Maxwell’s Equations

( )Gauss's law electric

0 Gauss's law in magnetism

Faraday's law

Ampere-Maxwell lawI

oS

S

B

Eo o o

qd

ε

d

dd

dtd

dμ ε μdt

× =

× =

Φ× = −

Φ× = +

∫

∫

∫

∫

E A

B A

E s

B s

Ñ

Ñ

Ñ

Ñ•The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gauss’s law for magnetism•Faraday’s law and the Ampere-Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the respective fluxes

• Gauss’s law (electrical):• The total electric flux through any

closed surface equals the net charge inside that surface divided by εo

• This relates an electric field to the charge distribution that creates it

• Gauss’s law (magnetism): • The total magnetic flux through

any closed surface is zero• This says the number of field lines

that enter a closed volume must equal the number that leave that volume

• This implies the magnetic field lines cannot begin or end at any point

• Isolated magnetic monopoles have not been observed in nature

oS

qd

ε× =∫ E AÑ

0S

d× =∫B AÑ

• Faraday’s law of Induction:• This describes the creation of an electric field by a

changing magnetic flux• The law states that the emf, which is the line

integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path

• One consequence is the current induced in a conducting loop placed in a time-varying B

• The Ampere-Maxwell law is a generalization of Ampere’s law

• It describes the creation of a magnetic field by an electric field and electric currents

• The line integral of the magnetic field around any closed path is the given sum

Bdd

dt

Φ× = −∫ E sÑ

I Eo o o

ddμ ε μ

dt

Φ× = +∫B sÑ

Maxwell’s Equation’s in integral form

A Vo o

Q 1E dA dV⋅ = = ρ

ε ε∫∫ ∫∫∫rr

Ò

AB dA 0⋅ =∫∫

rrÒ

Gauss’s Law

Gauss’s Law for Magnetism

B

C A

d dE d B dA

dt dt

Φ⋅ = − = − ⋅∫ ∫∫r rr rlÑ

Eo encl o o o oC A

d dEB d I J dA

dt dt

Φ⋅ = µ + µ ε = µ + ε ⋅

∫ ∫∫r

r rr rlÑ

Faraday’s Law

Ampere’s Law

Maxwell’s Equation’s in free space (no charge or current)

AE dA 0⋅ =∫∫

rrÒ

AB dA 0⋅ =∫∫

rrÒ

Gauss’s Law

Gauss’s Law for Magnetism

B

C A

d dE d B dA

dt dt

Φ⋅ = − = − ⋅∫ ∫∫r rr rlÑ

Eo o o oC A

d dB d E dA

dt dt

Φ× = µ ε = µ ε ×∫ ∫∫r rr rlÑ

Faraday’s Law

Ampere’s Law

Hertz’s Experiment• An induction coil is connected to a

transmitter• The transmitter consists of two spherical

electrodes separated by a narrow gap• The discharge between the electrodes

exhibits an oscillatory behavior at a very high frequency

• Sparks were induced across the gap of the receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitter

• In a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave properties

– Interference, diffraction, reflection, refraction and polarization

• He also measured the speed of the radiation

Implication • A magnetic field will be produced in empty space if there

is a changing electric field. (correction to Ampere)• This magnetic field will be changing. (originally there

was none!)• The changing magnetic field will produce an electric field.

(Faraday)• This changes the electric field.• This produces a new magnetic field.

• This is a change in the magnetic field.

An antenna

We have changed the magnetic field near the antenna

Hook up an AC source

An electric field results! This is the start of a “radiation field.”

Look at the cross section

E and B are perpendicular (transverse) We say that the waves are “polarized.”E and B are in phase (peaks and zeros align)

Called:“Electromagnetic Waves”

Accelerating electric charges give rise to electromagnetic waves

Angular Dependence of Intensity

• This shows the angular dependence of the radiation intensity produced by a dipole antenna

• The intensity and power radiated are a maximum in a plane that is perpendicular to the antenna and passing through its midpoint

