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)

Fig.aFig.b

In electrostatics we saw that positive and negative chargescan 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 barmagnet with well defined north and south poles. If we attempt to cut the magnet into pieces as is shown in fig.bwe do not get isolated north and south poles. Instead newpole 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 asouth pole. This result can be expr

magnetic dipoleessed 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)

iBr

ˆiniφ

∆Ai

1 2 3

The magnetic flux through a closed surfaceis determined as follows: First we dividethe surface into area element with areas

, , ,..., n

nA A A A∆ ∆ ∆ ∆

BMagnetic Flux Φ

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

φ

φ

∆Φ =r

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

φ

φ

= =

∆Φ =

→∞

Φ = = ⋅

∑ ∑

∫ ∫SI magnetic flux un

rrÑ Ñ

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

B B dAΦ = ⋅∫rr

Ñ

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 dAε

φΦ =

Φ = ⋅

⋅ =

=

=∫ ∫

∫r

r

r

rÑ

Ñ

Ñ

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

Φmagnetic chargeto the

net number of electric or magnetic field lines that either enter or exit the surface. Gauss' law for the magnetic fieldexpresses the fact that the magnetic field lines are closed. The number of magnetic field lines that enter any closedsurface is exactly equal to the number of lines that exit thesurface. Thus 0.BΦ =

(32 – 4)

0 B B dAΦ = ⋅ =∫rr

Ñ

Faraday's law states that: This law describes

how a changing magnetic field generates (induces) an electricfield. Ampere's law in its original form reads:

BdE dSdtΦ

⋅ = −∫

Induced magnetic fieldsrr

r

Ñ

. 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 i ddt

µ ε

µ

µ

⋅ =

⋅ =Φ

+

∫

∫

r

rr

Ñ

ÑM "

desrcibes how a changing electric field can generate a magnetic field. The electric field between the plates of thecapacitor in the figure changes with time . Thus the elet

axwell's law of induction

E

ctricflux through the red circle is also changing with and a non-vanishing magnetic field is predicted by Maxwell's lawof induction. Experimentaly it was verified thatthe predicted magnetic field

tΦ

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 idt

i

B dS i i

didt

µ µ ε

µ

ε

µ

Φ

Φ⋅ = +

⋅ =

=

+

∫

∫

The displacement current

rr

rr

Ñ

Ñxample of the figure we can show that

between the capacitor plates is equal to thecurrent that flows through the wires whichcharge the capacitor plates.

dii

Eo enc o o

dB dS idt

µ µ ε Φ⋅ = +∫

rrÑ

The electric flux through the capacitor plates .

1The displacement current

Eo o

Ed o o

o o o

qAE A

d q qi idt

σε ε

ε εε ε ε

Φ = = =

Φ= = = =

(32 – 6)

,o enc o d encB dS i iµ µ⋅ = +∫rr

Ñ

,

Consider the capacitor withcircular plates of radius In the space between the capacitorplates the term is equal to zeroThus Ampere's law becomes:

We will use Ampere's law to determ

o d encB dS

R

i

i

µ⋅ =∫rr

Ñ

ine the magnetic field.

The calculation is identical to that of a magnetic field generated by a long wireof radius . This calculation was carried out in chapter 29 for a point P at a distance

from the wire center. We wR

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)

×

rB

di

r

RC

rdS

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 di

B

B ds Bds B ds rB i i Br

π µ µµπ

⋅ = = = = = → =∫ ∫ ∫r r

Ñ Ñ Ñ(32 – 8)

×

rB

dir

R

rdS

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

rr 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 iR RrrB i B rR

iR

π µ

π

µπ

π

π µ =

⋅ = = = =

= =

= →

∫ ∫ ∫r r

Ñ Ñ Ñ

R

2o diR

µπ

r

B

O (32 – 9)

Below we summarize the four equations on which electromagnetic theoryis based on. We use here the complete form of Ampere's law as modified by Maxwell:

: E

Maxwell's equations

Gauss' law forr

:

These equations desc

ri

be a

g

0

enc

o

B

Eo enc o o

qE dA

B dA

dE dSdt

dB dS

B

idt

ε

µ µ ε

⋅ =

⋅ =

Φ⋅ = −

Φ⋅ = +

∫

∫

∫

∫

Gauss' law for

Faraday's law :

Ampere's law :

rr

rr

rr

r

r

r

Ñ

Ñ

Ñ

Ñroup 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 suddenth

Eo enc o o

dB dS idt

µ µ ε Φ⋅ = +∫A word of explanation :

rrÑ

e set is called after him. Why? The reason is that Maxwell manipulatedthe 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 speed1 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

vε µ

= = ×

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 experimentallythe existance of electromagnetic waves.

(32 – 11)

N

S

Fig.b : Side view

horizontalφ

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. Thedipole axis does not coincide exactly

The magnetism of earth.

with the rotation axis but the two axes forman angle of 11.5 , as shown in the figure. The direction of the earth's magnetic fieldat any location is described by two angles:

(see

°

Field declination θ fig.a) is defined as the angle between the geographic northand the horizontal component of the earth's magnetic field.

(see fig.b) is defined as the angle between the horizontal aField inclination φ nd theearth's magnetic field.

