation is a

typical action at a distance law. If the charge density changes suddenly at   then

the potential at  responds immediately. However, we shall see later that the full

time-dependent Maxwell's equations only allow information to propagate at the

speed of light (i.e., they do not violate relativity). How can these two statements

be reconciled? The crucial point is that the scalar potential cannot be measured

directly, it can only be inferred from the electric field. In the time-dependent case,

there are two parts to the electric field: that part which comes from the scalar

potential, and that part which comes from the vector potential . So, if the scalar

potential responds immediately to some distance rearrangement of charge density

it does not necessarily follow that the electric field also has an immediate

response. What actually happens is that the change in the part of the electric field

which comes from the scalar potential is balanced by an equal and opposite

change in the part which comes from the vector potential, so that the overall

electric field remains unchanged. This state of affairs persists at least until

sufficient time has elapsed for a light signal to travel from the distant charges to

the region in question. Thus, relativity is not violated, since it is the electric field,

and not the scalar potential, which carries physically accessible information.

It is clear that the apparent action at a distance nature of Eq.  is highly misleading.

This suggests, very strongly, that the Coulomb gauge is not the optimum gauge in

the time-dependent case. A more sensible choice is the so called Lorentz gauge: 

It can be shown, by analogy with earlier arguments , that it is always possible to

make a gauge transformation, at a given instance in time, such that the above

equation is satisfied. Substituting the Lorentz gauge condition into Eq., we obtain 

It turns out that this is a three-dimensional wave equation in which information

propagates at the speed of light. But, more of this later. Note that the magnetically

induced part of the electric field (i.e.,  ) is not purely solenoidal in the

Lorentz gauge. This is a slight disadvantage of the Lorentz gauge with respect to

the Coulomb gauge. However, this disadvantage is more than offset by other

advantages which will become apparent presently. Incidentally, the fact that the

part of the electric field which we ascribe to magnetic induction changes when we

change the gauge suggests that the separation of the field into magnetically

induced and charge induced components is not unique in the general time-varying

case (i.e., it is a convention).

Poynting's Theorem  :

In electrodynamics, Poynting's theorem is a statement of energy conservation for

the electromagnetic field, in the form of a partial differential equation, due to the

British physicist John Henry Poynting. Poynting's theorem is analogous to

the work-energy theorem in classical mechanics, and mathematically similar to

the continuity equation, because it relates the energy stored in the electromagnetic

field to the work done on a charge distribution (i.e. an electrically charged object),

through energy flux

Poynting's Theorem, Work and Energy

Recall from elementary physics that the rate at which work is done on an electric

charge by an electromagnetic field is: 

If one follows the usual method of constructing a current density made up of

many charges, it is easy to show that this generalizes to: 

for the rate at which an electric field does work on a current density throughout a

volume. The magnetic field, of course, does no work because the force it creates

is always perpendicular to   or  .

If we use AL to eliminate 

and integrate over a volume of space to compute the rate the electromagnetic field

is doing work within that volume: 

Using: 

(which can be easily shown to be true as an identity by distributing the

derivatives) and then use FL to eliminate  , one gets: 

It is easy to see that: 

from which we see that these terms are the time derivative of the electromagnetic

field energy density: 

Moving the sign to the other side of the power equation above, we get: 

as the rate at which power flows out of the volume   (which is arbitrary).

Equating the terms under the integral: 

where we introduce the Poynting vector 

This has the precise appearance of conservation law. If we apply the divergence

theorem to the integral form to change the volume integral of the divergence

of   into a surface integral of its flux: 

where   is the closed surface that bounds the volume  . Either the differential

or integral forms constitute the Poynting Theorem.

In words, the sum of the work done by all fields on charges in the volume, plus

the changes in the field energy within the volume, plus the energy that flows out

of the volume carried by the field must balance - this is a version of the work-

energy theorem, but one expressed in terms of the fields.

In this interpretation, we see that   must be the vector intensity of the

electromagnetic field - the energy per unit area per unit time - since the flux of the

Poynting vector through the surface is the power passing through it. It's

magnitude is the intensity proper, but it also tells us the direction of energy flow.

With this said, there is at least one assumption in the equations above that is not

strictly justified, as we are assuming that the medium is dispersionless and has no

resistance. We do not allow for energy to appear as heat, in other words, which

surely would happen if we drive currents with the electric field. We also used

the macroscopic field equations and energy densities, which involve a coarse-

grained average over the microscopic particles that matter is actually made up of -

it is their random motion that is the missing heat.

It seems, then, that Poynting's theorem is likely to be applicable in

a microscopic description of particles moving in a vacuum, where their individual

energies can be tracked and tallied: 

but not necessarily so useful in macroscopic media with dynamical dispersion that

we do not yet understand. There we can identify the   term as the rate at

which the mechanical energy of the charged particles that make up   changes

and write: 

(where   is, recall, an outward directed normal) so that this says that the rate at

which energy flows into the volume carried by the electromagnetic field equals

the rate at which the total mechanical plus field energy in the volume increases.

This is a marvelous result!

Momentum can similarly be considered, again in a microscopic description. There

we start with Newton's second law and the Lorentz force law: 

summing with coarse graining into an integral as usual: 

As before, we eliminate sources using the inhomogeneous MEs (this time starting

from the beginning with the vacuum forms): 

or 

Again, we distribute: 

or 

substitute it in above, and add  : 

 

     

   

Finally, substituting in FL: 

 

     

   

Reassembling and rearranging: 

 

     

The quantity under the integral on the left has units of momentum density. We

define: 

to be the field momentum density. Proving that the right hand side of this

interpretation is consistent with this is actually amazingly difficult. It is simpler to

just define the Maxwell Stress Tensor: 

In terms of this, with a little work one can show that: 

That is, for each component, the time rate of change of the total momentum (field

plus mechanical) within the volume equals the flux of the field momentum

through the closed surface that contains the volume.

Example Plane Parallel Capacitor

The plane parallel capacitor of Fig. is familiar from Example . The circular

electrodes are perfectly conducting, while the region between the electrodes is

free space. The system is driven by a voltage source distributed around the edges

of the electrodes. Between the electrodes, the electric field is simply the voltage

divided by the plate spacing

while the magnetic field that follows from the integral form of Ampère's law is

Figure  Plane parallel circular electrodes are driven by a distributed voltage

source. Poynting flux through surface denoted by dashed lines accounts for rate of

change of electric energy stored in the enclosed volume.

Consider the application of the integral version of (8) to the surface S enclosing

the region between the electrodes in Fig. First we determine the power flowing

into the volume through this surface by evaluating the left-hand side of (8). The

density of power flow follows from (11) and (12).

The top and bottom surfaces have normals perpendicular to this vector, so the

only contribution comes from the surface at r = b. Because S is constant on that

surface, the integration amounts to a multiplication.

where

Here the expression has been written as the rate of change of the energy stored in

the capacitor. With Eagain given by (11), we double-check the expression for the

time rate of change of energy storage.

From the field viewpoint, power flows into the volume through the surface at r =

b and is stored in the form of electrical energy in the volume between the plates.

In the quasistatic approximation used to evaluate the electric field, the magnetic

energy storage is neglected at the outset because it is small compared to the

electric energy storage. As a check on the implications of this approximation,

consider the total magnetic energy storage. From (12),

Comparison of this expression with the electric energy storage found in (15)

shows that the EQS approximation is valid provided that

For a sinusoidal excitation of frequency  , this gives

where c is the free space velocity of light

Example . Long Solenoidal Inductor

The perfectly conducting one-turn solenoid of Fig. is familiar from Example . In

terms of the terminal current i = Kd, the magnetic field intensity inside is ,

while the electric field is the sum of the particular and conservative homogeneous

parts [ for the particular part and Eh for the conservative part].

Figure One-turn solenoid surrounding volume enclosed by surface S denoted by

dashed lines. Poynting flux through this surface accounts for the rate of change of

magnetic energy stored in the enclosed volume.

Consider how the power flow through the surface S of the volume enclosed by the

coil is accounted for by the time rate of change of the energy stored. The Poynting

flux implied by (19) and (20) is

This Poynting vector has no component normal to the top and bottom surfaces of

the volume. On the surface at r = a, the first term in brackets is constant, so the

integration on S amounts to a multiplication by the area. Because Eh is

irrotational, the integral of Eh   ds = E  h rd  around a contour at r = a must be

zero. For this reason, there is no net contribution of Eh to the surface integral.

where

Here the result shows that the power flow is accounted for by the rate of change

of the stored magnetic energy. Evaluation of the right hand side of (8), ignoring

the electric energy storage, indeed gives the same result.

The validity of the quasistatic approximation is examined by comparing the

magnetic energy storage to the neglected electric energy storage. Because we are

only interested in an order of magnitude comparison and we know that the

homogeneous solution is proportional to the particular solution , the latter can be

approximated by the first term in (20).

We conclude that the MQS approximation is valid provided that the angular

frequency   is small compared to the time required for an electromagnetic wave

to propagate the radius a of the solenoid and that this is equivalent to having an

electric energy storage that is negligible compared to the magnetic energy storage.

A note of caution is in order. If the gap between the "sheet" terminals is made

very small, the electric energy storage of the homogeneous part of the E field can

become large. If it becomes comparable to the magnetic energy storage, the

structure approaches the condition of resonance of the circuit consisting of the gap

capacitance and solenoid inductance. In this limit, the MQS approximation breaks

down. In practice, the electric energy stored in the gap would be dominated by

that in the connecting plates, and the resonance could be described as the coupling

of MQS and EQS systems

The Lorentz force

In physics, particularly electromagnetism, the Lorentz force is the force on

a point charge due to electromagnetic fields. If a particle of charge q moves with

velocity v in the presence of an electric field E and a magnetic field B, then it will

experience a force

Variations on this basic formula describe the magnetic force on a current-carrying

wire (sometimes called Laplace force), the electromotive force in a wire loop

moving through a magnetic field (an aspect of Faraday's law of induction), and

the force on a particle which might be traveling near the speed of

light (relativistic form of the Lorentz force).

The first derivation of the Lorentz force is commonly attributed to Oliver

Heaviside in 1889, although other historians suggest an earlier origin in an 1865

paper by James Clerk Maxwell. Lorentz derived it a few years after Heaviside.

The Lorentz force

The flow of an electric current down a conducting wire is ultimately due to the

motion of electrically charged particles (in most cases, electrons) through the

conducting medium. It seems reasonable, therefore, that the force exerted on the

wire when it is placed in a magnetic field is really the resultant of the forces

exerted on these moving charges. Let us suppose that this is the case.

Let   be the (uniform) cross-sectional area of the wire, and let   be the number

density of mobile charges in the conductor. Suppose that the mobile charges each

have charge   and velocity  . We must assume that the conductor also contains

stationary charges, of charge   and number density   (say), so that the net

charge density in the wire is zero. In most conductors, the mobile charges are

electrons and the stationary charges are atomic nuclei. The magnitude of the

electric current flowing through the wire is simply the number of coulombs per

second which flow past a given point. In one second, a mobile charge moves a

distance  , so all of the charges contained in a cylinder of cross-sectional area   

and length   flow past a given point. Thus, the magnitude of the current

is  . The direction of the current is the same as the direction of motion of

the charges, so the vector current is . According to Eq. , the force

per unit length acting on the wire is 

However, a unit length of the wire contains   moving charges. So, assuming

that each charge is subject to an equal force from the magnetic field (we have no

reason to suppose otherwise), the force acting on an individual charge is 

We can combine this with Eq.  to give the force acting on a charge   moving

with velocity   in an electric field   and a magnetic field  : 

This is called the Lorentz force law, after the Dutch physicist Hendrik Antoon

Lorentz who first formulated it. The electric force on a charged particle is parallel

to the local electric field. The magnetic force, however, is perpendicular to both

the local magnetic field and the particle's direction of motion. No magnetic force

is exerted on a stationary charged particle.

The equation of motion of a free particle of charge   and mass   moving in

electric and magnetic fields is 

according to the Lorentz force law. This equation of motion was first verified in a

famous experiment carried out by the Cambridge physicist J.J. Thompson in

1897. Thompson was investigating cathode rays, a then mysterious form of

radiation emitted by a heated metal element held at a large negative voltage (i.e., a

cathode) with respect to another metal element (i.e., an anode) in an evacuated

tube. German physicists held that cathode rays were a form of electromagnetic

radiation, whilst British and French physicists suspected that they were, in reality,

a stream of charged particles. Thompson was able to demonstrate that the latter

view was correct. In Thompson's experiment, the cathode rays passed though a

region of ``crossed'' electric and magnetic fields (still in vacuum). The fields were

perpendicular to the original trajectory of the rays, and were also mutually

perpendicular.

