Chapter 32

Maxwell’s equations; Magnetism in matter

In this chapter we will discuss the following topics:

-Gauss’ law for magnetism -The missing term from Ampere’s law added by Maxwell -The magnetic field of the earth -Orbital and spin magnetic moment of the electron -Diamagnetic materials -Paramagnetic materials -Ferromagnetic materials

(32 – 1)

The magnetic flux through each of five faces of a die (singular of ''dice'') is given by ΦB = ±N Wb, where N (= 1 to 5) is the number of spots on the

face. The flux is positive (outward) for N even and negative (inward) for N odd. What is the flux (in Wb) through the sixth face of the die?

A.1 B.2 C.3 D.4 E.5

Fig.aFig.b

In electrostatics we saw that positive and negative charges

can be separated. This is not the case with magnetic poles,

as is shown in the figure. In fig.a we have a p

Gauss' Law for the magnetic field

ermanent bar

magnet with well defined north and south poles. If we

attempt to cut the magnet into pieces as is shown in fig.b

we do not get isolated north and south poles. Instead new

pole faces appear on the newly cut faces of the pieces and

the net result is that we end up with three smaller magnets,

each of which is a i.e. it has a north and a

south pole. This result can be expr

magnetic dipole

essed as follows:

The simplest magnetic structure that can exist is a magnetic dipole.

Magnetic monopoles do not exists as far as we know.

(32 – 2)

iB

ˆin

iΔAi

1 2 3

The magnetic flux through a closed surface

is determined as follows: First we divide

the surface into area element with areas

, , ,..., n

n

A A A A

BMagnetic Flux Φ

For each element we calculate the magnetic flux through it: cos

ˆHere is the angle between the normal and the magnetic field vectors

at the position of the i-th element. The inde

i i i i

i i i

B dA

n B

1 1

x runs from 1 to n

We then form the sum cos

Finally, we take the limit of the sum as

The limit of the sum becomes the integral:

cos

n n

i i i ii i

B

i

B dA

n

BdA B dA

SI magnetic flux un

2 T m known as the "Weber" (Wb)it :(32 – 3)

B B dA

Gauss' law for the magnetic field can be expressed

mathematically as follows: For any closed surface

Contrast this with Gauss' law for

cos

the electric field:

0B

encE

o

BdA B dA

qE dA

Gauss' law for the magnetic

field expresses the fact that there is no such a thing as a

" ". The flux of either the electric or

the magnetic field through a surface is proportional

magnetic charge

to the

net number of electric or magnetic field lines that either

enter or exit the surface. Gauss' law for the magnetic field

expresses the fact that the magnetic field lines are closed.

The number of magnetic field lines that enter any closed

surface is exactly equal to the number of lines that exit the

surface. Thus 0.B (32 – 4)

0 B B dA

Faraday's law states that: This law describes

how a changing magnetic field generates (induces) an electric

field. Ampere's law in its original form reads:

BdE dS

dt

Induced magnetic fields

. Maxwell using an elegant symmetry

argument guessed that a similar term exists in Ampere's law.

The new term is written in red :

This term, also known as "

Eo o

o enc

o enc

B dS i

B dS id

dt

M "

desrcibes how a changing electric field can generate a

magnetic field. The electric field between the plates of the

capacitor in the figure changes with time . Thus the elet

axwell's law of induction

E

ctric

flux through the red circle is also changing with and

a non-vanishing magnetic field is predicted by Maxwell's law

of induction. Experimentaly it was verified that

the predicted magnetic field

t

exists. (32 – 5)

,

Ampere's complete law has the form:

We define the displacement current

Using Ampere's law takes the form:

In t

he e

Ed o

Eo enc o o

d

o enc o d enc

dB dS i

dt

i

B dS i i

di

dt

The displacement current

xample of the figure we can show that

between the capacitor plates is equal to the

current that flows through the wires which

charge the capacitor plates.

di

i

Eo enc o o

dB dS i

dt

The electric flux through the capacitor plates .

