lecture note #6a chapter 6. magnetic fields in matter...

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Chapter 6. Magnetic Fields in MatterLecture Note #6A

6.1 Magnetization

6.2 The Field of a Magnetized Object

6.3 The Auxiliary Field H

6.4 Linear and Nonlinear Media

• 자성의 근원: 전자의 궤도 운동에 의한 자기 모멘트와 전자 스핀

- 외부 자기장이 가해지면 회전력 발생 → 자기 모멘트 변화

• 자기장이 가해지면 원자 내 전자 궤도 운동에 의한 자기모멘트 변화 발생 +

-R

spin angular

momentumorbital angular

momentumzevRnRInIAm ˆ

2

1ˆ ˆ 2

Bm

RezR

m

eRBezRvem

ee

4ˆ

22

1ˆ

2

1 2

- 상자성 (paramagnet), 반자성 (diamagnet) , 강자성 (ferromagnet)

- 자화도 (magnetization) = 단위 부피당 자기 모멘트V

mM

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일반물리 복습 (pages 2 ~ 9)

▣원자의 자기모멘트

◈전자의 궤도운동에 의한 자기모멘트

◈전자의 스핀 모멘트

◈원자핵의 자기 모멘트

▣자성 물질 : 자기 감수율에 따라 다음과 같이 분류

◈상자성 (paramagnet)

◈강자성 (ferromagnet)

◈반자성 (diamagnet)

+

-R

spin angular


momentum

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원자의 자기모멘트

▣원운동하는 전자의 궤도운동에 의한 자기 모멘트

◈원운동하는 입자의 각운동량 :

◈한바퀴 도는 주기 :

◈전류 :

◈자기 모멘트 :

◈각운동량으로 다시 표현하면 :

▣전자의 스핀 자기 모멘트 :

▣양자론에 의하면, 자기모멘트는 보어자자수 (Bohr magnetron)의 정수 배만 가

지며, 전자의 스핀 자기모멘트는 1 보어자자수이다.

v/2T r

vqrmL

r

qqi

2

v

T

2

v

2

v 2 rqr

r

qiAe

rv

q

qm

q

em

qL

2

e

sm

eS

Lm

e

2

bllZZ mmm

eL

m

e -

2-

2- gnetron Bohr Ma 1027.9

2

24-

TJ

m

eb

zBB ˆ

2

1zS

(보어자자수)lml ...., ,2 ,1 ,0

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자기화와 자기감수율

▣자기화 : 단위 부피당 자기 쌍극자 (magnetic dipole)

▣보통 금속인 경우

◈자기화는 외부의 자기장의 크기에 비례

◈ : 자기 감수율 (magnetic susceptibility)

▣자성물질의 분류

◈상자성 : 자기감수율이 양수

◈반자성 : 자기감수율이 음수

VM

m

o

m

BM

외부

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Magnetic Susceptibility of Some Elements & Minerals

http://www.jmu.edu/cisr/journal/13.1/rd/igel/Igel_Table1Web.jpg

m105 SI

m : magnetic

susceptibility

: density

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상자성 (paramagnetism)

◈알루미늄, 나트륨, 티타늄, 텅스텐 등

◈각 원자나 분자들은 독립적으로 자기모멘트를

갖고 있으나, 모두 제멋대로 배열되어 있기

때문에 평소에는 물질이 자성을 갖지 않는다

◈외부의 자기장을 받으면, 각

자기모멘트들은 같은 방향으로

배열을 하려하나, 열운동 때문에

완전히 나란하게 배열하지 못한다

◈온도가 매우 낮으면, 물질의 자기화는 외부의

자기장에 비례하며, 온도에 반비례한다

B외부

T

BCM 외부큐리법칙:

H

HB

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강자성 (Ferromagnetism)