Let us analyze Thompson's experiment. Suppose that the rays are originally

traveling in the  -direction, and are subject to a uniform electric field   in

the  -direction and a uniform magnetic field   in the  -direction. Let us

assume, as Thompson did, that cathode rays are a stream of particles of mass   

and charge  . The equation of motion of the particles in the  -direction is 

where   is the velocity of the particles in the  -direction. Thompson started off

his experiment by only turning on the electric field in his apparatus, and

measuring the deflection   of the ray in the  -direction after it had traveled a

distance   through the electric field. It is clear from the equation of motion that 

where the ``time of flight''   is replaced by  . This formula is only valid

if  , which is assumed to be the case. Next, Thompson turned on the

magnetic field in his apparatus, and adjusted it so that the cathode ray was no

longer deflected. The lack of deflection implies that the net force on the particles

in the  -direction was zero. In other words, the electric and magnetic forces

balanced exactly. It follows from Eq.  that with a properly adjusted magnetic field

strength 

Thus, Eqs.  and can be combined and rearranged to give the charge to mass ratio

of the particles in terms of measured quantities: 

Using this method, Thompson inferred that cathode rays were made up of

negatively charged particles (the sign of the charge is obvious from the direction

of the deflection in the electric field) with a charge to mass ratio

of   C/kg. A decade later, in 1908, the American Robert Millikan

performed his famous ``oil drop'' experiment, and discovered that mobile electric

charges are quantized in units of   C. Assuming that mobile

electric charges and the particles which make up cathode rays are one and the

same thing, Thompson's and Millikan's experiments imply that the mass of these

particles is   kg. Of course, this is the mass of an electron (the

modern value is   kg), and  C is the charge of an

electron. Thus, cathode rays are, in fact, streams of electrons which are emitted

from a heated cathode, and then accelerated because of the large voltage

difference between the cathode and anode.

Consider, now, a particle of mass   and charge   moving in a uniform

magnetic field,  . According, to Eq , the particle's equation of motion

can be written: 

This reduces to 

Here,   is called the cyclotron frequency. The above equations can

be solved to give 

and 

According to these equations, the particle trajectory is a spiral whose axis is

parallel to the magnetic field. The radius of the spiral is  , where   

is the particle's constant speed in the plane perpendicular to the magnetic field.

The particle drifts parallel to the magnetic field at a constant velocity,  .

Finally, the particle gyrates in the plane perpendicular to the magnetic field at the

cyclotron frequency.

Finally, if a particle is subject to a force   and moves a distance   in a time

interval  , then the work done on the particle by the force is 

The power input to the particle from the force field is 

where   is the particle's velocity. It follows from the Lorentz force law, Eq.  that

the power input to a particle moving in electric and magnetic fields is 

Note that a charged particle can gain (or lose) energy from an electric field, but

not from a magnetic field. This is because the magnetic force is always

perpendicular to the particle's direction of motion, and, therefore, does no work on

the particle [see Eq. ]. Thus, in particle accelerators, magnetic fields are often

used to guide particle motion (e.g., in a circle) but the actual acceleration is

performed by electric fields.

Conservation laws for a system of charges and electromagnetic fields :

conservation laws, in physics, basic laws that together determine which

processes can or cannot occur in nature; each law maintains that the total value of

the quantity governed by that law, e.g., mass or energy, remains unchanged

during physical processes. Conservation laws have the broadest possible

application of all laws in physics and are thus considered by many scientists to be

the most fundamental laws in nature.

Conservation of classical processes :

Most conservation laws are exact, or absolute, i.e., they apply to all possible

processes; a few conservation laws are only partial, holding for some types of

processes but not for others. By the beginning of the 20th cent. physics had

established conservation laws governing the following quantities: energy, mass

(or matter), linear momentum, angular momentum, and electric charge. When the

theory of relativity showed (1905) that mass was a form of energy, the two laws

governing these quantities were combined into a single law conserving the total of

mass and energy.

Conservation of elementary particle properties :

With the rapid development of the physics of elementary particles during the

1950s, new conservation laws were discovered that have meaning only on this

subatomic level. Laws relating to the creation or annihilation of particles

belonging to the baryon and lepton classes of particles have been put forward.

According to these conservation laws, particles of a given group cannot be created

or destroyed except in pairs, where one of the pair is an ordinary particle and the

other is an antiparticle belonging to the same group. Recent work has raised the

possibility that the proton, which is a type of baryon, may in fact be unstable and

decay into lighter products; the postulated methods of decay would violate the

conservation of baryon number. To date, however, no such decay has been

observed, and it has been determined that the proton has a lifetime of at least

1031 years. Two partial conservation laws, governing the quantities known as

strangeness and isotopic spin, have been discovered for elementary particles.

Strangeness is conserved during the so-called strong interactions and the

electromagnetic interactions, but not during the weak interactions associated with

particle decay; isotopic spin is conserved only during the strong interactions.

Conservation of natural symmetries :

One very important discovery has been the link between conservation laws and

basic symmetries in nature. For example, empty space possesses the symmetries

that it is the same at every location (homogeneity) and in every direction

(isotropy); these symmetries in turn lead to the invariance principles that the laws

of physics should be the same regardless of changes of position or of orientation

in space. The first invariance principle implies the law of conservation of linear

momentum, while the second implies conservation of angular momentum. The

symmetry known as the homogeneity of time leads to the invariance principle that

the laws of physics remain the same at all times, which in turn implies the law of

conservation of energy. The symmetries and invariance principles underlying the

other conservation laws are more complex, and some are not yet understood.

Three special conservation laws have been defined with respect to symmetries

and invariance principles associated with inversion or reversal of space, time, and

charge. Space inversion yields a mirror-image world where the "handedness" of

particles and processes is reversed; the conserved quantity corresponding to this

symmetry is called space parity, or simply parity, P. Similarly, the symmetries

leading to invariance with respect to time reversal and charge conjugation

(changing particles into their antiparticles) result in conservation of time

parity, T, and charge parity, C. Although these three conservation laws do not

hold individually for all possible processes, the combination of all three is thought

to be an absolute conservation law, known as the CPT theorem, according to

which if a given process occurs, then a corresponding process must also be

possible in which particles are replaced by their antiparticles, the handedness of

each particle is reversed, and the process proceeds in the opposite direction in

time. Thus, conservation laws provide one of the keys to our understanding of the

universe and its material basis.

Conservation laws (physics)

Principles which state that the total values of specified quantities remain constant

in time for an isolated system. Conservation laws occupy enormously important

positions both at the foundations of physics and in its applications.

Realization in classical mechanics

There are three great conservation laws of mechanics: the conservation of linear

momentum, often referred to simply as the conservation of momentum; the

conservation of angular momentum; and the conservation of energy.

The linear momentum, or simply momentum, of a particle is equal to the product

of its mass and velocity. It is a vector quantity. The total momentum of a system

of particles is simply the sum of the momenta of each particle considered

separately. The law of conservation of momentum states that this total momentum

does not change in time. See Conservation of momentum, Momentum

The angular momentum of a particle is more complicated. It is defined by the

vector product of the position and momentum vectors. The law of conservation of

angular momentum states that the total angular momentum of an isolated system

is constant in time. See Angular momentum

The conservation of energy is perhaps the most important law of all. Energy is a

scalar quantity, and takes two forms: kinetic and potential. The kinetic energy of a

particle is defined to be one-half the product of its mass and the square of its

velocity. The potential energy is loosely defined as the ability to do work. The

total energy is the sum of the kinetic and potential energies, and according to the

conservation law it remains constant in time for an isolated system.

The essential difficulty in applying the conservation of energy law can be

appreciated by considering the problem of two colliding bodies. In general, the

bodies emerge from the collision moving more slowly than when they entered.

This phenomenon seems to violate the conservation of energy, until it is

recognized that the bodies involved may consist of smaller particles. Their

random small-scale motions will require kinetic energy, which robs kinetic energy

from the overall coherent large-scale motion of the bodies that are observed

directly. One of the greatest achievements of nineteenth-century physics was the

recognition that small-scale motion within macroscopic bodies could be identified

with the perceived property of heat. See Conservation of energy, Energy, Kinetic

theory of matter

Position in modern physics

As physics has evolved, the great conservation laws have likewise evolved in both

form and content, but have never ceased to be important guiding principles.

In order to account for the phenomena of electromagnetism, it was necessary to

go beyond the notion of point particles, to postulate the existence of continuous

electric and magnetic fields filling all space. To obtain valid conservation laws,

energy, momentum, and angular momentum must be ascribed to the

electromagnetic fields. See Electromagnetic radiation, Maxwell's

equations, Poynting's vector

In the special theory of relativity, energy and momentum are not independent

concepts. Einstein discovered perhaps the most important consequence of special

relativity, that is, the equivalence of mass and energy, as a consequence of the

conservation laws. The “law” of conservation of mass is understood as an

approximate consequence of the conservation of energy. See Conservation of

mass, Relativity

A remarkable, beautiful, and very fruitful connection has been established

between symmetries and conservation laws. Thus the law of conservation of

linear momentum is understood as a consequence of the homogeneity of space,

the conservation of angular momentum as a consequence of the isotropy of space,

and the conservation of energy as a consequence of the homogeneity of

time. See Symmetry laws (physics)

The development of general relativity, the modern theory of gravitation,

necessitates attention to a fundamental question for the conservation laws: The

laws refer to an “isolated system,” but it is not clear that any system is truly

isolated. This is a particularly acute problem for gravitational forces, which are

long range and add up over cosmological distances. It turns out that the symmetry

of physical laws is actually a more fundamental property than the conservation

laws themselves, for the symmetries remain valid while the conservation laws,

strictly speaking, fail.

In quantum theory, the great conservation laws remain valid in a very strong

sense. Generally, the formalism of quantum mechanics does not allow prediction

of the outcome of individual experiments, but only the relative probability of

different possible outcomes. One might therefore entertain the possibility that the

conservation laws were valid only on the average. However, momentum, angular

momentum, and energy are conserved in every experiment. See Quantum

mechanics, Quantum theory of measurement

Conservation laws of particle type

There is another important class of conservation laws, associated not with the

motion of particles but with their type. Perhaps the most practically important of

these laws is the conservation of chemical elements. From a modern viewpoint,

this principle results from the fact that the small amount of energy involved in

chemical transformations is inadequate to disrupt the nuclei deep within atoms. It

is not an absolute law, because some nuclei decay spontaneously, and at

sufficiently high energies it is grossly violated. See Radioactivity

Several conservation laws in particle physics are of the same character: They are

useful even though they are not exact because, while known processes violate

them, such processes are either unusually slow or require extremely high energy

UNIT IV : Electromagnetic Waves

Plane waves in non-conducting media - Linear and circular

polarization, reflection and refraction at a plane interface -

Fresnel’s law, interference, coherence and diffraction -Waves in a

conducting medium - Propagation of waves in a rectangular wave

guide - Inhomogeneous wave equation and retarded potentials -

Radiation from a localized source - Oscillating electric dipole.

Plane waves in non-conducting media : Propagation in LIH non-

conducting media

Propagation velocity and refractive index The refractive index n of a given

material is defined as the speed of light in vacuum divided by that in the material.

Hence Because  always has a value greater than one, the speed of light in a

material is always less than in a vacuum.

As both Pr and n may vary strongly with frequency, discrepancies may arise when

comparing values of Pf (often the static, DC value is used) with n2 (generally the

value appropriate to optical frequencies).

Relationship between E and B In vacuum E=cB. In a material this is modified to

E=cmB=cB/n

and because

H=BXV

Reflection and refraction at the interface between two different

materials

The aim is to establish the properties of electromagnetic waves when they

encounter a plane interface separating two different non-conducting materials

(generally two different dielectrics). In particular equations will be derived which

give the fractions of the incident wave which are reflected and transmitted at the

interface

Mathematical description of a plane wave not propagating along one of the

principal axes

So far we have only considered waves which propagate along one of the principal

axes. For example a plane wave propagating along the x-axis is described by an

equation of the formE=E0sin(kx-t), one propagating along the y-axis

by E=E0sin(ky-t) etc.

In the following we will need to consider plane waves which do not propagate

along one of these principal axes. However we can always resolve the direction of

the wave onto two or more of the principal axes.

For example in the figure below the wave lies in the x/y-plane and propagates in a

direction which makes an angle  to the y-axis. There are hence components

cos along the y-axis and sin along the x-axis. The equation for this plane wave

is hence of the form

E=E0sin(k(xsin$+ycos$)-wt)

 

The wave vector, k, in the above equations is given by 2/ where  is the

wavelength of the wave. In addition if the wave propagates in a material with a

velocity cm then

k=nw/c

Boundary conditions for E, D, B and H at an interface

In the lectures on dielectrics and magnetic materials it was shown that, in the

absence of free surface charge and conduction currents, the following boundary

conditions existed for E,D, B and H

D and B: Normal components continuous

E and H: Tangential components continuous

These boundary conditions will be used below for electromagnetic waves incident

at the boundary between two materials.

Frequency and direction of reflected and refracted waves

In the figure below a plane electromagnetic wave travelling in a material of

refractive index n1 is incident on the plane boundary with a second material of

refractive index n2. The incident wave propagates at an angle i to the normal of

the boundary and has E- and H-fields of magnitude Ei and Hi. The E-field of the

incident wave lies in the plane of incidence (the x/y-plane), this is known as the E

parallel configuration.

For this configuration the B- and H-fields must be normal to the plane of

incidence.

In general there will be, in addition to the incident wave, a reflected wave at an

angle r and a refracted or transmitted wave at an angle t. Each of these waves

will have E- and H-fields as shown with amplitudes related by the

expression H=nE/0c where n is the refractive index of the appropriate material.

The E-fields of the three waves are given by the following expressions

Ei=Ei0sin(k1(xsini-ycosi)-1t) (A)

Er=Er0sin(k1(xsinr+ycosr)-1t)   (B)

Et=Et0sin(k2(xsint-ycost)-2t) (C)

Where Ei0, Er0 and Et0 are the amplitudes of the E-fields and k1 and

k2 and 1 and 2 are the wavevectors and angular frequencies of the waves in the

two materials.

