Chapter 5 Steady Magnetic Fields

Magnetic Flux Density, Field Equations Boundary Conditions

1. Magnetic Flux Density, Flux, and Field Lines

2. Equations for Steady Magnetic Fields in Free Space

3. Vector & Scalar Magnetic Potentials

4. Magnetization of Media

5. Equations for Steady Magnetic Fields in A Medium

6. Boundary Conditions for Steady Magnetic Fields

1. Magnetic Flux Density, Flux, and Field Lines

A magnetic field exerts a force on a moving charge. Hence, the

force acting on the moving charges, the current element, or the

torque acting on a small current loop can be used to quantify the

magnetic fields.

Experiments show that the magnetic force acting on a moving

charge is related not only to the magnitude and the speed of the

charge, but also to the direction of motion.

F

B

v

In-line Direction

The magnetic force will be maximum when the charge is moving

along a certain direction, and will be zero when the motion is

perpendicular to it. We define the direction in which the force is zero

as the in-line direction, as shown in the following figure.

Assuming the maximum force is Fm, if

the angle between the direction of charge

motion and the in-line direction is , the

force will besinmF

The magnitude of the force F is proportional to the product of

the magnitude of the charge q and the magnitude of the velocity v.

This force is called Lorentz force.

We define a vector B whose magnitude is with the direction

being the in-line direction. The relationship between the vector B,

the charge q, the velocity v, and the force F is

qv

Fm

BvF q

Where vector B is called magnetic flux

density, and the unit is tesla ( T ).

Lorentz force is always perpendicular to the direction of charge

motion. Consequently, the Lorenz force can only change the

direction of the charge in motion and there is no work done in this

action.

In-line Direction

F

B

v

The current element is a segment of current-carrying wire.

The magnitude of the line element vector dl stands for the length of

the current element I , and the direction is that of the current I.

F

BIdl If the current flowing in the current element

is I,then

qqtt

qI dd

d

dd

d

dd v

lll

And the force F acting on the current element in a magnetic field

with magnetic flux density B is

BlF dI

if the current is parallel to the magnetic flux density B, the force will

be zero. If it is perpendicular to B, the force is maximum.

The direction of the magnetic force on a current is always perpen-

dicular to the direction of the current flow.

The torque on a small current loop.

cd

ba F

FB

S

When viewed from a large distance, the

current loop may be considered a magnetic

dipole.

ISBBIlIlBlFlT 2

where S is the area of the frame.

The magnetic field in the plane of the

frame current can be taken to be a uniform.

The small current loop is a plane square frame with four sides of

length l each, and the direction of flow current is shown in figure.

If the magnetic flux density B is parallel to the plane of the

frame, no force will act on the sides ab and cd, while the directions of

the forces on the sides ad and bc are opposite. The magnitude of the

torque T on the frame current is

F

d c

ba

F

F

FB

S

d c

ba F

F

BBn

Bt

F

F

S

If B is perpendicular to the plane of the

frame, the forces on the four sides are

directed outside and will cancel each other.

The torque acting on the frame current is

zero.

If the angle between the vector B and the

normal to the plane of the frame is , the

vector B may be resolved into two

components Bn and Bt . Then, the magnitude

of the torque T on the current loop is

sint ISBISBT

Requiring the direction of the directed surface S and the

direction of the current to obey the right hand rule, the above

equation can be written in the following vector form as

)( BST I

It is valid for any small current loop. In general, the product IS is

called the magnetic moment of the current loop, and it is denoted as

m, so thatSm I

The above equation can be written as

BmT

which states that if the magnetic moment m is parallel to the magnetic

flux density B, the torque acting on the frame is zero. If they are

perpendicular to each other, the torque is maximum.

The magnetic flux density can also be described using a set of curves.

The tangential direction at a point on the curve stands for the direction

of magnetic flux density, and these curves are called magnetic field lines.

The vector equation for the magnetic field line is

0d lB

The magnetic field lines cannot also be intersected.

The flux of the magnetic flux density B through a directed surface

S is called magnetic flux, and it is denoted as , given by

SB d S

The unit of magnetic flux is weber (Wb).