◈철, 코발트, 니켈, 또는 디스프로슘, 가돌리니움

등의 희토류 금속이나 합금처럼 외부자기장이 없더라도

자기화 되어있는 물질

◈교환상호작용때문이며, 전기적 쿨롱 상호작용 중 양자역학

적인 효과이다

◈큐리(Curie) 온도 이상에서는 상자성 상태가 된다

◈자기 구역들로 이루어져 있으며, 정렬된 정도에

따라 총 자기모멘트가 달라진다

http://en.wikipedia.org/wiki/Curie_temperaturehttp://www.phys.aoyama.ac.jp/~w3-

jun/achievements/study/oo/fig4-3_eng.gif

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자기이력 현상

◈ a-b-c-d-e-f-g-b-c…

◈ a : 처음에 자기화가 되어 있지 않은 상태

◈ b : 포화상태

◈ c : 외부자기장이 없더라도 잔류 자기가 남아있는 상태

◈ d : 잔류 자기를 없애기 위한 반대방향의 외부 자기장

◈ e : 반대방향의 포화상태

◈ f : 상태 c와 동일

◈ g : 상태 d와 동일

iniBo

oM BBB

솔레노이드 내부의 자기장

여기에서

: 전류 세기에 따른 솔레노이드에생성되는 내부 자기장.

BM : 솔레노이드 내부에 있는 강자성체(철심)에 의한 자기장

자석 상태

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반자성 (Diamagnetism)

◈구리, 비스무스 등과 같이 자석을 갖다

대면 약하게 반발한다

◈시계 반대 방향으로 돌고 있는 왼쪽 입

자에 B외부이 그림과 같이 주어지면, 렌츠

법칙에 의해 B외부의 반대방향으로 자기

선속이 증가해야 하므로, 입자의 속도가

증가한다

◈마찬가지로 시계 방향으로 돌고 있는 오

른쪽 입자는 자기 선속이 감소해야 하므

로 입자의 속도가 감소 한다

두 입자의 총 자기모멘트는 렌츠법칙에 의해 B외부와 반대방향으로 생긴다.

외부자기장이 없는 경우

외부자기장이 있는 경우

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6.1 Magnetization (자화)

Paramagnet:

6.1.1 Diamagnets (반자성체), Paramagnets(상자성체), Ferromagnets(강자성체)

외부 자기장이 없는 평상 상태 (내부 원자의자기쌍극자가 각기 무질서한 방향으로 배열되어 있는 상태 자기적 특성이 없음)

H

HB

외부 자기장이 상자성체에 걸리면, 원자의 자기쌍극자가 외부 자기장에 따라 나란히 배열됨.

Diamagnet:H

외부 자기장이 반자성체에 걸리면, 원자의 자기쌍극자에 의한 자기장이 외부 자기장과 서로 밀어내는 방향으로 배열됨.

(상자성체)

(반자성체)

Ferromagnet:

(강자성체)외부 자기장이 없는 평상 상태에서도 내부 원자의 자기쌍극자가 모두 한 방향으로 나란하게 배열되어 있는 상태 자기적 특성을 보임)

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I I

I

I

Fb3

Fb1

Fa4

Fa2


For a current flowing rectangular loop placed at an

angle with respect a magnetic field B in the z-direction,

6.1.2 Torques and Forces on Magnetic Dipoles

0ˆ cosˆ cos42 xIaBxIaBFF aa

- Net forces acting on the sloping sides of the loop :

Current flowing

infinitesimal rectangles

Magnetic field

in z-direction

When the current flowing

infinitesimal rectangle is

tilted at an angle from

the z axis,- A Torque acting on the loop :

0ˆ ˆ 31 yIbByIbBFF bb

- Net forces acting on the horizontal sides of the loop :

BmxBIabxIbBa

xIbBa

xIbBa

yFza

yFza

Fr bb

ˆ sinˆ sin

ˆ sin2

ˆ sin2

ˆˆ sin2

ˆˆ sin2

13

where nIAnIabm ˆ ˆ

: the magnetic dipole moment of the current loop

( : a unit vector along the normal direction of the surface area A = ab)n

Bm

: Torque acting on the magnetic dipole moment

in a uniform magnetic field

(6.1)

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For an electric dipole in an electric field

6.1.2 Torques and Forces on Magnetic Dipoles

From Chapter 4

No net torque

nIAnIabm ˆ ˆ

This torque

accounts for

“paramagnetism”.