The boundary condition for E requires that the tangential component (the

component parallel to the boundary) be continuous. Hence if we evaluate this

component on both sides of the boundary we must obtain the same result.

The tangential component of each E-field is given by the magnitude of the E-field

multiplied by cos, where  is the appropriate angle.

On the side of the boundary in material 1 the total tangential component of the E-

field is the difference of the components due to the incident and reflected waves

(see above figure). In material 2 there is only the transmitted wave.

Hence the requirement that the tangential component of the E-field be continuous

can be written 

Eicosiy=0 - Ercosry=0=Etcosty=0                    (D)

Where y=0 indicates that the preceding expression is evaluated at the

boundary y=0.

Substituting in (D) the expressions for Ei, Er and Et given by (A), (B) and (C) and

evaluated for y=0

Ei0cosisin(k1xsini-1t)-Er0cosrsin(k1xsinr-1t)=Et0costsin(k2xsint-2t)      

           (E)

This equation must hold at all times and for all values of x. This is only possible if

all the coefficients of x and all the coefficients of t are equal. This requires

1=2 and k1sini=k1sinr=k2sint

Hence

The above results would have been obtained if E were polarized normal to the

plane of incidence (H polarised parallel to the plane of incidence). These results

hence apply to any incident wave.

Amplitudes of the reflected and refracted waves

The above procedure provided information on the frequencies and directions of

the incident, reflected and transmitted waves. The next step is to derive

expressions which give their relative amplitudes.

Returning to equation (E) above, which is valid when E is parallel to the plane of

incidence, the spatial and time dependent components have been shown to be

equal. Hence (E) reduces to

Ei0cosi - Er0cosi=Et0cost             (F)

Where r has been replaced by i.

In addition the tangential component of the H-field must be continuous at the

boundary between the two materials. For the present case H is normal to the

incident plane so that the tangential component of H is simply H.

For the tangential component of H to be continuous we must therefore have

Hi0+Hr0=Ht0

But H=nE/0c so the above can be written as

n1Ei0+n1Er0=n2Et0                  (G)

Equations (F) and (G) can now be used to eliminate either Er0 or Et0.

Eliminating Et0:

From (G) Et0=(Ei0+Er0)n1/n2

Now eliminating Er0. From (G)

Er0=(n2Et0-n1Ei0)/n1

Substituting into (F)

Ei0cosi-(n2Et0-n1Ei0)/n1cosi=Et0cost

Ei0n1cosi-(n2Et0-n1Ei0)cosi=Et0n1cost

2Ei0n1cosi=Et0(n1cost+n2cosi)

r// and t//, which relate the amplitudes of the reflected and transmitted E-fields to

that of the incident E-field, are known as the reflection and transmission

coefficients for E parallel to the plane of incidence.

The polarisation of E can also be aligned normal to the plane of incidence (H is

now parallel to the plane of incidence). The boundary conditions that the

tangential components of E and H are continuous now require:

Ei0+Er0=Et0

Properties of the Fresnel Equations

The properties can be more easily seen if we consider a special case where one of

the materials is air (n=1) and the other has a refractive index n.

Consider the case where the wave is incident from air. Hence n1=1 and n2=n. The

Fresnel equations become

with sini/sint=n

The above figure (plotted for n=2) shows a number of points:

The reflection coefficients tend to 1 (total reflection) and the magnitude of the

transmission coefficients tend to zero (zero transmission).

For a certain angle B, known as the Brewster angle, r// becomes zero whereas

r remains non-zero. For this angle light can only be reflected with E

perpendicular to the plane of incidence.

If unpolarised light is incident at the Brewster angle then the reflected light will

be polarised.

The Brewster angle is found by setting r//=0.

and hence

  Waves propagating from a material into air

We now have n1=n and n2=1. Fresnel’s equations have the form and nsini=sint

In this case the equations for r// and r and for t// and t are interchanged compared

to those for when the wave is incident from air.

However because nsini=sint t>i. For i equal to a certain value, the

critical angle c, sint becomes equal to unity and hence t=90.

At this point both reflection coefficients become equal to one and the waves are

totally reflected. As the value of sint can not exceed unity, for i>c the

reflection coefficients remain equal to unity.

Hence for i>c all the incident light is reflected. This is known as ‘total internal

reflection’.

Power reflection and transmission coefficients

The reflection and transmission coefficients derived above give the amplitudes of

the electric fields associated with the waves.

However in general it is the power of the wave which is measured experimentally.

the power density of an electromagnetic wave is given by the Poynting

vector N=ExH.

This has a magnitude EH = nE2/(c0), using H=nE/(c0).

Hence the energy density nE2

The coefficients which give the fraction of energy reflected or transmitted at a

boundary between two materials equal the appropriate values of r2 or t2 with the

inclusion of the appropriate value(s) of n.

If R is the power reflection coefficient at an interface between a material of

refractive index n1 and one of n2 then where r is the appropriate reflection

coefficient given by the Fresnel relationships. Similarly if T is the power

transmission coefficient because energy must always be conserved R+T=1

   Power reflection and transmission coefficients

Linear and circular polarization :

Circular Polarization :

In electrodynamics, circular polarization of an electromagnetic wave is

a polarization in which the electric field of the passing wave does not change

strength but only changes direction in a rotary manner.

The electric field vectors of a traveling circularly polarized electromagnetic wave.

In electrodynamics the strength and direction of an electric field is defined by

what is called an electric field vector. In the case of a circularly polarized wave, as

seen in the accompanying animation, the tip of the electric field vector, at a given

point in space, describes a circle as time progresses. If the wave is frozen in time,

the electric field vector of the wave describes a helix along the direction of

propagation.

Circular polarization is a limiting case of the more general condition of elliptical

polarization. The other special case is the easier-to-understand linear polarization.

The phenomenon of polarization arises as a consequence of the fact

that light behaves as a two-dimensional transverse wave.

On the right is an illustration of the electric field vectors of a circularly polarized

electromagnetic wave.The electric field vectors have a constant magnitude but

their direction changes in a rotary manner. Given that this is a plane wave, each

vector represents the magnitude and direction of the electric field for an entire

plane that is perpendicular to the axis. Specifically, given that this is a circularly

polarized plane wave, these vectors indicate that the electric field, from plane to

plane, has a constant strength while its direction steadily rotates. It is considered

to be right-hand, clockwise circularly polarized if viewed by the receiver. Since

this is anelectromagnetic wave each electric field vector has a corresponding, but

not illustrated, magnetic field vector that is at a right angle to the electric field

vector and proportional in magnitude to it. As a result, the magnetic field vectors

would trace out a second helix if displayed.

Circular polarization is often encountered in the field of optics and in this section,

the electromagnetic wave will be simply referred to as light.

The nature of circular polarization and its relationship to other polarizations is

often understood by thinking of the electric field as being divided into

two components which are at right angles to each other. Refer to the second

illustration on the right. The vertical component and its corresponding plane are

illustrated in blue while the horizontal component and its corresponding plane are

illustrated in green. Notice that the rightward (relative to the direction of travel)

horizontal component leads the vertical component by one quarter of

a wavelength. It is thisquadrature phase relationship which creates the helix and

causes the points of maximum magnitude of the vertical component to correspond

with the points of zero magnitude of the horizontal component, and vice versa.

The result of this alignment is that there are select vectors, corresponding to the

helix, which exactly match the maxima of the vertical and horizontal components.

(To minimize visual clutter these are the only helix vectors displayed.)

To appreciate how this quadrature phase shift corresponds to an electric field that

rotates while maintaining a constant magnitude, imagine a dot traveling clockwise

in a circle. Consider how the vertical and horizontaldisplacements of the dot,

relative to the center of the circle, vary sinusoidally in time and are out of phase

by one quarter of a cycle. The displacements are said to be out of phase by one

quarter of a cycle because the horizontal maximum displacement (toward the left)

is reached one quarter of a cycle before the vertical maximum displacement is

reached. Now referring again to the illustration, imagine the center of the circle

just described, traveling along the axis from the front to the back. The circling dot

will trace out a helix with the displacement toward our viewing left, leading the

vertical displacement. Just as the horizontal and vertical displacements of the

rotating dot are out of phase by one quarter of a cycle in time, the magnitude of

the horizontal and vertical components of the electric field are out of phase by one

quarter of a wavelength.

Left-handed/counter-clockwise circularly polarized light displayed with and

without the use of components. This would be considered right-handed/clockwise

circularly polarized if defined from the point of view of the source rather than the

receiver.

The next pair of illustrations is that of left-handed, counter-clockwise circularly

polarized light when viewed by the receiver. Because it is left-handed, the

rightward (relative to the direction of travel) horizontal component is

nowlagging the vertical component by one quarter of a wavelength rather than

leading it.

One should appreciate that our choice to focus on the horizontal and vertical

components was arbitrary. Given the symmetry of circularly polarized light, we

could have in fact selected any other two orthogonal components and found the

same phase relationship between them.

To convert a given handedness of polarized light to the other handedness one can

use a half-wave plate. A half-wave plate shifts a given component of light one

half of a wavelength relative to the component to which it is orthogonal.

The handedness of polarized light is also reversed when it is reflected off of a

mirror. Initially, as a result of the interaction of the electromagnetic field with the

conducting surface of the mirror, both orthogonal components are effectively

shifted by one half of a wavelength. However as a result of the change in

direction, a mirror image of the wave is created and the two components' phase

relationship is reversed.

For a better appreciation of the nature of circularly polarized light one may find it

useful to read how circularly polarized light is converted to and from linearly

polarized light in the circular polarizer article.

Left / Right hardness convention :

Circular polarization may be referred to as right-handed or left-handed, and

clockwise or counter-clockwise, depending on the direction in which the electric

field vector rotates. Unfortunately, two opposing historical conventions exist.

From the point of view of the source

Using this convention, polarization is defined from the point of view of the

source. When using this convention, left or right handedness is determined by

pointing one's left or right thumb away from the source, in the same direction that

the wave is propagating, and matching the curling of one's fingers to the direction

of the temporal rotation of the field at a given point in space. When determining if

the wave is clockwise or counter-clockwise circularly polarized, one again takes

the point of view of the source, and while looking away from the source and in

the same direction of the wave’s propagation, one observes the direction of the

field’s temporal rotation.

Using this convention, the electric field vector of a right handed circularly

polarized wave is as

follows: 

As a specific example, refer to the circularly polarized wave in the first animation.

Using this convention that wave is defined as right-handed because when one

points one's right thumb in the same direction of the wave’s propagation, the

fingers of that hand curl in the same direction of the field’s temporal rotation. It is

considered clockwise circularly polarized because from the point of view of the

source, looking in the same direction of the wave’s propagation, the field rotates

in the clockwise direction. The second animation is that of left-handed or counter-

clockwise light using this same convention.

This convention is in conformity with the Institute of Electrical and Electronics

Engineers (IEEE) standard and as a result it is generally used in the engineering

community.

Quantum physicists also use this convention of handedness because it is

consistent with their convention of handedness for a particle’s spin. In quantum

mechanics the direction of spin of a photon is tied to the handedness of the

circularly polarized light and the spin of a beam of photons is similar to the spin

of a beam of particles, such as electrons.

Many radio astronomers also use this convention.

From the point of view of the receiver

In this alternative convention, polarization is defined from the point of view of the

receiver. Using this convention, left or right handedness is determined by pointing

one’s left or right thumb toward the source, against the direction of propagation,

and then matching the curling of one's fingers to the temporal rotation of the field.

When using this convention, in contrast to the other convention, the defined

handedness of the wave matches the handedness of the screw type nature of the

field in space. Specifically, if one freezes a right-handed wave in time, when one

curls the fingers of one’s right hand around the helix, the thumb will point in the

direction which the helix progresses given that sense of rotation. Note that it is the

nature of all screws and helices that it does not matter in which direction you

point your thumb when determining its handedness.

When determining if the wave is clockwise or counter-clockwise circularly

polarized, one again takes the point of view of the receiver and, while

looking toward the source, against the direction of propagation, one observes the

direction of the field’s temporal rotation.

Many optics textbooks use this second convention.

Uses of the two conventions

As stated earlier, there is significant confusion with regards to these two

conventions. As a general rule the engineering and quantum physics community

use the first convention where the wave is observed from the point of view of the

source.In many physics textbooks dealing with optics the second convention is

used where the light is observed from the point of view of receiver.

To avoid confusion, it is good practice to specify “as defined from the point of

view of the source” or "as defined from the point of view of the receiver" when

discussing polarization matters.

The archive of the US Federal Standard 1037C proposes two contradictory

conventions of handedness

Circularly polarized luminance :

Circularly polarized luminescence (CPL) can occur when either a luminophore or

an ensemble of luminophores is chiral. The extent to which emissions are

polarized is quantified in the same way it is for circular dichroism, in terms of

the dissymmetry , also sometimes referred to as the anisotropy factor. This value

is given by

where   corresponds to the quantum yield of left-handed circularly polarized

light, and   to that of right-handed light. The maximum absolute value of gem,

corresponding to purely left- or right-handed circular polarization, is therefore 2.

Meanwhile the smallest absolute value that gem can achieve, corresponding to

linearly polarized or unpolarized light, is zero.