As the electric field lines, the density of the magnetic field lines can

describe the intensity of the magnetic field. A larger density of

magnetic field lines stands for stronger magnetic field intensity.

2. Equations for Steady Magnetic Fields in Free Space

The magnetic flux density B of a steady magnetic field in

vacuum satisfies the following equations

Il 0 d lB

S 0dSB

Left equation is called Ampere’s circuital law, where 0 is the

permeability of vacuum, H/m , and I is the current

enclosed by the closed curve.

70 10π4

Ampere’s circuital law: The circulation of the magnetic flux

density in vacuum around a closed curve is equal to the current

enclosed by the curve multiplied by the permeability of vacuum.

The magnetic field lines are closed everywhere, with no beginning or

end. This may be called the principle of magnetic flux continuity.

Right equation shows that the total magnetic flux through a closed

surface is equal to zero.

From Stokes’ theorem we have Sl

SBlB d)(d

And considering S

I SJ d

0d)( 0 S SJB

Then, from Ampere’s circuital law we have

Since the above equation holds for any surface, the integrand

should be zero, leading to

JB 0

which states that the curl of the magnetic flux density of a steady

magnetic field at a point in vacuum is equal to the product of the

current density at the point and the permeability of vacuum.

From the divergence theorem we have

VS

V

d d BSB

V

V

0d B

0 B

Since the equation holds everywhere, the integrand should be zero,

i.e.

which states that the divergence of the magnetic flux density of a steady

magnetic field is equal to zero everywhere.

Consequently, we find the differential form of the equations for the

steady magnetic field in vacuum as

JB 0 0 B

The steady magnetic field in vacuum is a solenoidal field.

Considering , we obtain 0d

SSB

Based on Helmholtz’s theorem, the magnetic flux density B

should be )()()( rArrB

VV

d

)(

π4

1)(

rr

rBr V

V

d

)(

π4

1)(

rr

rBrA

Where

0)( r VV

d

)(

4π)(

0

rr

rJrA

which shows that the magnetic flux density of a steady magnetic field

at a point in vacuum is equal to the curl of the vector function A at the

point.

Considering , we haveJB 0 0 B

and )()( rArB

If the distribution of the current is known, the vector magnetic

potential A at a point can be found, and we can calculate the

magnetic flux density at the point.

VV

d

) ()(

4π)( 3

0

rr

rrrJrB

which is called the Biot-Savart’s law.

The current can be distributed in a volume, on a surface, or in a

line, and they are called volume current, surface current, and line

current, respectively.

lJJ ddd ISV S

The relationship between the magnetic flux density and the

current is

An equivalent relation among these currents is as

Where JS is the surface current density (A/m), and the direction of dl

is the flow direction of the line current.

The vector magnetic potentials and the magnetic flux densities

caused by a surface current and a line current are, respectively

SS

S

d

)(

π4)( 0

rr

rJrA

S

S

S

d)()(

π4)(

30

rr

rrrJrB

l rr

lrA

d

π4)( 0 I

l rr

rrlrB

30 )(d

π4)(

I

For some steady magnetic fields, it will be simple to calculate

the magnetic flux density based on Ampere’s circuital law.

For this, we need to find a closed curve along which the magnitude of

the magnetic flux density is constant everywhere, and the direction

coincides with the tangential direction of the curve. Then the vector

integral becomes a scalar integral, B can be taken out of the integral,

and it can be determined.

Il 0 d lB

Example 1. Calculate the magnetic flux density of an infinitely

long line current of I.

rO

z

y

x

dl

I

r ′ r – r ′

e

Solution: Select the cylindrical coordinate

system, and let the line current be along the z-

axis, then the direction of the vector

is that of B. And the direction of the cross

product vector is that of the unit vector e ,

and the direction of B is that of e , i.e.

)(d rrl

eB B

which states that the magnetic field lines are a set of the circles with

the center at the z-axis, and the magnetic flux densities is independent

of the variable .

Since the line current is infinitely long, the fields must have no

dependence on z.

In this case, the circulation of the magnetic flux density around

the magnetic field line of radius r is

rB π2d lB

Based on Ampere’s circuital law, we have

r

IB

π2 0

We can prove that this equation is valid for

the magnetic fields outside the cylindrical

conducting wire of certain cross section and

carrying a current I.