Bm

- continued (1)

The torque acting on an electric dipole moment

in a uniform field E :

Ep

The magnetic dipole moment

The electric dipole moment

dqp

For a magnetic dipole in a magnetic field

The torque acting on a magnetic dipole moment

in a uniform field B :

+

-

-

ⓔn

nⓔ

ⓔ

n The paramagnetic materials

require odd number of electrons

For an infinitesimal loop, with magnetic dipole m, in a magnetic

field B, the net force on the loop is BmF

For a electric dipole p, in an electric field E, the net

force on the dipole is EpF

See next

page

: an analogy to that in the magnetic

case (See next page)

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R

m


6.1.2 Torques and Forces on Magnetic Dipoles - continued (2)

- the net force on a current loop is

0 BldIBldIF

For a magnetic dipole in a uniform magnetic field

- the net force on a circular current loop is

BldIF

For a magnetic dipole in a nonuniform magnetic field

F

Fhor

Fdown

F

Fhor

Fdown

- the net downward force on the circular current loop is

R

Bm

R

Bm

R

BIR

BRIF

cos cos

cos 2

2

nIRnIAm ˆ ˆ 2

For an infinitesimal loop, with magnetic dipole m, in a

magnetic field B, the net force on the loop is

BmF

(6.2)

(6.3)

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6.1.2 Torques and Forces on Magnetic Dipoles - continued (3)

Electric dipole

+

p

-

N

m

S

Magnetic dipole

m

I

Magnetic dipole

(Gilbert model)

If we break them,

+

-

N

S

Is this correct?

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Consider a circular motion of the electron around the nucleus

(without considering the electron spin yet)

6.1.3 Effect of a Magnetic Field on Atomic Orbits

v

RT

2- Period of the circular motion :

- The orbital dipole moment :

R

ev

T

eI

2

- The current caused by the electron motion :

zevRnRInIAm ˆ 2

1ˆ ˆ 2

For a magnetic field B perpendicular to the plane of the orbit,

Bm

(Tilting the entire orbit with this torque is hard, but change of the electron’s speed is relatively easy.)

+

-R

- The torque acting the orbital dipole moment in a magnetic field B :

- The electrical attraction force between the electron and proton = the centripetal force :

R

vm

R

ee

2

2

2

04

1

R

vmBve

R

ee

2

2

2

04

1

(6.4)

(6.5)

(6.6)

Under the situation of the magnetic field B applied, vv

spin angular


momentum

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Consider a circular motion of the electron around the nucleus

(without considering the electron spin yet)

6.1.3 Effect of a Magnetic Field on Atomic Orbits

Eq. (6.5) Eq. (6.6) :

When

The electron speeds up when B is turned on.

+

-R

- The magnetic dipole moment change becomes

vvvvR

mvv

R

mBve ee 22

(6.7)

(6.8)

The change in m is opposite to the direction of the field B.

- continued (1)

,smallvvv vvR

mBve e 2

em

eRBv

2

zevRm ˆ

2

1

Bm

RezR

m

eRBezRvem

ee

4ˆ

22

1ˆ

2

1 2

The increment of m is antiparallel to the field B.

The mechanism is responsible for diamagetism.

This is observed mainly in atoms with even numbers of electrons.

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6.1.4 Magnetization

(6.9)

: magnetization (자화)

Paramagnetism: The dipole associated with the spins of unpaired electrons

experience a torque tending to line them up parallel to the field.

H

HB

Diamagnetism:

(analogues to the polarization P in electrostatics)

(상자성)

(반자성)H

The orbital speed of the electrons is altered in such a way as to

change the orbital dipole moment in a direction opposite to the field.