Mathamatical Discription :

The classical sinusoidal plane wave solution of the electromagnetic wave

equation for the electric and magnetic fields is

where k is the wavenumber,

is the angular frequency of the wave,   is an orthogonal   matrix

whose columns span the transverse x-y plane and   is the speed of light.

Here

is the amplitude of the field and

is the Jones vector in the x-y plane.

If   is rotated by   radians with respect to   and the x amplitude equals the

y amplitude the wave is circularly polarized. The Jones vector is

where the plus sign indicates left circular polarization and the minus sign

indicates right circular polarization. In the case of circular polarization, the

electric field vector of constant magnitude rotates in the x-y plane.

If basis vectors are defined such that

and

then the polarization state can be written in the "R-L basis" as

where

and

Linear Polarization :

In electrodynamics, linear polarization or plane polarization of electromagnetic

radiation is a confinement of the electric field vector or magnetic field vector to a

given plane along the direction of propagation. See polarization for more

information.

The orientation of a linearly polarized electromagnetic wave is defined by the

direction of the electric field vector. For example, if the electric field vector is

vertical (alternately up and down as the wave travels) the radiation is said to be

vertically polarized.

Mathematical description of linear polarization :

The classical sinusoidal plane wave solution of the electromagnetic wave

equation for the electric and magnetic fields is (cgs units)

for the magnetic field, where k is the wavenumber,

is the angular frequency of the wave, and   is the speed of light.

Here

is the amplitude of the field and

is the Jones vector in the x-y plane.

The wave is linearly polarized when the phase angles   are equal,

.

This represents a wave polarized at an angle   with respect to the x axis. In that

case the Jones vector can be written

.

The state vectors for linear polarization in x or y are special cases of this state

vector.

If unit vectors are defined such that

and

then the polarization state can written in the "x-y basis" as

Light in the form of a plane wave in space is said to be linearly polarized. Light is

a transverse electromagnetic wave, but natural light is generally unpolarized, all

planes of propagation being equally probable. If light is composed of two plane

waves of equal amplitude by differing in phase by 90°, then the light is said to be

circularly polarized. If two plane waves of differing amplitude are related in phase

by 90°, or if the relative phase is other than 90° then the light is said to be

elliptically polarized.

Reflection and Refraction at a Plane Interface :

The reflection and refraction of a beam of light is treated. Approximate solutions

of Maxwell’s equations are used to describe the electromagnetic field of the beam

being limited in the transverse direction. The point of departure is the classical

paper by Schaefer and Pich. The laws of reflection and refraction are derived.

Fresnel’s formulas and their corrections are presented for both polarizations. The

case of total reflection is investigated for E polarization in greater detail. The

electromagnetic fields and the time-average Poynting vectors are explicitly

derived for both the optically dense and less-dense media. The flow of energy at

total reflection is studied extensively. It is shown that, due to the flow of energy in

the less-dense medium, the center of gravity of the reflected beam is displaced, as

was suggested by v. Fragstein. This leads to a shift of the totally reflected beam

with respect to a geometrically reflected beam . . 

Suppose a plane wave is incident upon a plane surface that is an interface between

two materials, one with   and the other with  .

Figure : Geometry for reflection and refraction at a plane interface between two media, one

with permittivity/permeability  , one with permittivity/permeability  .

In order to derive an algebraic relationship between the intensities of the incoming

wave, the reflected wave, and the refracted wave, we must begin by defining the

algebraic form of each of these waves in terms of the wave numbers. The

reflected wave and incident wave do not leave the first medium and hence retain

speed  ,  ,   and  . The refracted

wave changes to speed  ,  ,  .

Note that the frequency of the waves is the same in both media as a kinematic

constraint! Why is that?

This yields the following forms for the various waves:

Incident Wave

Refracted Wave

Reflected Wave

Our goal is to completely understand how to compute the reflected and refracted

wave from the incident wave. This is done by matching the wave across the

boundary interface. There are two aspects of this matching - astatic or kinematic

matching of the waveform itself and a dynamic matching associated with the

(changing) polarization in the medium. These two kinds of matching lead to two

distinct and well-known results.

Fresnel’s Equations :

This article is about the Fresnel equations describing reflection and refraction of

light at uniform planar interfaces. For the diffraction of light through an aperture,

see Fresnel diffraction. For the thin lens and mirror technology, see Fresnel lens.

Partial transmission and reflection amplitudes of a wave travelling from a low to

high refractive index medium.

The Fresnel equations (or Fresnel conditions), deduced by Augustin-Jean

Fresnel , describe the behaviour of light when moving between media of

differing refractive indices. The reflection of light that the equations predict is

known as Fresnel reflection.

Overview :

When light moves from a medium of a given refractive index n1 into a second

medium with refractive index n2, both reflection and refraction of the light may

occur. The Fresnel equations describe what fraction of the light is reflected and

what fraction is refracted (i.e., transmitted). They also describe the phase shift of

the reflected light.

The equations assume the interface is flat, planar, and homogeneous, and that the

light is a plane wave.

Definitions and power equations

Variables used in the Fresnel equations.

In the diagram on the right, an incident light ray IO strikes the interface between

two media of refractive indices n1 and n2 at point O. Part of the ray is reflected as

ray OR and part refracted as ray OT. The angles that the incident, reflected and

refracted rays make to the normal of the interface are given as θi, θr and θt,

respectively.

The relationship between these angles is given by the law of reflection:

and Snell's law:

The fraction of the incident power that is reflected from the interface is given by

the reflectance R and the fraction that is refracted is given by the transmittance T.

The media are assumed to be non-magnetic.

The calculations of R and T depend on polarisation of the incident ray. Two cases

are analyzed:

The incident light is s-polarized. That means its electric field is in the

plane of the interface (perpendicular to the plane of the diagram

above).

The incident light is p-polarized. That means its electric field is in a

perpendicular direction to s-polarized above (in the plane of the

diagram above).

For the s-polarized light, the reflection coefficient is given by

,

where the second form is derived from the first by eliminating θt using Snell's

law and trigonometric identities.

For the p-polarized light, the R is given by

.

As a consequence of the conservation of energy, the transmission coefficients are

given by 

and

These relationships hold only for power coefficients, not for amplitude

coefficients as defined below.

If the incident light is unpolarised (containing an equal mix of s- and p-

polarisations), the reflection coefficient is

For common glass, the reflection coefficient is about 4%. Note that reflection by a

window is from the front side as well as the back side, and that some of the light

bounces back and forth a number of times between the two sides. The combined

reflection coefficient for this case is 2R/(1 + R), when interference can be

neglected (see below).

The discussion given here assumes that the permeability μ is equal to the vacuum

permeability μ0 in both media. This is approximately true for

most dielectric materials, but not for some other types of material. The completely

general Fresnel equations are more complicated.

Special angles

At one particular angle for a given n1 and n2, the value of Rp goes to zero and a p-

polarised incident ray is purely refracted. This angle is known as Brewster's angle,

and is around 56° for a glass medium in air or vacuum. Note that this statement is

only true when the refractive indices of both materials are real numbers, as is the

case for materials like air and glass. For materials that absorb light,

like metals and semiconductors, n is complex, and Rp does not generally go to

zero.

When moving from a denser medium into a less dense one (i.e., n1 > n2), above an

incidence angle known as the critical angle, all light is reflected and Rs = Rp = 1.

This phenomenon is known as total internal reflection. The critical angle is

approximately 41° for glass in air.

Amplitude equations

Equations for coefficients corresponding to ratios of the electric field complex-

valued amplitudes of the waves (not necessarily real-valued magnitudes) are also

called "Fresnel equations". These take several different forms, depending on the

choice of formalism and sign convention used. The amplitude coefficients are

usually represented by lower case r and t.

Amplitude ratios: air to glass

Amplitude ratios: glass to air

Conventions used here

In this treatment, the coefficient r is the ratio of the reflected

wave's complex electric field amplitude to that of the incident wave. The

coefficient t is the ratio of the transmitted wave's electric field amplitude to that of

the incident wave. The light is split into s and p polarizations as defined above.

(In the figures to the right, s polarization is denoted " " and p is denoted " ".)

For s-polarization, a positive r or t means that the electric fields of the incoming

and reflected or transmitted wave are parallel, while negative means anti-parallel.

For p-polarization, a positive r or t means that the magnetic fields of the waves are

parallel, while negative means anti-parallel.

Formulas

Using the conventions above,

Notice that   but  .

Because the reflected and incident waves propagate in the same medium and

make the same angle with the normal to the surface, the amplitude reflection

coefficient is related to the reflectance R by 

.

The transmittance T is generally not equal to |t|2, since the light travels with

different direction and speed in the two media. The transmittance is related

to t by.

.

The factor of n2/n1 occurs from the ratio of intensities (closely related

to irradiance). The factor of cos θt/cos θi represents the change in area m of

the pencil of rays, needed since T, the ratio of powers, is equal to the ratio of

(intensity × area). In terms of the ratio of refractive indices,

,

and of the magnification m of the beam cross section occurring at the interface,

.

Multiple surfaces

When light makes multiple reflections between two or more parallel surfaces, the

multiple beams of light generally interfere with one another, resulting in net

transmission and reflection amplitudes that depend on the light's wavelength. The

interference, however, is seen only when the surfaces are at distances comparable

to or smaller than the light's coherence length, which for ordinary white light is

few micrometers; it can be much larger for light from a laser.

An example of interference between reflections is the iridescent colours seen in

a soap bubble or in thin oil films on water. Applications include Fabry–Pérot

interferometers, antireflection coatings, and optical filters. A quantitative analysis

of these effects is based on the Fresnel equations, but with additional calculations

to account for interference.

The transfer-matrix method, or the recursive Rouard method  can be used to solve

multiple-surface problems.

Interference of Waves

Interference (wave propagation) , in physics (optics), the superposition of

two or more waves resulting in a new wave pattern

Thin-film interference . a special kind of optical interference

Interference (communication) , anything which alters, modifies, or disrupts

a message as it travels along a channel

Electromagnetic interference  (EMI)

Co-channel interference  (CCI), also known as crosstalk

Adjacent-channel interference  (ACI), interference caused by

extraneous power from a signal in an adjacent channel

Intersymbol interference  (ISI), distortion of a signal in which one

symbol interferes with subsequent symbols

Inter-carrier interference  (ICI), caused by doppler shift

in OFDM modulation

Vaccine interference , may occur when two or more vaccines are mixed in

the same formulation

Interference fit , two items attempting to occupy the same space

Interference engine , an engine whose pistons can strike and damage the

piston valves when the engine is cranked while the timing belt is broken

Interference theory  in psychology

Statistical interference , a method of determining when one distribution of

values exceeds another

Interference (genetic) , a phenomenon by which a chromosomal

crossover in one interval decreases the probability that additional crossovers

will occur nearby

RNA interference , a process within living cells that moderates the activity

of their genes

What happens when two waves meet while they travel through the same medium?

What effect will the meeting of the waves have upon the appearance of the

medium? Will the two waves bounce off each other upon meeting (much like two

billiard balls would) or will the two waves pass through each other? These

questions involving the meeting of two or more waves along the same medium

pertain to the topic of wave interference.

Wave interference is the phenomenon that occurs when two waves meet while

traveling along the same medium. The interference of waves causes the medium

to take on a shape that results from the net effect of the two individual waves

upon the particles of the medium. To begin our exploration of wave interference,

consider two pulses of the same amplitude traveling in different directions along

the same medium. Let's suppose that each displaced upward 1 unit at its crest and

has the shape of a sine wave. As the sine pulses move towards each other, there

will eventually be a moment in time when they are completely overlapped. At that

moment, the resulting shape of the medium would be an upward displaced sine

pulse with an amplitude of 2 units. The diagrams below depict the before and

during interference snapshots of the medium for two such pulses. The individual

sine pulses are drawn in red and blue and the resulting displacement of the

medium is drawn in green.

This type of interference is sometimes called constructive

interference. Constructive interference is a type of interference that occurs at

any location along the medium where the two interfering waves have a

displacement in the same direction. In this case, both waves have an upward

displacement; consequently, the medium has an upward displacement that is

greater than the displacement of the two interfering pulses. Constructive

interference is observed at any location where the two interfering waves are

displaced upward. But it is also observed when both interfering waves are

displaced downward. This is shown in the diagram below for two downward

displaced pulses.

In this case, a sine pulse with a maximum displacement of -1 unit (negative means

a downward displacement) interferes with a sine pulse with a maximum

displacement of -1 unit. These two pulses are drawn in red and blue. The resulting

shape of the medium is a sine pulse with a maximum displacement of -2 units.

Destructive interference is a type of interference that occurs at any location

along the medium where the two interfering waves have a displacement in the

opposite direction. For instance, when a sine pulse with a maximum displacement

of +1 unit meets a sine pulse with a maximum displacement of -1 unit, destructive

interference occurs. This is depicted in the diagram below.