I B

Example 2. Calculate the magnetic flux density produced by a

current loop of radius a and carrying the current I.

r

z

y

x

a

r'r - r'

e'

'

x

yO

a

r''

'e'–ex

ey

e'

Solution: Employing the cylindrical

coordinate system, let the center of the

loop to be at the origin, and the plane

of the loop in the xy-plane. Because of

the symmetry of the structure, the

fields must be independent of .

Finally, we obtain

20

π4

sin)(

r

IS erA

where is the area of the loop.2 π aS

For the convenience of calculation,

let the field points be in the xz-plane.

Consider the magnetic moment of the loop current is ,

the above equation becomes

ISzem

30

π4)(

r

rmrA

Using ， we

have

AB

sincos2 π4

)(3

0 eerB rr

IS

The above results show that the vector magnetic potential A caused

by a current loop is inversely proportional to the square of the distance

r, the magnetic flux density B is inversely proportional to the cube of

the distance r, and both are related to the elevation angle.

which is correct for any magnetic

dipole with the magnetic moment m

and at the origin.

m

rA(r)

x

z

y

3. Vector & Scalar Magnetic Potentials

The relationship between the vector magnetic potential A and

the magnetic flux density B is

AB

The vector magnetic potential is different from the scalar electric

potential, but it is mainly an auxiliary function （辅助函数） for

deriving the field quantities.

Since , we have0 A AA 2 A

BA 2

In view of this, the vector magnetic potential A satisfies vector Poisson’s

equation.

when the distribution of the current is unknown, we have to solve the

equations for the steady magnetic fields based on the boundary conditions

（边界条件） . For this reason, we need to derive the differential

equations for the vector magnetic potential.

JA 0 2

In the source-free region, the above equation becomes Laplace’s

equation. 02 A

In rectangular coordinate system, Both Poisson’s and Laplace’s

equations can be decomposed into three scalar equations for the three

coordinate components, respectively. Therefore, both the method of

Green’s function and the method of separation of variables can be used

to solve the scalar Poisson’s and Laplace’s equations for all rectangular

components of the vector magnetic potential.

S

Φ SA d)(We know , then S

Φ

dSB

Using Stokes’ theorem, we obtain l

Φ

d lA

Obviously, it is very simple to calculate the magnetic flux by using

the vector magnetic potential.

In addition, the method of images can also be used .

In a source-free region, J = 0, and . Thus the magnetic

flux density B is irrotational （无旋） . In this case, the magnetic

flux density can be expressed in terms of the gradient of a scalar field

by letting

0 B

m0 B

where m is called the scalar magnetic potential.

In view of this, the scalar magnetic potential satisfies Laplace’s

equation. In this way, from the boundary conditions we can solve

Laplace’s equation to obtain the scalar magnetic potential, and the

magnetic flux density can be determined.

Nevertheless, the scalar magnetic potential is used only for source-

free regions.

0m2 0 BDue to , we have

4. Magnetization （磁化） of Media

In an atom the electrons are continuously orbited around the

nucleus, and closed loop currents are formed. Such a current loop

represents a magnetic dipole.

Under the influence of an external magnetic field, these dipoles

are rearranged, and a macroscopic( 宏 观 ) magnetic moment is

resulted in. This phenomenon is called magnetization.

In general, the arrangement of these magnetic dipoles is random,

due to thermal motion （热运动） , leading to zero overall magnetic

moment, or the lack of magnetization.

On the other hand, the electrons and the atomic nuclei themselves

are in spinning motion （自旋运动） , and they also form magnetic

dipoles with magnetic moments.

Medium

Composite field Ba+ Bs

Applied field Ba

Unlike the polarization of dielectric, the composite magnetic field

in the medium could be larger or smaller than the external

magnetic field, while the internal electric field is always smaller.

Magnetization Process

Secondary field Bs

Magnetization

Based on the magnetization process of the media, the behavior of

magnetization can be classified into three types: diamagnetic 抗磁性 ,

paramagnetic 顺词性 , ferromagnetic 铁磁性 and ferrimagnetic 亚铁磁性 media.