1The displacement current

Eo o

Ed o o

o o o

qAE A

d q qi i

dt

(32 – 6)

,o enc o d encB dS i i

,

Consider the capacitor with

circular plates of radius

In the space between the capacitor

plates the term is equal to zero

Thus Ampere's law becomes:

We will use Ampere's law to

determ

o d encB dS

R

i

i

ine the magnetic field.

The calculation is identical to that of a magnetic field generated by a long wire

of radius . This calculation was carried out in chapter 29 for a point P at a distance

from the wire center. We w

R

r

ill repeat the calculation for points outside

as well as inside the capacitor plates. In this example is the distance

of the point P from the capacitor center C.

r R

r R r

(32 – 7)

B

di

r

RC

dS

P

We choose an Amperian loop that reflects the cylindrical symmetry of the problem.

The loop is a circle of radius that has its center at the capacitor platr

Magnetic field outside the capacitor plates :

,

e center C.

The magnetic field is tangent to the loop and has a constant magnitude .

cos 0 22o d

o d enc o d

i

B

B ds Bds B ds rB i i Br

(32 – 8)

B

dir

R

dS

P

C

We assume that the distribution of

within the cross-section of the capacitor plate is uniform.

We choose an Amperian loop is a circle of radius

( ) that

di

r

r R

Magnetic field inside the capacitor plates

,

2 2

, 2 2

2

2

2

has its center at C. The magnetic field is

tangent to the loop and has a constant magnitude .

cos 0

2

2

2

do enc

d enc d d

o do d

B

B ds Bds B ds rB i

r ri i i

R R

rrB i B r

R

i

R

R

2o di

R

r

B

O (32 – 9)

Below we summarize the four equations on which electromagnetic theory

is based on. We use here the complete form of Ampere's law as modified by

Maxwell:

: E

Maxwell's equations

Gauss' law for

:

These equations desc

ri

be a

g

0

enc

o

B

Eo enc o o

qE dA

B dA

dE dS

dt

dB dS

B

idt

Gauss' law for

Faraday's law :

Ampere's law :

roup of diverse phenomena and devices based

on them such as the magnetic compass,electric motors, electric generators,

radio, television, radar, x-rays, and all of optical effects.

All these in just four equations! (32 – 10)

In this section I will discuss a question which many of you may have.

Maxwell added just one term in one out of four equations, and all of a sudden

th

Eo enc o o

dB dS i

dt

A word of explanation :

e set is called after him. Why? The reason is that Maxwell manipulated

the four equations (with Ampere's law now containing histerm) and he got

solutions that described waves that could travel in vacu

8

um with a speed

1 3 10 m/s.

This happens to be the speed of light in vacuum measured a few years earlier

by Fizeau. It was natural for Maxwell to contemplate whether light,

whose nature was not

oo

v

clear could be such an electromagnetic wave.

Maxwell died soon after this and was not able to verify his hypothesis.

This task was carried out by Hertz who verified experimentally

the existance of electromagnetic waves.

(32 – 11)

N

S

Fig.b : Side view

horizontal

Compass needle

Earth has a magnetic field that can be approximated

as the field of a very large bar magnet that straddles the center of the planet. The

dipole axis does not coincide exactly

The magnetism of earth.

with the rotation axis but the two axes form

an angle of 11.5 , as shown in the figure. The direction of the earth's magnetic field

at any location is described by two angles:

(see

Field declination fig.a) is defined as the angle between the geographic north

and the horizontal component of the earth's magnetic field.

(see fig.b) is defined as the angle between the horizontal aField inclination nd the

earth's magnetic field.

N

S Fig.a : Top view

Geographic North

Compass needle

(32 – 12)

There are three ways in which electrons can generate

a magnetic field. We have already encountered the

first method. Moving electrons constitute a current

which according to Ampe

Magnetism and electrons

re's law generates a

magnetic field in its vicinity. An electron can also

generate a magnetic field because it acts as a magnetic

dipole. There are two mechanisms involved.