In the diagram above, the interfering pulses have the same maximum

displacement but in opposite directions. The result is that the two pulses

completely destroy each other when they are completely overlapped. At the

instant of complete overlap, there is no resulting displacement of the particles of

the medium. This "destruction" is not a permanent condition. In fact, to say that

the two waves destroy each other can be partially misleading. When it is said that

the two pulses destroy each other, what is meant is that when overlapped, the

effect of one of the pulses on the displacement of a given particle of the medium

is destroyed or canceled by the effect of the other pulse. Recall fromLesson 1 that

waves transport energy through a medium by means of each individual particle

pulling upon its nearest neighbor. When two pulses with opposite displacements

(i.e., one pulse displaced up and the other down) meet at a given location, the

upward pull of one pulse is balanced (canceled or destroyed) by the downward

pull of the other pulse. Once the two pulses pass through each other, there is still

an upward displaced pulse and a downward displaced pulse heading in the same

direction that they were heading before the interference. Destructive interference

leads to only a momentary condition in which the medium's displacement is less

than the displacement of the largest-amplitude wave.

The two interfering waves do not need to have equal amplitudes in opposite

directions for destructive interference to occur. For example, a pulse with a

maximum displacement of +1 unit could meet a pulse with a maximum

displacement of -2 units. The resulting displacement of the medium during

complete overlap is -1 unit.

This is still destructive interference since the two interfering pulses have opposite

displacements. In this case, the destructive nature of the interference does not lead

to complete cancellation.

Interestingly, the meeting of two waves along a medium does not alter the

individual waves or even deviate them from their path. This only becomes an

astounding behavior when it is compared to what happens when two billiard balls

meet or two football players meet. Billiard balls might crash and bounce off each

other and football players might crash and come to a stop. Yet two waves will

meet, produce a net resulting shape of the medium, and then continue on doing

what they were doing before the interference.

 

The task of determining the shape of the resultant demands that the principle of

superposition is applied. Theprinciple of superposition is sometimes stated as

follows:

When two waves interfere, the resulting displacement of the medium at

any location is the algebraic sum of the displacements of the individual

waves at that same location.

In the cases above, the summing the individual displacements for locations of

complete overlap was made out to be an easy task - as easy as simple arithmetic:

Displacement of Pulse 1 Displacement of Pulse 2 =Resulting Displacement

+1 +1 =+2

-1 -1 =-2

+1 -1 =0

+1 -2 =-1

In actuality, the task of determining the complete shape of the entire medium

during interference demands that the principle of superposition be applied for

every point (or nearly every point) along the

medium. As an example of the complexity of

this task, consider the two interfering waves

at the right. A snapshot of the shape of each

individual wave at a particular instant in time

is shown. To determine the precise shape of

the medium at this given instant in time, the

principle of superposition must be applied to

several locations along the medium. A short

cut involves measuring the displacement from equilibrium at a few strategic

locations. Thus, approximately 20 locations have been picked and labeled as A, B,

C, D, etc. The actual displacement of each individual wave can be counted by

measuring from the equilibrium position up to the particular wave. At position A,

there is no displacement for either individual wave; thus, the resulting

displacement of the medium at position will be 0 units. At position B, the smaller

wave has a displacement of approximately 1.4 units (indicated by the red dot); the

larger wave has a displacement of approximately 2 units (indicated by the blue

dot). Thus, the resulting displacement of the medium will be approximately 3.4

units. At position C, the smaller wave has a displacement of approximately 2

units; the larger wave has a displacement of approximately 4 units; thus, the

resulting displacement of the medium will be approximately 6 units. At position

D, the smaller wave has a displacement of approximately 1.4 units; the larger

wave has a displacement of approximately 2 units; thus, the resulting

displacement of the medium will be approximately 3.4 units. This process can be

repeated for every position. When finished, a dot (done in green below) can be

marked on the graph to note the displacement of the medium at each given

location. The actual shape of the medium can then be sketched by estimating the

position between the various marked points and sketching the wave. This is

shown as the green line in the diagram below.

 

Coherence :  

In physics, coherence is an ideal property of waves that enables stationary (i.e.

temporally and spatially constant) interference. It contains several distinct

concepts, which are limit cases that never occur in reality but allow an

understanding of the physics of waves, and has become a very important concept

in quantum physics. More generally, coherence describes all properties of the

correlation between physical quantities of a single wave, or between several

waves or wave packets.

Interference is nothing more than the addition, in the mathematical sense, of wave

functions. In quantum mechanics, a single wave can interfere with itself, but this

is due to its quantum behavior and is still an addition of two waves (see Young's

slits experiment). This implies that constructive or destructive interferences are

limit cases, and that waves can always interfere, even if the result of the addition

is complicated or not remarkable.

When interfering, two waves can add together to create a wave of greater

amplitude than either one (constructive interference) or subtract from each other

to create a wave of lesser amplitude than either one (destructive interference),

depending on their relative phase. Two waves are said to be coherent if they have

a constant relative phase. The degree of coherence is measured by the interference

visibility, a measure of how perfectly the waves can cancel due to destructive

interference.

Spatial coherence describes the correlation between waves at different points in

space. Temporal coherence describes the correlation or predictable relationship

between waves observed at different moments in time. Both are observed in

the Michelson–Morley experiment and Young's interference experiment. Once the

fringes are obtained in the Michelson–Morley experiment, when one of the

mirrors is moved away gradually, the time for the beam to travel increases and the

infringes become dull and finally are lost, showing Temporal Coherence.

Similarly, if in Young's double slit experiment the space between the two slits is

increased, the coherence dies gradually and finally the infringes disappear,

showing spatial coherence

Introduction

Coherence was originally conceived in connection with Thomas Young's double-

slit experiment in optics but is now used in any field that involves waves, such

as acoustics, electrical engineering,neuroscience, and quantum mechanics. The

property of coherence is the basis for commercial applications such

as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence

tomography and telescope interferometers (astronomical optical

interferometers and radio telescopes).

Coherence and correlation

The coherence of two waves follows from how well correlated the waves are as

quantified by the cross-correlation function.  The cross-correlation quantifies the

ability to predict the value of the second wave by knowing the value of the first.

As an example, consider two waves perfectly correlated for all times. At any time,

if the first wave changes, the second will change in the same way. If combined

they can exhibit complete constructive interference/superposition at all times, then

it follows that they are perfectly coherent. As will be discussed below, the second

wave need not be a separate entity. It could be the first wave at a different time or

position. In this case, the measure of correlation is the autocorrelation function

(sometimes called self-coherence). Degree of correlation involves correlation

functions.

Examples of wave-like states

These states are unified by the fact that their behavior is described by a wave

equation or some generalization thereof.

Waves in a rope (up and down) or slinky (compression and expansion)

Surface waves in a liquid

Electric signals (fields) in transmission cables

Sound

Radio waves and Microwaves

Light waves (optics)

Electrons , atoms and any other object (such as a baseball, as described

by quantum physics)

In most of these systems, one can measure the wave directly. Consequently, its

correlation with another wave can simply be calculated. However, in optics one

cannot measure the electric fielddirectly as it oscillates much faster than any

detector’s time resolution. Instead, we measure the intensity of the light. Most of

the concepts involving coherence which will be introduced below were developed

in the field of optics and then used in other fields. Therefore, many of the standard

measurements of coherence are indirect measurements, even in fields where the

wave can be measured directly.

Temporal coherence

Figure : The amplitude of a single frequency wave as a function of time t (red)

and a copy of the same wave delayed by τ(green). The coherence time of the wave

is infinite since it is perfectly correlated with itself for all delays τ.

Figure : The amplitude of a wave whose phase drifts significantly in time τc as a

function of time t (red) and a copy of the same wave delayed by 2τc(green). At

any particular time t the wave can interfere perfectly with its delayed copy. But,

since half the time the red and green waves are in phase and half the time out of

phase, when averaged over t any interference disappears at this delay.

Temporal coherence is the measure of the average correlation between the value

of a wave and itself delayed by τ, at any pair of times. Temporal coherence tells

us how monochromatic a source is. In other words, it characterizes how well a

wave can interfere with itself at a different time. The delay over which the phase

or amplitude wanders by a significant amount (and hence the correlation

decreases by significant amount) is defined as the coherence time τc. At τ=0 the

degree of coherence is perfect whereas it drops significantly by delay τc.

The coherence length Lc is defined as the distance the wave travels in time τc.

One should be careful not to confuse the coherence time with the time duration of

the signal, nor the coherence length with the coherence area (see below).

The relationship between coherence time and bandwidth

It can be shown that the faster a wave decorrelates (and hence the smaller τ c is)

the larger the range of frequencies Δf the wave contains. Thus there is a tradeoff:

.

Formally, this follows from the convolution theorem in mathematics, which

relates the Fourier transform of the power spectrum (the intensity of each

frequency) to its autocorrelation.

Examples of temporal coherence

We consider four examples of temporal coherence.

A wave containing only a single frequency (monochromatic) is perfectly

correlated at all times according to the above relation. (See Figure 1)

Conversely, a wave whose phase drifts quickly will have a short coherence

time. (See Figure 2)

Similarly, pulses (wave packets) of waves, which naturally have a broad

range of frequencies, also have a short coherence time since the amplitude

of the wave changes quickly. (See Figure 3)

Finally, white light, which has a very broad range of frequencies, is a

wave which varies quickly in both amplitude and phase. Since it

consequently has a very short coherence time (just 10 periods or so), it is

often called incoherent.

The most monochromatic sources are usually lasers; such high monochromaticity

implies long coherence lengths (up to hundreds of meters). For example, a

stabilized helium-neon laser can produce light with coherence lengths in excess of

5 m . Not all lasers are monochromatic, however (e.g. for a mode-locked Ti-

sapphire laser, Δλ ≈ 2 nm - 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and

tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter

coherence times than the most monochromatic lasers.

Holography requires light with a long coherence time. In contrast, Optical

coherence tomography uses light with a short coherence time.

Measurement of temporal coherence

Figure : The amplitude of a wavepacket whose amplitude changes

significantly in time τc (red) and a copy of the same wave delayed by

2τc(green) plotted as a function of time t. At any particular time the red and

green waves are uncorrelated; one oscillates while the other is constant and so

there will be no interference at this delay. Another way of looking at this is

the wavepackets are not overlapped in time and so at any particular time there

is only one nonzero field so no interference can occur.

Figure : The time-averaged intensity (blue) detected at the output of an

interferometer plotted as a function of delay τ for the example waves in

Figures . As the delay is changed by half a period, the interference switches

between constructive and destructive. The black lines indicate the interference

envelope, which gives the degree of coherence. Although the waves in

Figures have different time durations, they have the same coherence time.

In optics, temporal coherence is measured in an interferometer such as

the Michelson interferometer or Mach–Zehnder interferometer. In these

devices, a wave is combined with a copy of itself that is delayed by time τ. A

detector measures the time-averaged intensity of the light exiting the

interferometer. The resulting interference visibility (e.g. see Figure ) gives the

temporal coherence at delay τ. Since for most natural light sources, the

coherence time is much shorter than the time resolution of any detector, the

detector itself does the time averaging. Consider the example shown in Figure

. At a fixed delay, here 2τc, an infinitely fast detector would measure an

intensity that fluctuates significantly over a time tequal to τc. In this case, to

find the temporal coherence at 2τc, one would manually time-average the

intensity.

Spatial coherence

In some systems, such as water waves or optics, wave-like states can extend

over one or two dimensions. Spatial coherence describes the ability for two

points in space, x1 and x2, in the extent of a wave to interfere, when averaged

over time. More precisely, the spatial coherence is the cross-

correlation between two points in a wave for all times. If a wave has only 1

value of amplitude over an infinite length, it is perfectly spatially coherent.

The range of separation between the two points over which there is significant

interference is called the coherence area, Ac. This is the relevant type of

coherence for the Young’s double-slit interferometer. It is also used in optical

imaging systems and particularly in various types of astronomy telescopes.

Sometimes people also use “spatial coherence” to refer to the visibility when

a wave-like state is combined with a spatially shifted copy of itself.

Examples of spatial coherence

Spatial coherence

Figure 5: A plane wave with an infinite coherence length.

Figure 6: A wave with a varying profile (wavefront) and infinite coherence

length.

Figure 7: A wave with a varying profile (wavefront) and finite coherence

length.

Figure 8: A wave with finite coherence area is incident on a pinhole (small

aperture). The wave will diffract out of the pinhole. Far from the pinhole the

emerging spherical wavefronts are approximately flat. The coherence area is

now infinite while the coherence length is unchanged.

Figure 9: A wave with infinite coherence area is combined with a spatially

shifted copy of itself. Some sections in the wave interfere constructively and

some will interfere destructively. Averaging over these sections, a detector

with length D will measure reducedinterference visibility. For example a

misaligned Mach–Zehnder interferometer will do this.

Consider a tungsten light-bulb filament. Different points in the filament emit

light independently and have no fixed phase-relationship. In detail, at any

point in time the profile of the emitted light is going to be distorted. The

profile will change randomly over the coherence time  . Since for a white-

light source such as a light-bulb   is small, the filament is considered a

spatially incoherent source. In contrast, a radio antenna array, has large spatial

coherence because antennas at opposite ends of the array emit with a fixed

phase-relationship. Light waves produced by a laser often have high temporal

and spatial coherence (though the degree of coherence depends strongly on

the exact properties of the laser). Spatial coherence of laser beams also

manifests itself as speckle patterns and diffraction fringes seen at the edges of

shadow.

Holography requires temporally and spatially coherent light. Its

inventor, Dennis Gabor, produced successful holograms more than ten years

before lasers were invented. To produce coherent light he passed the

monochromatic light from an emission line of a mercury-vapor lamp through

a pinhole spatial filter.