Diamagnetic: In these media, the composite magnetic moment is

zero under normal condition. When an external

magnetic field is applied, besides the spinning and

the orbital motion of the atom, the atomic orbit will

rotated about the applied magnetic field. This

motion is called precession 进动 .

The direction of the magnetic moment caused by

the atom is always opposite to the applied magnetic

field, and it results in a lower composite magnetic

field.

Bt

Silver, copper, bismuth, zinc, lead, mercury, and

other metals are diamagnetic.

Paramagnetic: The macroscopic composite magnetic moment is

zero due to thermal motion. Under the influence of applied magnetic

field, the direction of the magnetic moment of magnetic dipole is

turned toward the direction of the applied magnetic field. Hence, the

composite magnetic field will be increased. Aluminum, tin,

magnesium, wolfram, platinum, palladium, and others are

paramagnetic. Ferromagnetic: “Magnetic domains 磁 畴 ” exist in these

media. Under the influence of an applied magnetic field, a number

of magnetic domains will be turned. The directions of the magnetic

domains will give rise to a net magnetization that produces a

stronger total magnetic field. These media are called ferromagnetic.

Iron, cobalt, nickel, and others are ferromagnetic.

These ferromagnetic media often exhibit memory effect and

acquiring spontaneous magnetization 天然磁性 .

Ferrimagnetic: It is a metal oxide 金属氧化物 , and the magnetism

is weaker than that of ferromagnetic media, while the conductivity 导电 率 is very small. These media are called ferrimagnetic materials,

such as ferrites 铁氧体 .

The magnetization will generate the magnetic moments in the

medium. In order to measure the magnetization of the medium, we

define the vector sum of the magnetic moment per unit volume as the

magnetization intensity 磁化强度 , and it is denoted as M, given by

V

N

i

1

imM

where mi is the magnetic moment of the i-th magnetic dipole in .

is an infinitesimal volume.V

V

Ferrites are applied in microwave devices.

After magnetization, the resulted magnetic moment is produced by

the new currents in the media, and these currents are called magnetizing

currents 磁化电流 . The density of magnetizing current is denoted as J '.

SV

SV d)(

4πd

)(

4π)( n0 0

rr

erM

rr

rMrA

x

P

z

yr

dV'

O

V'r'

r - r'

S'

Where the first term stands for the vector

magnetic potential produced by the volume

magnetizing currents, and the second term is

that by the surface magnetizing currents.

The relationship between the density of magnetizing current and the

magnetization intensity M will be derived below.

After some derivations, we obtain

MJ neMJ SHence, we find

Example. A magnetic cylinder of radius a and length l is magnetized

uniformly along the axis of the cylinder. If the magnetization intensity is

M, find the magnetic flux density produced by the magnetizing current

at the point P of the axis and at a distance much larger than the radius of

the cylinder.

x

y

z

l

P(0,0, z)

O

a

SJ

Solution: Select cylindrical coordinate

system, and let the z-axis coincide with the

axis of the cylinder.

Since the magnetization is uniform, the

magnetization intensity is independent of the

coordinate variables. Hence,

0 MJ

We know the surface magnetizing current density is

neMJ S

where en is the unit vector of the outward normal to the surface.

eeeeMJ MM rzS n

x

y

z

l

P(0,0, z)

zdz'

O

a

SJ

These surface magnetizing currents form

loop currents on the lateral surface. The loop

current of width dz at z is ( dz) , and the

magnetic flux density dB caused by the current

loop at the point P(0, 0, z) (z >> a) is

SJ

zzz

Maz

d

)(2d

3

20

eB

The composite magnetic flux density produced by all of the

magnetizing currents at the point P is

zzz

Ma l

z

d)(

1

2

0 3

20 eB

22

20 1

)(

1

4 zlz

Maz

e

Since , the surface magnetizing currents exist only on the

lateral surface 侧面 of the cylinder, while the magnetizing currents

on the upper and the lower end faces are zero. Therefore, we have

MzeM

SJ

5. Equations for Steady Magnetic Fields 恒定磁场 in A Medium

In the magnetized medium, the magnetic field can be considered as

that produced by the conducting current I and the magnetizing

current I in vacuum. In this way, the circulation of the magnetic flux

density B around a closed curve is

)(d 0 IIl

lB

Il

lM

Bd

0 Considering , we have

lI

dlM

Let , thenHMB

0

Il

lH d

where H is called magnetic field intensity 磁场强度 , with a unit of

A/m. The above equation is called Ampere’s circuital law for a

medium, and it shows that in the medium the circulation of the

magnetic field intensity around a closed curve is equal to the

conducting current enclosed by the curve.