. An electron in an atom moves around the nucleus

as shown in the figure. For simplicity we assume a circular orbit of radius with

period . This constitutes an elect

r

T

Orbital magnetic dipole moment

2 2

ric current . The resulting2 / 2

magnetic dipole moment 2 2 2 2

In vector form: The negative sign is due to the negative charge

of t

2

he

orb or

orb orb

b

e e eviT r v r

e mvrev evr er i r L

r m meL

m

electron.

2

orb orb

eL

m

(32 – 13)

S

eS

m

In addition to the orbital angular momentum an electron

has what is known as " " or " " angular

momentum . Spin is a quantum relativistic effect. One

can give

S

Spin magnetic dipole moment

intrinsic spin

a simple picture by viewing the electron as a spinning

charge sphere. The corresponding magnetic dip

ole moment is

given by the equation:

Unlike classical mechanics in whi

S

eS

m

Spin quantization.

ch the

angular momemntum can take any value, spin and orbital

angular momentum can only have certain discreet values.

Furthermore, we cannot measure the vectors or but only

their projections

S

L

S L

along an axis (in this case defined by ).

These apparently strange rules result from the fact that at the

microscopic level classical mechanics do not apply and we must

use .

B

quantum mechanics

(32 – 14)

z- axis

SzS

B

z- axis

SzS

B

34

The quantized values of the spin angular momentum are:

The constant 6.63 10 J s is 2

known as " ". It is the yardstick by which

we can tell whethe

z S

hS m h

Spin quantization

Planck's constant

,

,

r a system is described by classical or by

1quantum mechanics. The term can take the values +

21

or . Thus the z-component of can take the values2

. The energy of the electron4

S

S z

S z

m

eh

m

,

24

The constant 9.27 10 J/T is known as4 4

the electron " " (symbol ). The electron energy

can be expressed as:

S S z

B

B

U B B

ehB ehU

m m

U U B

Bohr magneton

S

eS

m

2z S

hS m

, 4S z

eh

m

4

ehBU

m

(32 – 15)

Materials can be classified on the basis of their magnetic

properties into three categories: , , and .

Below we discuss briefly each catecory.

Magnetic Materials.

Diamagnetic paramagnetic ferromagnetic

2

3

Magnetic materials are characterized by

the magnetization vector defined as the magnetic moment per unit volume.

A m A

:

Di

m m

amag

netMV

M

Diamagneti

S

s

I unit for M

m.

netism occurs in materials composed of atoms that have electrons whose

magnetic moments are antiparallel in pairs and thus result in a zero net magnetic

moment. When we apply an external magnetic field , diamagnetic materials acquire

a weak magnetic moment which is directed opposite to . If is

inhomogeneous, the diamagnetic material is

to regines o

B

B Brepelled from regions of stronger

field

f weaker . All materials exhibit diamanetism but in

paramagnetic and ferromagnetic materials ths weak diamagnetism is masked

by the much stronger paramagnetism or ferromagnetism.

B

(32 – 16)

A model for a diamagnetic material is shown in the figure.

Two electrons move on identical orbits of radius with angular

speed . The electron in the top figure moves in the

whileo

r

counterclockwise that in the lower figure moves in the

direction. When the magnetic field 0 the

magnetic moments for each orbit are antiparallel and thus

the net magnetic moment 0. When a magnetic

B

clockwise

field

is applied, the top electron speeds up while the elecron in the

bottom orbit slows down. The corresponding angular speeds

are: , The magnetic dipole2 2

moment for

o o

B

Be Be

m m

22 2the electons is:

2 2

e eri r r

2

2

2 2 2

2 2 2 2 2 2

The negative sign indicates that are antiparall2

el

o o

netnet

er er Be er e

er

r Be

m m

Bm

(32 – 17)

. B

+v

F

BF

C e

+ω

.

-v

F

BFCe

-ω.