In February 2011, Dr Andrew Truscott, leader of a research team at the ARC

Centre of Excellence for Quantum-Atom Optics at Australian National

University in Canberra, Australian Capital Territory, showed

that helium atoms cooled to near absolute zero / Bose-Einstein

condensate state, can be made to flow and behave as a coherent beam as

occurs in a laser.

Spectral coherence

Figure : Waves of different frequencies (i.e. colors) interfere to form a pulse if

they are coherent.

Figure : Spectrally incoherent light interferes to form continuous light with a

randomly varying phase and amplitude

Waves of different frequencies (in light these are different colours) can interfere

to form a pulse if they have a fixed relative phase-relationship (see Fourier

transform). Conversely, if waves of different frequencies are not coherent, then,

when combined, they create a wave that is continuous in time (e.g. white light

or white noise). The temporal duration of the pulse   is limited by the spectral

bandwidth of the light   according to:

,

which follows from the properties of the Fourier transform and results

in Küpfmüller's uncertainty principle (for quantum particles it also results in

the Heisenberg uncertainty principle).

If the phase depends linearly on the frequency (i.e.  ) then the pulse

will have the minimum time duration for its bandwidth (a transform-

limited pulse), otherwise it is chirped (see dispersion).

Measurement of spectral coherence

Measurement of the spectral coherence of light requires a nonlinear optical

interferometer, such as an intensity optical correlator,frequency-resolved optical

gating (FROG), or Spectral phase interferometry for direct electric-field

reconstruction (SPIDER).

Polarization coherence

Light also has a polarization, which is the direction in which the electric field

oscillates. Unpolarized light is composed of incoherent light waves with random

polarization angles. The electric field of the unpolarized light wanders in every

direction and changes in phase over the coherence time of the two light waves. An

absorbing polarizer rotated to any angle will always transmit half the incident

intensity when averaged over time.

If the electric field wanders by a smaller amount the light will be partially

polarized so that at some angle, the polarizer will transmit more than half the

intensity. If a wave is combined with an orthogonally polarized copy of itself

delayed by less than the coherence time, partially polarized light is created.

The polarization of a light beam is represented by a vector in the Poincare sphere.

For polarized light the end of the vector lies on the surface of the sphere, whereas

the vector has zero length for unpolarized light. The vector for partially polarized

light lies within the sphere

Applications

Holography

Coherent superpositions of optical wave fields include holography. Holographic

objects are used frequently in daily life in bank notes and credit cards.

Non-optical wave fields

Further applications concern the coherent superposition of non-optical wave

fields. In quantum mechanics for example one considers a probability field, which

is related to the wave function  (interpretation: density of the probability

amplitude). Here the applications concern, among others, the future technologies

of quantum computing and the already available technology of quantum

cryptography. Additionally the problems of the following subchapter are treated.

Quantum coherence

In quantum mechanics, all objects have wave-like properties (see de Broglie

waves). For instance, in Young's Double-slit experiment electrons can be used in

the place of light waves. Each electron's wave-function goes through both slits,

and hence has two separate split-beams that contribute to the intensity pattern on a

screen. According to standard wave theory [Fresnel, Huygens] these two

contributions give rise to an intensity pattern of bright bands due to constructive

interference, interlaced with dark bands due to destructive interference, on a

downstream screen. (Each split-beam, by itself, generates a diffraction pattern

with less noticeable, more widely spaced dark and light bands.) This ability to

interfere and diffract is related to coherence (classical or quantum) of the wave.

The association of an electron with a wave is unique to quantum theory.

When the incident beam is represented by a quantum pure state, the split beams

downstream of the two slits are represented as a superposition of the pure states

representing each split beam. (This has nothing to do with two particles or Bell's

inequalities relevant to an entangled state: a 2-body state, a kind of coherence

between two 1-body states.) The quantum description of imperfectly coherent

paths is called a mixed state. A perfectly coherent state has a density matrix (also

called the "statistical operator") that is a projection onto the pure coherent state,

while a mixed state is described by a classical probability distribution for the pure

states that make up the mixture.

Large-scale (macroscopic) quantum coherence leads to novel phenomena, the so-

called macroscopic quantum phenomena. For instance,

the laser, superconductivity, and superfluidity are examples of highly coherent

quantum systems, whose effects are evident at the macroscopic scale. The

macroscopic quantum coherence (Off-Diagonal Long-Range Order, ODLRO)

[Penrose & Onsager (1957), C. N. Yang (1962)] for laser light, and superfluidity,

is related to first-order (1-body) coherence/ODLRO, while superconductivity is

related to second-order coherence/ODLRO. (For fermions, such as electrons, only

even orders of coherence/ODLRO are possible.) Superfluidity in liquid He4 is

related to a partial Bose–Einstein condensate. Here, the condensate portion is

described by a multiply occupied single-particle state. [e.g., Cummings &

Johnston (1966)]

On the other hand, the Schrödinger's cat thought experiment highlights the fact

that quantum coherence cannot be arbitrarily applied to macroscopic situations. In

order to have a quantum superposition of dead and alive cat, one needs to have

pure states associated with aliveness and pure states associated with death, which

are then superposed. Given the problem of defining death (absence of EEG, heart

beat, ...) it is hard to imagine a set of quantum parameters that could be used in

constructing such superposition. In any case, this is not a good topic for a

description of quantum coherence.

Regarding the occurrence of quantum coherence at a macroscopic level, it is

interesting to note that the classical electromagnetic field exhibits macroscopic

quantum coherence. The most obvious example is carrier signals for radio and

TV. They satisfy Glauber's quantum description of coherence.

Diffraction :

Diffraction refers to various phenomena which occur when a wave encounters an

obstacle. In classical physics, the diffraction phenomenon is described as the

apparent bending of waves around small obstacles and the spreading out of waves

past small openings. Similar effects occur when a light wave travels through a

medium with a varying refractive index, or a sound wave travels through one with

varying acoustic impedance. Diffraction occurs with all waves,

including sound waves, water waves, and electromagnetic waves such as visible

light, X-rays and radio waves. As physical objects have wave-like properties (at

the atomic level), diffraction also occurs with matter and can be studied according

to the principles of quantum mechanics.

[N]o-one has ever been able to define the difference between interference and

diffraction satisfactorily. It is just a question of usage, and there is no specific,

important physical difference between them.

He suggested that when there are only a few sources, say two, we call it

interference, as in Young's slits, but with a large number of sources, the process

be labelled diffraction.

While diffraction occurs whenever propagating waves encounter such changes, its

effects are generally most pronounced for waves whose wavelength is roughly

similar to the dimensions of the diffracting objects. If the obstructing object

provides multiple, closely spaced openings, a complex pattern of varying intensity

can result. This is due to the superposition, or interference, of different parts of a

wave that travels to the observer by different paths (see diffraction grating).

The formalism of diffraction can also describe the way in which waves of finite

extent propagate in free space. For example, the expanding profile of a laser

beam, the beam shape of a radar antenna and the field of view of an ultrasonic

transducer can all be analysed using diffraction equations.

Examples

Solar glory at the steam from hot springs. A glory is an optical phenomenon

produced by light backscattered (a combination of

diffraction, reflection and refraction) towards its source by a cloud of uniformly

sized water droplets.

The effects of diffraction are often seen in everyday life. The most striking

examples of diffraction are those involving light; for example, the closely spaced

tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow

pattern seen when looking at a disk. This principle can be extended to engineer a

grating with a structure such that it will produce any diffraction pattern desired;

the hologram on a credit card is an example.Diffraction in the atmosphere by

small particles can cause a bright ring to be visible around a bright light source

like the sun or the moon. A shadow of a solid object, using light from a compact

source, shows small fringes near its edges. The speckle pattern which is observed

when laser light falls on an optically rough surface is also a diffraction

phenomenon. All these effects are a consequence of the fact that light propagates

as a wave.

Diffraction can occur with any kind of wave. Ocean waves diffract

around jetties and other obstacles. Sound waves can diffract around objects,

which is why one can still hear someone calling even when hiding behind a tree.

Diffraction can also be a concern in some technical applications; it sets

afundamental limit to the resolution of a camera, telescope, or microscope.

Mechanism

Photograph of single-slit diffraction in a circular ripple tank

Diffraction arises because of the way in which waves propagate; this is described

by the Huygens–Fresnel principle and the principle of superposition of waves.

The propagation of a wave can be visualized by considering every point on a

wavefront as a point source for a secondary spherical wave. The wave

displacement at any subsequent point is the sum of these secondary waves. When

waves are added together, their sum is determined by the relative phases as well

as the amplitudes of the individual waves so that the summed amplitude of the

waves can have any value between zero and the sum of the individual amplitudes.

Hence, diffraction patterns usually have a series of maxima and minima.

There are various analytical models which allow the diffracted field to be

calculated, including the Kirchhoff-Fresnel diffraction equation which is derived

from wave equation, the Fraunhofer diffraction approximation of the Kirchhoff

equation which applies to the far field and the Fresnel diffraction approximation

which applies to the near field. Most configurations cannot be solved analytically,

but can yield numerical solutions through finite element and boundary

element methods.

It is possible to obtain a qualitative understanding of many diffraction phenomena

by considering how the relative phases of the individual secondary wave sources

vary, and in particular, the conditions in which the phase difference equals half a

cycle in which case waves will cancel one another out.

The simplest descriptions of diffraction are those in which the situation can be

reduced to a two-dimensional problem. For water waves, this is already the case;

water waves propagate only on the surface of the water. For light, we can often

neglect one direction if the diffracting object extends in that direction over a

distance far greater than the wavelength. In the case of light shining through small

circular holes we will have to take into account the full three dimensional nature

of the problem.

Diffraction of light

Some examples of diffraction of light are considered below.

Single-slit diffraction

Numerical approximation of diffraction pattern from a slit of width equal to

wavelength of an incident plane wave in 3D spectrum visualization

Numerical approximation of diffraction pattern from a slit of width equal to

five times the wavelength of an incident plane wave in 3D spectrum

visualization

Diffraction of red laser beam on the hole

Numerical approximation of diffraction pattern from a slit of width four

wavelengths with an incident plane wave. The main central beam, nulls, and phase

reversals are apparent.

Graph and image of single-slit diffraction.

A long slit of infinitesimal width which is illuminated by light diffracts the light

into a series of circular waves and the wavefront which emerges from the slit is a

cylindrical wave of uniform intensity.

A slit which is wider than a wavelength produces interference effects in the space

downstream of the slit. These can be explained by assuming that the slit behaves

as though it has a large number of point sources spaced evenly across the width of

the slit. The analysis of this system is simplified if we consider light of a single

wavelength. If the incident light is coherent, these sources all have the same

phase. Light incident at a given point in the space downstream of the slit is made

up of contributions from each of these point sources and if the relative phases of

these contributions vary by 2π or more, we may expect to find minima and

maxima in the diffracted light. Such phase differences are caused by differences

in the path lengths over which contributing rays reach the point from the slit.

We can find the angle at which a first minimum is obtained in the diffracted light

by the following reasoning. The light from a source located at the top edge of the

slit interferes destructively with a source located at the middle of the slit, when

the path difference between them is equal to λ/2. Similarly, the source just below

the top of the slit will interfere destructively with the source located just below the

middle of the slit at the same angle. We can continue this reasoning along the

entire height of the slit to conclude that the condition for destructive interference

for the entire slit is the same as the condition for destructive interference between

two narrow slits a distance apart that is half the width of the slit. The path

difference is given by  so that the minimum intensity occurs at an

angle θmin given by

where

d is the width of the slit,

is the angle of incidence at which the minimum intensity occurs, and

is the wavelength of the light

A similar argument can be used to show that if we imagine the slit to be divided

into four, six, eight parts, etc., minima are obtained at angles θn given by

where

n is an integer other than zero.

There is no such simple argument to enable us to find the maxima of the

diffraction pattern. The intensity profile can be calculated using the Fraunhofer

diffraction equation as

where

is the intensity at a given angle,

is the original intensity, and

the sinc function is given by sinc(x) = sin(πx)/(πx) if x ≠ 0, and

sinc(0) = 1

This analysis applies only to the far field, that is, at a distance much larger than

the width of the slit.

2-slit (top) and 5-slit diffraction of red laser light

Diffraction of a red laser using a diffraction grating.

A diffraction pattern of a 633 nm laser through a grid of

150 slits

Diffraction grating

A diffraction grating is an optical component with a regular pattern. The form of

the light diffracted by a grating depends on the structure of the elements and the

number of elements present, but all gratings have intensity maxima at angles

θm which are given by the grating equation

where

θi is the angle at which the light is incident,

d is the separation of grating elements, and

m is an integer which can be positive or negative.

The light diffracted by a grating is found by summing the light diffracted from

each of the elements, and is essentially a convolution of diffraction and

interference patterns.

The figure shows the light diffracted by 2-element and 5-element gratings where

the grating spacings are the same; it can be seen that the maxima are in the same

position, but the detailed structures of the intensities are different.

A computer-generated image of an Airy disk.

Computer generated light diffraction pattern from a circular aperture of diameter

0.5 micrometre at a wavelength of 0.6 micrometre (red-light) at distances of

0.1 cm – 1 cm in steps of 0.1 cm. One can see the image moving from the Fresnel

region into the Fraunhofer region where the Airy pattern is seen.