By using Stokes’ theorem, from the above equation, we obtain

JH

which is the differential form of Ampere’s circuital law in a medium ,

and it states that in a medium the curl of the magnetic field intensity at

a point is equal to the density of the conduction current at the point.

The magnetizing currents do not affect that the magnetic field lines

are closed everywhere, hence, in medium the flux of the magnetic flux

density through a closed surface is still zero, and the divergence of the

magnetic flux density is zero everywhere, i.e.

S

0dSB 0 B

Since the magnetic field intensity is only related to the conducting

current 传导电流 , the magnetic field intensity simplifies the calculation

of the magnetic fields in medium just likes the electric flux density

simplified that of the electrostatic fields in dielectric.

For most of the materials, the magnetization intensity M is

proportional 成比例的 to the magnetic field intensity H, i.e.

HM m

where m is called magnetic susceptibility 磁化率 , and it could be a

positive or negative real number.

Considering , we haveHMB

0

HB )1( m0

Let , then )1( m0 HB

where is called permeability 磁导率 , and it is usually expressed by

the relative value as

m0

r 1

Nevertheless, whether it is the diamagnetic or the paramagnetic

material, magnetization is very weak. Hence, usually their relative

permeability can be considered to be equal to 1.

After a diamagnetic 抗磁性 material is magnetized the composite

magnetic field will be decreased, thus

1 , ,0 r 0 m

A paramagnetic 顺磁性 material is magnetized, the composite

magnetic field will be increased, thus1 , ,0 r 0 m

Recently, a macromolecule 高分子 magnetic material is developed,

and the relative permeability could be on the same order as the

permittivity 介电常数 .

The magnetization in ferromagnetic materials is very strong, so that

the value of the relative permeability could be very large.

The relative permabilities of

three kinds of magnetic materials

Materials r

Gold 0.9996

Silver 0.9998

Copper 0.9999

Materials r

Aluminum 1.000021

Magnesium 1.000012

Titanium 1.000180

Materials r

Nickel 250

Iron 4000

Mu-metal 磁性合金 105

Diamagnetic FerromagneticParamagnetic

Similar to the electrical properties of the dielectrics, the

magnetic properties of the media can also be homogeneous or

inhomogeneous, linear or non-linear, and isotropic or anisotropic.

HB

33 23 31

32 22 21

31 12 11

If the permeability does not vary with space, and the medium is

called homogeneous for the magnetic property, otherwise, it is

inhomogeneous. If the permeability is independent of the magnitude

and the direction of applied magnetic field, and the magnetic flux

density is proportional to the magnetic field intensity, it is called a

linear isotropic medium for its magnetic property.

The permeability of an anisotropic medium has 9 components, and

the relationship between B and H is

For homogeneous, linear, and isotropic magnetic media, we have

l

I

d lB JB

S

0d SH 0 H

Because of , from Helmholtz’s theorem we obtain JB

VV

d

)(

π4)(

rr

rJrA

which satisfies the following differential equation

JA 2

The upper equation is the special solution 特解 of the lower equation.

The above results show that for homogeneous, linear, and isotropic

magnetic media, the equations in free space can be used, provided the

permeability of vacuum 0 is replaced by that of the medium 。

6. Boundary Conditions for Steady Magnetic Fields

The derivation of the boundary conditions is similar to that of

electrostatic fields, and here we only list the results as follows:

1

2

B2

H1B1

H2en

(a) If there is no surface current at boundary,

then we find2t1t HH

For linear isotropic media, the above equation

can be rewritten as

2

2t

1

1t

BB

(b) The normal components of the magnetic flux density are

continuous. 2n1n BB

For linear isotropic media, we have

n22 n11 HH

Il

lH d S

0dSB

Both the magnitude and the direction of the magnetic field intensity

and magnetic flux density will be discontinuous. In fact, this

discontinuity is resulted by the surface magnetizing current at the

boundary.