2

2 2

2 2 2 2

1/ 2

2 2

2

1

1 12 2

o B o

net B o

o o oo

o o oo o

net B o

F m F evB e rB

F F F m evB m r

Bem r m r e rB

m

Be Be Be

m m m

F F F m r evB m r

m

Top electron :

Bottom electron :

2 2 2

1/ 2

1

1 12 2

o o oo

o o oo o

Ber m r e rB

m

Be Be Be

m m m

(32 – 18)

. B

+v

F

BF

C e

+ω

.

-v

F

BFCe

-ω.

The atoms of paramagnetic materials

have a net magnetic dipole moment

in the absence of an external magnetic

field. This moment is the vector sum

of the electron magnetic moments.

Paramagnetism

In the presence of a magnetic field each dipole has energy cos . Here

is the angle between and . The potential energy is minimum when 0.

The magnetic field partially aligns the momen

U B

B U

t of each atom. Thermal motion

opposes the alignment. The alignment improves when the temperature is lowered

and/or when the magnetic field is large. The resulting magnetization is parallel

to the

M

field . When a paramagnetic material is placed in an inhomogeneous field

it moves in the region where is stronger.

B

B

(32 – 18)

Curie's Law

When the ratio is below 0.5 the magnetization of a paramagnetic material

follows

The constant is known as the Curie constant

When 0.5 Curie's law breaks down and a diffe

BM

T

BM C C

TB

T

Curie's law :

rent approach is required.

For very high magnetic fields and/or low temperatures, all magnetic moments

are parallel to and the magnetization

Here the ratio is the number of paramagnetic a

sat

NB M

VN

V

toms per unit volume.

B

M CT

(32 – 19)

Feromagnetism is exhibited by Iron, Nickel, Cobalt,

Gadolinium, Dysprosium and their alloys.

Ferromagnetism is abserved even in the absence of a

magnetic field (the familiar permanent ma

Ferromagnetism

gnets).

Ferromagnetism disappears when the temperature

exceeds the Curie temperature of the material.

Above its Curie temperature a ferromagnetic

material becomes paramagnetic.

Ferromagnetism is due to a quantum effect known as "exchange coupling" which

tends to align the magnetic dipole moments of neighboring atoms

The magnetization of a ferromagnetic material can be measured using a Rowland ring.

The ring consists of two parts. A prinary coil in the from of a toroid which generates

the external magnetic field . A secondary coil which measures the total magnetic fieloB d

. The amagnetic material forms the core of the torroid. The net field

Here is the contribution of the ferromagnetic core. is proportional to the

sample magnetization

o M

M M

B B B B

B B

M

(32 – 20)

Below the Curie temperature all magnetic moments

in a ferromagnetic material are perfectly aligned.

Yet the magnetization is not saturated. The reason

is that the ferromagnetic material

Magnetic domains

contains regions

" ". The magnetization is each domain is

saturated but the domains are aligned in such a

way so as to have at best a small net magnetic

moment. In the presence of an external ma

domains

gnetic

field two effects are observed:

The domains whose magnetization is aligned

with grow at the expence of those domains that

are not aligned.

The magnetization of the non-aligned dom

o

o

B

B

1.

2.

ains

turns and becomes parallel with . oB

(32 – 21)

If we plot the net field as function of the applied

field we get the loop shown in the figure known as

a " " loop. If we start with a unmagnetized

ferromagnetic material the cu

M

o

B

B

Hysteresis

hysteresis

rve follows the path from

point to point , where the magnetization saturates.

If we reduce the curve follows the path which is

different from the original path . Furtermore, even

when is

o

o

a b

B bc

ab

B switched off, we have a non-zero magnetic

field. Similar effects are observed if we reverse the

direction of . This is the familiar phenomenon of

permanent magnetism and forms the basis of magneticoB

data recording. Hysteresis is due to the fact that the

domain reorientation is not totally revesrsible and

that the domains do not return completely to their

original configuration.

(32 – 22)