Circular aperture

The far-field diffraction of a plane wave incident on a circular aperture is often

referred to as the Airy Disk. The variation in intensity with angle is given by

,

where a is the radius of the circular aperture, k is equal to 2π/λ and J1 is a Bessel

function. The smaller the aperture, the larger the spot size at a given distance, and

the greater the divergence of the diffracted beams.

General aperture

The wave that emerges from a point source has amplitude   at location r that is

given by the solution of thefrequency domain wave equation for a point source

(The Helmholtz Equation),

where   is the 3-dimensional delta function. The delta function has only radial

dependence, so the Laplace operator (aka scalar Laplacian) in thespherical

coordinate system simplifies to (see del in cylindrical and spherical coordinates)

By direct substitution, the solution to this equation can be readily shown to be the

scalar Green's function, which in the spherical coordinate system(and using the

physics time convention  ) is:

This solution assumes that the delta function source is located at the origin. If the

source is located at an arbitrary source point, denoted by the vector   and the

field point is located at the point  , then we may represent the scalar Green's

function (for arbitrary source location) as:

Therefore, if an electric field, Einc(x,y) is incident on the aperture, the field

produced by this aperture distribution is given by the surface integral:

On the calculation of Fraunhofer region

fields

where the source point in the aperture is given by the vector

In the far field, wherein the parallel rays approximation can be employed, the

Green's function,

simplifies to

as can be seen in the figure to the right (click to enlarge).

The expression for the far-zone (Fraunhofer region) field becomes

Now, since

and

the expression for the Fraunhofer region field from a planar aperture now

becomes,

Letting,

and

the Fraunhofer region field of the planar aperture assumes the form of a Fourier

transform

In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of

the aperture distribution. Huygens' principle when applied to an aperture simply

says that the far-field diffraction pattern is the spatial Fourier transform of the

aperture shape, and this is a direct by-product of using the parallel-rays

approximation, which is identical to doing a plane wave decomposition of the

aperture plane fields (see Fourier optics).

Propagation of a laser beam

The way in which the profile of a laser beam changes as it propagates is

determined by diffraction. The output mirror of the laser is an aperture, and the

subsequent beam shape is determined by that aperture. Hence, the smaller the

output beam, the quicker it diverges.

Paradoxically, it is possible to reduce the divergence of a laser beam by first

expanding it with one convex lens, and then collimating it with a second convex

lens whose focal point is coincident with that of the first lens. The resulting beam

has a larger aperture, and hence a lower divergence.

Diffraction-limited imaging

Main article: Diffraction-limited system

The Airy disk around each of the stars from the 2.56 m telescope aperture can be

seen in this lucky image of the binary star zeta Boötis.

The ability of an imaging system to resolve detail is ultimately limited

by diffraction. This is because a plane wave incident on a circular lens or mirror is

diffracted as described above. The light is not focused to a point but forms

an Airy disk having a central spot in the focal plane with radius to first null of

where λ is the wavelength of the light and N is the f-number (focal length divided

by diameter) of the imaging optics. In object space, the correspondingangular

resolution is

where D is the diameter of the entrance pupil of the imaging lens (e.g., of a

telescope's main mirror).

Two point sources will each produce an Airy pattern – see the photo of a binary

star. As the point sources move closer together, the patterns will start to overlap,

and ultimately they will merge to form a single pattern, in which case the two

point sources cannot be resolved in the image. The Rayleigh criterion specifies

that two point sources can be considered to be resolvable if the separation of the

two images is at least the radius of the Airy disk, i.e. if the first minimum of one

coincides with the maximum of the other.

Thus, the larger the aperture of the lens, and the smaller the wavelength, the finer

the resolution of an imaging system. This is why telescopes have very large lenses

or mirrors, and why optical microscopes are limited in the detail which they can

see.

Speckle patterns

The speckle pattern which is seen when using a laser pointer is another diffraction

phenomenon. It is a result of the superpostion of many waves with different

phases, which are produced when a laser beam illuminates a rough surface. They

add together to give a resultant wave whose amplitude, and therefore intensity

varies randomly.

Patterns

The upper half of this image shows a diffraction pattern of He-Ne laser

beam on an elliptic aperture. The lower half is its 2D Fourier transform

approximately reconstructing the shape of the aperture.

Several qualitative observations can be made of diffraction in general:

The angular spacing of the features in the diffraction pattern is

inversely proportional to the dimensions of the object causing the

diffraction. In other words: The smaller the diffracting object, the

'wider' the resulting diffraction pattern, and vice versa. (More

precisely, this is true of the sines of the angles.)

The diffraction angles are invariant under scaling; that is, they depend

only on the ratio of the wavelength to the size of the diffracting

object.

When the diffracting object has a periodic structure, for example in a

diffraction grating, the features generally become sharper. The third

figure, for example, shows a comparison of a double-slit pattern with

a pattern formed by five slits, both sets of slits having the same

spacing, between the center of one slit and the next.

Particle diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular,

massive particles can interfere and therefore diffract. Diffraction of electrons and

neutrons stood as one of the powerful arguments in favor of quantum mechanics.

The wavelength associated with a particle is the de Broglie wavelength

where h is Planck's constant and p is the momentum of the particle (mass ×

velocity for slow-moving particles).

For most macroscopic objects, this wavelength is so short that it is not meaningful

to assign a wavelength to them. A sodium atom traveling at about 30,000 m/s

would have a De Broglie wavelength of about 50 pico meters.

Because the wavelength for even the smallest of macroscopic objects is extremely

small, diffraction of matter waves is only visible for small particles, like electrons,

neutrons, atoms and small molecules. The short wavelength of these matter waves

makes them ideally suited to study the atomic crystal structure of solids and large

molecules like proteins.

Relatively larger molecules like buckyballs were also shown to diffract

Propagation of waves in a rectangular wave guide :

In electromagnetics and communications engineering, the term waveguide may

refer to any linear structure that conveys electromagnetic waves between its

endpoints. However, the original and most common meaning is a hollow metal

pipe used to carry radio waves. This type of waveguide is used as atransmission

line mostly at microwave frequencies, for such purposes as connecting

microwave transmitters and receivers to their antennas, in equipment such

as microwave ovens, radar sets, satellite communications, and microwave radio

links.

A dielectric waveguide employs a solid dielectric rod rather than a hollow pipe.

An optical fibre is a dielectric guide designed to work at optical

frequencies.Transmission lines such as microstrip, coplanar

waveguide, stripline or coaxial may also be considered to be waveguides.

The electromagnetic waves in (metal-pipe) waveguide may be imagined as

travelling down the guide in a zig-zag path, being repeatedly reflected between

opposite walls of the guide. For the particular case of rectangular waveguide, it

is possible to base an exact analysis on this view. Propagation in dielectric

waveguide may be viewed in the same way, with the waves confined to the

dielectric by total internal reflection at its surface. Some structures, such as Non-

radiative dielectric waveguide and the Goubau line, use both metal walls and

dielectric surfaces to confine the wave.

Short length of rectangular waveguide (WG17 with UBR120 connection-

flanges)

Rectangular Waveguides

Rectangular waveguides are important for two reasons. First of all, the Laplacian

operator separates nicely in Cartesian coordinates, so that the boundary value

problem that must be solved is both familiar and straightforward. Second, they are

extremely common in actual application in physics laboratories for piping e.g.

microwaves around as experimental probes.

In Cartesian coordinates, the wave equation becomes: 

This wave equation separates and solutions are products of sin, cos or exponential

functions in each variable separately. To determine which combination to use it

suffices to look at the BC's being satisfied. For TM waves, one solves

for   subject to  , which is automatically true if: 

where and are the dimensions of the and sides of the boundary

rectangle and where in principle .

However, the wavenumber of any given mode (given the frequency) is

determined from: 

where   for a ``wave'' to exist to propagate at all. If either index   or   

is zero, there is no wave, so the first mode that can propagate has a dispersion

relation of: 

so that: 

Each combination of permitted and is associated with a cutoff of this sort -

waves with frequencies greater than or equal to the cutoff can support propogation

in all the modes with lower cutoff frequencies.

If we repeat the argument above for TE waves (as is done in Jackson, which is

why I did TM here so you could see them both) you will be led by nearly identical

arguments to the conclusion that the lowest frequency mode cutoff occurs

for  ,   and   to produce the   solution

to the wave equation above. The cutoff in this case is: 

There exists, therefore, a range of frequencies in between where only one TE

mode is supported with dispersion: 

Note well that this mode and cutoff corresponds to exactly one-half a free-space

wavelength across the long dimension of the waveguide. The wave solution for

the right-propagating TE mode is: 

We used and to get the

second of these, and ) to get the last one.

There is a lot more one can study in Jackson associated with waveguides, but we

must move on at this time to a brief look at resonant cavities (another important

topic) and multipoles.

Inhomogeneous electromagnetic wave equation

Localized time-varying charge and current densities can act as sources of

electromagnetic waves in a vacuum. Maxwell's equations can be written in the

form of a inhomogeneous electromagnetic wave equation (or often "non

homogeneous electromagnetic wave equation") with sources. The addition of

sources to the wave equations makes the partial differential

equations inhomogeneous.

SI units

Maxwell's equations in a vacuum with charge   and current   sources can be

written in terms of the vector and scalar potentials as

where

and

.

If the Lorenz gauge condition is assumed

then the nonhomogeneous wave equations become

 .

CGS and Lorentz–Heaviside units

In cgs units these equations become

with

and the Lorenz gauge condition

.

For Lorentz–Heaviside units, sometimes used in high dimensional relativistic

calculations, the charge and current densities in cgs units translate as

.

Covariant form of the inhomogeneous wave equation

Time dilation in transversal motion. The requirement that the speed of light is

constant in every inertial reference frame leads to the theory of relativity

The relativistic Maxwell's equations can be written in covariant[disambiguation

needed] form as

 

 

where J is the four-current

,

is the 4-gradient and the electromagnetic four-potential is

 

 

with the Lorenz gauge condition

.

Here

 is the d'Alembert operator.

Curved spacetime

The electromagnetic wave equation is modified in two ways in curved spacetime,

the derivative is replaced with the covariant derivative and a new term that

depends on the curvature appears (SI units).

where

is the Ricci curvature tensor. Here the semicolon indicates covariant

differentiation. To obtain the equation in cgs units, replace the permeability

with  .

Generalization of the Lorenz gauge condition in curved spacetime is assumed

.

Solutions to the inhomogeneous electromagnetic wave equation

Retarded spherical wave. The source of the wave occurs at time t'. The wavefront

moves away from the source as time increases for t>t'. For advanced solutions, the

wavefront moves backwards in time from the source t<t'.

In the case that there are no boundaries surrounding the sources, the solutions (cgs

units) of the nonhomogeneous wave equations are

and

where

is a Dirac delta function.

For SI units

.

For Lorentz–Heaviside units,

.

These solutions are known as the retarded Lorenz gauge potentials. They

represent a superposition of spherical light waves traveling outward from the

sources of the waves, from the present into the future.

There are also advanced solutions (cgs units)

and

.

These represent a superposition of spherical waves travelling from the future into

the present.

Solution of the inhomogeneous wave equation

Equations  all have the general form

Can we find a unique solution to the above equation? Let us assume that the

source function  can be expressed as a Fourier integral

The inverse transform is

Similarly, we may write the general potential  as a Fourier integral

with the corresponding inverse

Fourier transformation of Eq.  yields

where  .

The above equation, which reduces to Poisson's equation in the limit  , and

is called Helmholtz's equation, is linear, so we may attempt a Green's function

method of solution. Let us try to find a function  such that

The general solution is then

The ``sensible'' spatial boundary conditions which we impose are

that   as  . In other words, the field goes to zero a long

way from the source. Since the system we are solving is spherically symmetric

about the point  it is plausible that the Green's function itself is spherically

symmetric. It follows that

where   and  . The most general solution to the above equation

in the region   is1

We know that in the limit  the Green's function for Helmholtz's equation must

tend towards that for Poisson's equation, which is

This is only the case if  .

Reconstructing   from Eqs. , we obtain

It follows from Eq. that

Now, the real space Green's function for the inhomogeneous wave equation satisfies

Hence, the most general solution of this equation takes the form

Comparing Eqs. we obtain

where

and  .

The real space Green's function specifies the response of the system to a point

source at position   which appears momentarily at time  . According to

the retarded Green's function   the response consists of a spherical wave,

centred on  , which propagates forward in time. In order for the wave to reach

position   at time   it must have been emitted from the source at   at

the retarded time  . According to the advanced Green's

function   the response consists of a spherical wave, centred on  , which

propagates backward in time. Clearly, the advanced potential is not consistent with our

ideas about causality, which demand that an effect can never precede its cause in time.

Thus, the Green's function which is consistent with our experience is

We are able to find solutions of the inhomogeneous wave equation which

propagate backward in time because this equation is time symmetric (i.e., it is

invariant under the transformation  ).

In conclusion, the most general solution of the inhomogeneous wave equation which

satisfies sensible boundary conditions at infinity and is consistent with causality is

This expression is sometimes written

where the rectangular bracket symbol   denotes that the terms inside the bracket

are to be evaluated at the retarded time  . Note, in particular, from

Eq.  that if there is no source (i.e.,  ) then there is no field

(i.e.,  ). But, is the above solution really unique? Unfortunately, there

is a weak link in our derivation, between Eqs. (2.110) and (2.111), where

we assume that the Green's function for the Helmholtz equation subject to the

boundary condition   as   is spherically symmetric. Let

us try to fix this problem.