SJBB 0 2t1t

Applying the right hand rule on the direction of the path and the

directed surface bound by the path, the above equation can be

rewritten in the following vector form

SJeMM n21 )(

1

2

en et

1M

2M

SJ

At the boundary, the relationship between the tangential components

of magnetic flux densities and the magnetizing current is

SJMM 2t1t

Considering , we have Il

lM d

Il 0 d lB

A medium with infinite permeability is called a perfect magnetic

conductor 理想导磁体 . The magnetic field intensity 磁场强度 should

be zero in a perfect magnetic conductor.

Otherwise, from we can see that infinite magnetic flux

density , an infinite electric current, and infinite energy are required.

HB

The tangential component of the magnetic field

intensity is continuous at the boundary. Hence, it

cannot exist on the surface of a perfect magnetic

conductor.

H

In other words, the magnetic field intensity must be perpendicular

to the surface of a perfect magnetic conductor.

Example 1. A loop magnetic core 环型磁芯 with a gap is closely

wound by a coil with N turns, as shown in the figure. When the coil

carries a current I, and the leakage 漏 magnetic flux outside the coil

is neglected, find the magnetic flux density and the magnetic field

intensity in the core and the gap.

Solution: Since the leakage magnetic flux

is neglected, the direction of the magnetic

flux density is around the circle, and it is

perpendicular to the end faces of the gap.

From the boundary condition, we know that

the magnetic flux density Bg in the gap is

equal to Bf in the core, i.e.

fg0fg HHBB

Since r0 >> a , the magnetic field in the core can be considered

to be uniform. Using Ampere’s circuital law in media, and taking

the circle of radius r0 as the integral path, then we have NI lH d

NIdrB

dB

) π2( 0f

0

g

Considering , we have fg BB

) π2(

00

0 fg drd

NI

eBB

Then) π2(

00 0

gg drd

NI

eB

H In the gap

) π2(

00

0 ff drd

NI

eB

H In the core

Example 2. An infinitely long wire carrying a direct electric

current 直流电 I is placed parallel to an infinite perfect magnetic

conducting plane. The distance between the wire and the plane is h;

find the magnetic field intensity in the upper half-space.

X

h

y

x =

0

I

O

Solution: The method of images 镜像法 is used. An infinitely long

electric current I is placed at the image position. The resultant

magnetic field intensity in the upper half-space is found as

r

I

r

I

π2 π221 eeHHH

r'

h

h

Py

x 0

I

H1H2

H1H2

H

O

r

I'

'

0

Since the tangential component 切向分量 of the magnetic field

intensity must be zero on the surface of the perfect magnetic conductor,

we haveII

Hence, the resultant 合成的 magnetic field intensity is

xy hyx

hy

hyx

hy

hyx

x

hyx

xIeeH

22222222 )()()()(π2

For any point on the boundary, y = 0 ， we

obtainyhx

xIeH

)( π

22

we can see that the magnetic field intensity to be perpendicular

垂直的 to the boundary.

Example 3. An infinite line current I is nearby an infinite interface

formed between two media, and the permeabilities of the media are 1

and 2, respectively. Find the magnetic field intensities in the two

media.

I

2

1 =

Solution: Assume the current I is placed in medium 2, as shown in

the following figure.

I

H2

I '

H'

e '

e

+

1

I " e

H"

I

2

1 = +

I

H2

I '

H'

e '

e

1

I " e

H"

Uniqueness theorem 唯一性定理 shows that the field depends on the

source and the boundary. After making the above postulates, the upper

half-space is still a region with source, and the lower half-space is still a

source-free region.

II

II

2 1

2

2 1

2 1

2

In order to maintain the original boundary conditions, the

resultant fields should obey the boundary conditions, .2n1n2t1t , BBHH

III

III

1 2 2 We find

eeH)( π2

)(

π2 2 1

2 1 2 r

I

r

I22 2 HB

eH)( π 2 1

2 1

r

I11 1 HB Then

01 H eHB

r

I

π2

11 1

In this case, the image currents are . These results are

the same as before.

0, III

Obviously, if medium 1 is a perfect magnetic conductor, i.e. ,

then

1