N

S Fig.a : Top view

Geographic North

θ

Compass needle

(32 – 12)

There are three ways in which electrons can generatea magnetic field. We have already encountered the first method. Moving electrons constitute a currentwhich according to Ampe

Magnetism and electrons

re's law generates a magnetic field in its vicinity. An electron can alsogenerate a magnetic field because it acts as a magneticdipole. There are two mechanisms involved.

. An electron in an atom moves around the nucleusas 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 Lr m m

e Lm

π π

µ π ππ

µ

= = =

= =

= −

= = =

rr

electron.

2

orb orbe Lm

µ = −rr

(32 – 13)

Se Sm

µ = −rr

In addition to the orbital angular momentum an electronhas what is known as " " or " " angular

momentum . Spin is a quantum relativistic effect. Onecan give

S

Spin magnetic dipole moment

intrinsic spinr

a simple picture by viewing the electron as a spinningcharge sphere. The corresponding magnetic dip

ole moment is

given by the equation:

Unlike classical mechanics in whi

Se Sm

µ = −

Spin quantization.

rr

( )( )

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

r

r

r r

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 mustuse .

B

quantum mechanics

r

(32 – 14)

rSzS

rB

rSzS

rB

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 whichwe can tell whethe

z ShS m hπ

−= = × ⋅

Spin quantization

Planck's constant

,

,

r a system is described by classical or by1quantum mechanics. The term can take the values +2

1or . Thus the z-component of can take the values2

. The energy of the electron4

S

S z

S z

m

ehm

µ

µπ

−

= ± ,

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 ehUm m

U U B

µ µ

π πµ

µ

−

= − ⋅ = −

= ± = ×

= ±

rr

Bohr magneton

Se Sm

µ = −rr

2z ShS mπ

=

, 4S zehm

µπ

= ±

4 ehBU

mπ= ±

(32 – 15)

Materials can be classified on the basis of their magneticproperties 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

Mµ ⋅

= =

Diamagneti

S

s

I unit for M

m.

r

rr

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 Bµrepelled from regions of stronger

field

r

r rr

f weaker . All materials exhibit diamanetism but in paramagnetic and ferromagnetic materials ths weak diamagnetism is maskedby the much stronger paramagnetism or ferromagnetism.

Br

(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

rω

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

µ

=

=

clockwise r

r field is applied, the top electron speeds up while the elecron in thebottom orbit slows down. The corresponding angular speeds

are: , The magnetic dipole2 2

moment for

o o

B

Be Bem m

ω ω ω ω+ −= + = −

r

22 2the electons is:

2 2e eri r rωµ π π ωπ

= = =

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 Bem m

Bm

µ ω ω µ ω

µ µ µ

ω

µ

+ + −

+ −

−

= −

= = + = = −

−

=r

(32 – 17)

. rB

r+v

rF

rBF

C e

+ω

×µ+r

.

r-v

rF r

BFCe

-ω.

µ−r

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 rBm

Be Be Bem m m

F F F m r evB m r

m

ω ω

ω ω

ω ω ω ω ωω

ω ω ω ωω ω

ω ω

ω

+

+ +

+

−

−

= = =

= + = + =

= + → = +

= + + = +

= − = − =

Top electron :

Bottom electron :

;

2 2 2

1/ 2

1

1 12 2

o o oo

o o oo o

Ber m r e rBm

Be Be Bem m m

ω ω ω ωω

ω ω ω ωω ω

−

−

= − → = −

= − − = −

;

(32 – 18)

. rB

r+v

rF

rBF

C e

+ω

×µ+r

.

r-v

rF r

BFCe

-ω.

µ−r

The atoms of paramagnetic materialshave a net magnetic dipole moment in the absence of an external magneticfield. This moment is the vector sum of the electron magnetic moments.

µ

Paramagnetism

r

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

µ θ

θ µ θ

= −

=r r

t of each atom. Thermal motionopposes 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

Mr

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

it moves in the region where is stronger.

B

B

r

r

(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

B MT

BM C CTBT

=

>

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

satNB MV

NV

µ=r

toms per unit volume.

BM 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 ferromagneticmaterial becomes paramagnetic.

Ferromagnetism is due to a quantum effect known as "exchange coupling" whichtends 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 BB B

M

= +

(32 – 20)

Below the Curie temperature all magnetic momentsin a ferromagnetic material are perfectly aligned.Yet the magnetization is not saturated. The reasonis 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 thatare not aligned.

The magnetization of the non-aligned dom

o

o

B

B

1.

2.

r

r

ains

turns and becomes parallel with . oBr

(32 – 21)

If we plot the net field as function of the appliedfield we get the loop shown in the figure known as a " " loop. If we start with a unmagnetized ferromagnetic material the cu

M

o

BB

Hysteresis

hysteresisrve follows the path from

point to point , where the magnetization saturates.If we reduce the curve follows the path which isdifferent from the original path . Furtermore, evenwhen is

o

o

a bB bc

abB switched off, we have a non-zero magnetic

field. Similar effects are observed if we reverse the direction of . This is the familiar phenomenon ofpermanent magnetism and forms the basis of magnetic

oB

data recording. Hysteresis is due to the fact that thedomain reorientation is not totally revesrsible and that the domains do not return completely to theiroriginal configuration.

(32 – 22)