• The intensity varies as (sin2 θ) / r2

Harmonic Plane Waves

x

At t = 0

At x = 0

λ

λ = spatial period or wavelength

Τ Τ = temporal period

2v f

T T 2 k

λ π λ ω= = λ = =π

t

Er

Er

Applying Faraday to radiation

B

C

dE d

dt

Φ⋅ = −∫rrlÑ

( )C

E d E dE y E y dE y⋅ = + ∆ − ∆ = ∆∫rrlÑ

Bd dBdx y

dt dt

Φ = ∆

dBdE y dx y

dt∆ = − ∆

dE dB

dx dt= −

Applying Ampere to radiation

Eo oC

dB d

dt

Φ⋅ = µ ε∫rrlÑ

( )C

B d B z B dB z dB z⋅ = ∆ − + ∆ = − ∆∫rrlÑ

Ed dEdx z

dt dt

Φ = ∆

o o

dEdB z dx z

dt− ∆ = µ ε ∆

o o

dB dE

dx dt= −µ ε

Fields are functions of both position (x) and time (t)

o o

dB dE

dx dt= −µ ε

dE dB

dx dt= − E B

x t

∂ ∂= −∂ ∂

o o

B E

x t

∂ ∂= −µ ε∂ ∂

2

2

E B

x x t

∂ ∂ ∂= −∂ ∂ ∂

2

o o 2

B E

t x t

∂ ∂ ∂= −µ ε∂ ∂ ∂

Partial derivatives are appropriate

2 2

o o2 2

E E

x t

∂ ∂= µ ε∂ ∂

This is a wave equation!

The Trial Solution

• The simplest solution to the partial differential equations is a sinusoidal wave:– E = Emax cos (kx – ωt)

– B = Bmax cos (kx – ωt)

• The angular wave number is k = 2π/λ– λ is the wavelength

• The angular frequency is ω = 2πƒ– ƒ is the wave frequency

The trial solution

( )y oE E E sin kx t= = − ω2 2

o o2 2

E E

x t

∂ ∂= µ ε∂ ∂

( )2

2o2

EE sin kx t

t

∂ = −ω − ω∂

( )2

2o2

Ek E sin kx t

x

∂ = − − ω∂

( ) ( )2 2o o o ok E sin kx t E sin kx t− − ω = −µ ε ω − ω

2

2o o

1

k

ω =µ ε

The speed of light (or any other electromagnetic radiation)

o o

1v c

k

ω= = =µ ε

2v f

T T 2 k

λ π λ ω= = λ = =π

The electromagnetic spectrum

2v f

T T 2 k

λ π λ ω= = λ = =π

Another lookdE dB

dx dt= −

( )y oE E E sin kx t= = − ω

( ) ( )o o

d dE sin kx t B sin kx t

dx dt− ω = − − ω

( ) ( )o oE k cos kx t B cos kx t− ω = ω − ω

o

o o o

E 1c

B k

ω= = =µ ε

( )z oB B B sin kx t= = − ω

Energy in Waves

2 20

0

1 1u E B

2 2= ε +

µ

o

o o o

E 1c

B k

ω= = =µ ε

20u E= ε

2

0

1u B=

µ

0

0

u EBε=µ

Poynting Vector

• Poynting vector points in the direction the wave moves• Poynting vector gives the energy passing through a unit

area in 1 sec.• Units are Watts/m2

( )0

1S E B= ×

µ

r r r

S cu=r

= = =2 2

o o o

E cBEBS

μ μ c μ

Intensity

• The wave intensity, I, is the time average of S (the Poynting vector) over one or more cycles

• When the average is taken, the time average of cos2(kx - ωt) = ½ is involved

= = = = =2 2

max max max maxav 2 2 2

I aveo o o

E B E cBS cu

μ μ c μ

Radiation Pressure

Up

c

∆∆ =

F 1 dpP

A A dt= =

aveS1 dUP

Ac dt c= =

r

Maxwell showed: (Absorption of radiation by an object)

What if the radiation reflects off an object?

Pressure and Momentum

• For a perfectly reflecting surface, p = 2U/c and P = 2S/c

• For a surface with a reflectivity somewhere between a perfect reflector and a perfect absorber, the momentum delivered to the surface will be somewhere in between U/c and 2U/c

• For direct sunlight, the radiation pressure is about 5 x 10-6 N/m2