With the benefit of hindsight, we can see that the Green's function

corresponds to the retarded solution in real space and is, therefore, the correct physical

Green's function. The Green's function

corresponds to the advanced solution in real space and must, therefore, be rejected. We

can select the retarded Green's function by imposing the following boundary condition

at infinity

This is called the Sommerfeld radiation condition; it basically ensures that sources

radiate waves instead of absorbing them. But, does this boundary

condition uniquely select the spherically symmetric Green's function as the solution

of

Here,   are spherical polar coordinates. If it does then we can be sure that

Eq.  represents the unique solution of the wave equation which is consistent with

causality.

Let us suppose that there are two solutions of Eq.  which satisfy the boundary

condition and revert to the unique Green's function for Poisson's equation in the

limit  . Let us call these solutions   and  , and let us form the

difference  . Consider a surface   which is a sphere of arbitrarily

small radius centred on the origin. Consider a second surface   which is a

sphere of arbitrarily large radius centred on the origin. Let   denote the volume

enclosed by these surfaces. The difference function  satisfies the homogeneous

Helmholtz equation,

throughout  . According to Green's theorem

where   denotes a derivative normal to the surface in question. It is clear

from Eq. that the volume integral is zero. It is also clear that the first surface

integral is zero, since both   and   must revert to the Green's function for

Poisson's equation in the limit  . Thus,

Equation can be written

where  is the spherical harmonic operator

The most general solution of Eq.  takes the form

Here, the   and   are arbitrary coefficients, the   are spherical

harmonics,2 and

where   are Hankel functions of the first and second kind.  It can be

demonstrated that4

where

and  . Note that the summations in Eqs.  terminate after   terms.

The large   behaviour of the   is clearly inconsistent with the Sommerfeld

radiation condition . It follows that all of the  in Eq. are zero. The most general

solution can now be expressed in the form

where the  are various weighted sums of the spherical harmonics. Substitution

of this solution into the differential equation yields

Replacing the index of summation   in the first term of the parentheses by 

we obtain

which gives us the recursion relation

It follows that if   then all of the   are equal to zero.

Let us now consider the surface integral . Since we are interested in the

limit   we can replace  by the first term of its expansion in , so

where   is a unit of solid angle. It is clear that  . This implies

that   and, hence, that  . Thus, there is only one solution of

Eq. which is consistent with the Sommerfeld radiation condition, and this is given

by Eq. We can now be sure that Eq.  is a unique solution of Eq.  subject to the

boundary condition . This boundary condition basically says that infinity is an

absorber of radiation but not an emitter, which seems entirely reasonable.

Retarded potentials

Equations have the same form as the inhomogeneous wave equation , so we can

immediately write the solutions to these equations as

Moreover, we can be sure that these solutions are unique, subject to the

reasonable proviso that infinity is an absorber of radiation but not an emitter. This

is a crucially important point. Whenever the above solutions are presented in

physics textbooks there is a tacit assumption that they are unique. After all, if they

were not unique why should we choose to study them instead of one of the other

possible solutions? The uniqueness of the above solutions has a physical

interpretation. It is clear from Eqs.  that in the absence of any charges and currents

there are no electromagnetic fields. In other words, if we observe an

electromagnetic field we can be certain that if we were to trace it backward in

time we would eventually discover that it was emitted by a charge or a current. In

proving that the solutions of Maxwell's equations are unique, and then finding a

solution in which all waves are emitted by sources, we have effectively ruled out

the possibility that the vacuum can be ``unstable'' to the production of

electromagnetic waves without the need for any sources.

Equations can be combined to form the solution of the 4-vector wave equation ,

Here, the components of the 4-potential are evaluated at some event   in space-

time,   is the distance of the volume element   from  , and the square

brackets indicate that the 4-current is to be evaluated at the retarded time; i.e., at a

time   before  .

But, does the right-hand side of Eq.  really transform as a contravariant 4-vector?

This is not a trivial question since volume integrals in 3-space are not, in general,

Lorentz invariant due to the length contraction effect. However, the integral in

Eq.  is not a straightforward volume integral because the integrand is evaluated at

the retarded time. In fact, the integral is best regarded as an integral over events in

space-time. The events which enter the integral are those which intersect a

spherical light wave launched from the event   and evolved backwards in time.

In other words, the events occur before the event   and have zero interval with

respect to  . It is clear that observers in all inertial frames will, at least, agree on

which events are to be included in the integral, since both the interval between

events and the absolute order in which events occur are invariant under a general

Lorentz transformation.

We shall now demonstrate that all observers obtain the same value of   for

each elementary contribution to the integral. Suppose that   and   are two

inertial frames in the standard configuration. Let unprimed and primed symbols

denote corresponding quantities in S and S’, respectively. Let us assign

coordinates

 (o,o,o,o,)  to r  and  (x,y,z,d) 

to the retarded event Q for which r and dv  are evaluated. Using the standard

Lorentz transformation , the fact that the interval between events P and Q is zero,

and the fact that both   and  are negative, we obtain

where   is the relative velocity between frames   and  ,   is the Lorentz

factor, and  , etc. It follows that

where   is the angle (in 3-space) subtended between the line   and the  -axis.

We now know the transformation for  . What about the transformation for  ?

We might be tempted to set  , according to the usual length

contraction rule. However, this is wrong. The contraction by a factor   only

applies if the whole of the volume is measured at the same time, which is not the

case in the present problem. Now, the dimensions of   along the   and   

axes are the same in both   and  , according to Eqs. (2.19). For the  -

dimension these equations give  . The extremities of   are

measured at times differing by  , where5

Thus,

giving

It follows from Eqs. that  . This result will clearly remain valid

even when   and   are not in the standard configuration.

Thus,   is an invariant and, therefore,   is a contravariant 4-vector.

For linear transformations, such as a general Lorentz transformation, the result of

adding 4-tensors evaluated at different 4-points is itself a 4-tensor. It follows that

the right-hand side of Eq.  is a contravariant 4-vector. Thus, this 4-vector equation

can be properly regarded as the solution to the 4-vector wave equation .

Radiation from a localized source :

While wave propagation in gyromagnetic media has been investigated extensively

in the technical literature, little attention has been given to the excitation of such

waves by a prescribed source. This problem is treated in the present paper. To

render the analysis tractable, the source function is taken as a time‐harmonic line

source of electric currents with a rapidly varying phase, embedded in an infinite

homogeneous gyromagnetic medium whose axis of magnetization is

perpendicular to the source direction. For further simplification, the operating

frequency is taken to be near gyromagnetic resonance. The resulting model

incorporates relevant anomalies of the radiation process, and also serves as a

prototype for more practical thin slab configurations. The analysis is performed

for permeability tensors in which spin exchange effects are ignored and included.

In the former case, the refractive index surface may have open branches and gives

rise to an infinity in the total radiated power (infinity catastrophe); this anomaly is

removed when spin exchange is accounted for. Detailed study of the radiation

fields and radiated power densities shows that except for certain initial directions,

the outward power flow is almost entirely in the electromagnetic waves, and the

total radiated power in the electromagnetic waves far exceeds that in the spin

exchange waves, even when the source dimensions tend to zero. These results

imply that a localized electromagnetic current source does not strongly excite the

spin exchange waves.

1 General Results

If a localized current is given as a function of space and time, the vector potential

in the Lorentz gauge is

( ,t) = d 3r' .

The retardation is often simpler if the current is Fourier transformed in time,

( ) = dt ( ,t)e i t

so that

( ,t) = ( )e - i t .

Since the current is real, the complex conjugate of the last equation shows that 

( ) =  *-  ( ) . The current can be written in terms of positive frequencies

alone as

( ,t) = Re ( )e - i t

which therefore simply looks like a sum (i.e. the integral) of terms each with an e -

i t time dependence.

Plugging this form in, the vector potential is

( ,t) = Re e - i t .

To extract the radiation fields, the distance from the source to observation point is

expanded assuming the source is localized and the observation point goes to r   

 . The vector potential is

( ,t) = Re d 3r' ( )e - i

Notice that the current integral is just the ``on energy shell'' space-time Fourier

transform of the current. That is defining the Fourier transform as

( , ) = d 3rdt ( ,t)e - i + i t

the vector potential in the radiation zone is

( ,t) = Re

There are two cases of interest. The first is where the current (or at least the

radiating part) is localized in time. In that case the total radiated energy in a small

frequency range is finite. This of course is the only physical situation. However,

in many cases, the motion is periodic for a very long time, and it is the energy

radiated per cycle or equivalently the power radiated that is desired. If the motion

continues indefinitely, the total energy radiated is infinite, so the mathematics

needs to be slightly modified as shown in the spectral resolution notes.

The magnetic field in the radiation zone is given by taking the curl of   , and to

give a r - 1 dependence, the gradient must operate on the exponent. Similarly, the

electric field is given for a harmonic time dependence as

x = i .

The fields are therefore 

( ,t)

=

Re

  

( ,t)

=

Re

The   x (  x  ) is the transform of the transverse component of the current.

The Poynting vector is   x   , and with direction along   and magnitude

given by c| |2/4  .

As shown in the spectral resolution notes, if the time integral of the product of

two functions of time 

f (t) =

Re ( )

  

g(t) =

Re ( )

is called I , the differential I per frequency range d is

= Re ( ) ( ) .

so that the energy radiated in frequency interval d in solid angle increment d

around is

= 2

where is the detected polarization which must be perpendicular to . Summing

over polarizations will gives the total energy radiated in the frequency interval

and solid angle interval. Compare this result to Jackson Eq.

Again applying results of the spectral resolution notes to the case where the

motion is periodic with period T =   , the power radiated per solid angle at the

frequency n  is

= 2

where

( ) = dt d 3rJ( ,t)e - i + in t .

Essentially, if you imagine that you have a signal that repeats with fundamental

frequency   , but in each period is localized to time much smaller than the

period, then the intensity of radiation for one period would be given by d 2I/d d

 where the current integration would be over just one period. Since the motion

repeats, the power radiated in a frequency n  is the energy radiated at that

frequency   = n  divided by the period 2 /  to get the power, and

multiplied by   to convert from d  to dn . The overall factor is  /2  which

is the difference in the prefactors of the two expressions above.

2. Long Wavelength Approximations

If the current is localized in a region or a set regions, each which has spatial

extent much smaller than a wavelength,   = 2 c/  , the origin of integration

can be translated to the approximate center of a region, and the exponential

expanded in a power series. The terms in the power series are roughly powers of

the extent of the region of nonzero current divided by the wavelength. The

exponential becomes

e - i = 1 - i + ...

and the current integrals needed to this order are 

I (0)k =

d 3r'J  k( )

  

I (1)jk =

d 3r'x'jJ  k( ) .

Just as in magnetostatics, some integrations by parts using the divergence theorem

are useful, 

(xk ) = xk + J  k   

  =i xk + J  k

  

(xjxk ) = xjxk + [xjJ  k + xkJ  j]   

  =i xjxk + [xjJ  k + xkJ  j] .

The symmetry in j and k of the last term suggests writing I (1) as

I (1)jk =

and the integrals become 

I (0)k =

- i d 3r'x'k ( ) = - i d  k

  

I (1)jk =

.

The I (0)k integral is proportional to the k component of the electric dipole moment.

The I (1)jk has two terms. The second looks like the magnetic dipole moment

d 3r'[x'jJ  k( ) - x'jJ  k( )] = 2c m .

Noting that in the field equation, one component is dotted with  , while the other

is crossed with  , a constant diagonal element can be added to it without

changing the result. That is

rjri = 0 .

so that the replacement

d 3r'x'jx'k ( ) d 3r' ( ) = Q  jk

does not change the fields. The latter expression is the quadrupole moment Qjk .

The three terms are then identified as the electric dipole, magnetic dipole and

quadrupole terms. Explicitly we can make the replacements 

=

- i + i x - + ...

or 

n

=

- in n + in x n - n + ...

where the subscript n indicates the Fourier integral over one period of of e in

t.

As a check against Jackson's results, look at the case where only the diagonal

quadrupole terms contribute with Q33 = Q0 , and Q11 = Q22 = - Q0/2 each oscillating

as cos( t) , so that only the n = 1 term survives, and the time integration gives

1 = .

The current transform is then 

1

=

-

The power radiated per solid angle is 

=

2

  

  =   

  =

which agrees with Jackson Eq. after unit conversion.

Radiation from an Oscillating Electric Dipole

 

For a charge to radiate electromagentic waves in free space, it must be either

accelerated or decelerated. Charges undergoing oscillatory motion are

continuously accelerated and decelerated and if the velocity of charges is

nonrelativistic, electromagentic waves are radiated at the oscillation frequency.

Radiation from antennas can be analyzed by superposing elementary radiation

fields emitted by individual electrons .

 

Electric field lines of static dipole.

    Animation shows the electric field lines due to an electric dipole oscillating

vertically at the origin. Near the dipole, the field lines are essentially those of a

static dipole leaving a positive charge and ending up at a negative charge.

However, at a distance of the order of half wavelength (lambda = 2Pi is assumed

here) or greater, the field lines are completely detached from the dipole. This

detachment characterizes radiation fields which propagate freely (without being

attached to charges) in free space at the speed c.