+

-

R

spin angular

momentum

orbital angular

momentum

M magnetic dipole moment per unit volume

State of magnetic polarization:

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Chapter 6. Magnetic Fields in Matter

6.1 Magnetization




• 각 원자들의 전자 궤도 운동에 의한 전류 : Bound currents

- Volume current :

- Surface current :

MJb

nMKbˆ

- 이들 전류에 의한 벡터 포텐셜 :

''

4'v

'

4

00 das

rKd

s

rJrA bb

• 균일하게 자화된 구에 의한 자기장RM

MB

03

2

MRm 3

3

4 mrrm

rrBdip

ˆ ˆ3

1

4 3

0

(구 내부)

(구 외부)

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For a single magnetic dipole m :

6.2.1 Bound Currents

the vector potential at point p is 2

0ˆ

4 r

rmrAdip

m

I

r

p

(6.10)

For a piece of magnetized material :

the vector potential at point p is

(with the magnetic dipole moments per unit volume M )

'vˆ'

4 2

0 ds

srMrA

s

p

'vd 'rM

(6.11)

Since

Appendix 1

, ˆ1

'2s

s

s

''1

4'v ''

1

4

'v '

''v ''1

4

'v 1

''4

00

0

0

adrMs

drMs

ds

rMdrM

s

ds

rMrA

Vector product rule

From Chapter 1,

fAAfAf

AfAffA

Appendix 2

SadrM

sd

s

rM''

1'v

''

v

From Appendix 2,

(6.12)MJb

nMKb

ˆ

Vector potential of a volume current Vector potential of a surface current ndaad ˆ

from Chapter 5 (5.85)

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For a piece of magnetized material :

the volume current :

s

p

'vd 'rM

(6.13)

''

4'v

'

4

00 das

rKd

s

rJrA bb

For the electric field of polarized object

MJb

nMKbˆ

- continued

the surface current : (6.14)

where is the normal unit vector of the surface element. n

Thus, the vector potential can be written as

(6.15)

where nPbˆ

: surface charge density

Pb

: volume charge density

(4.11)

(4.12)

v'

0S

0

v'4

1

4

1d

rda'

rrV bb

(4.13)

: Bound

current

A

p

r

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Let us choose the z axis along the direction of M.

For a rotating spherical shell of uniform surface charge ,

the corresponding surface current density becomes

(6.16)

[Example 6.1]

(Solution)

0 MJb

ˆ sinˆ MnMKb

ˆ sin RvK

n

n

z

v

sin v

RM

From the comparison of above two equations, we can get

From [Example 5.11], the magnetic field of the spinning spherical shell is

MRB 003

2

3

2

MB

03

2

Since the volume current density ,0 MJb

the field of a uniformly magnetized sphere is

identical to the field of a spinning spherical shell.

inside the sphere.

The magnetic field outside the spherical shell is the same as that of a pure dipole. MRm 3

3

4

Rv

Appendix 3

Appendix 4

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Comparison between Magnetization & Polarization



4.2 The Field of a Polarized Object

•For a uniformly polarized sphere•For a uniformly mangnetized sphere

n

0 MJb

ˆ sinˆ MnMKb

ˆ sin RvK

RM

MRB 003

2

3

2

MB

03

2

Inside the sphere.

Outside the sphere.

MRm 3

3

4

,3

4 3PRp

cosˆ PnPb

: the total dipole moment of the sphere.

0 Pb

: identical to that of a perfect dipole at the origin.

RrPzzP

dz

dVE

for

3

1ˆ

3 00

Since r cos = z , the field inside the sphere is uniform :

where

Rr for

The potential outside the sphere becomes

ˆ sinˆ cos2 4 2

0 rr

mArBdip

mrrmr

rBdip

ˆ ˆ3

1

4 3

0

ˆ3

sinˆ

3

cos23

0

3

3

0

3

r

PRr

r

PRE

Rrr

rpV

for

ˆ

4

12

0

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6.2.2 Physical Interpretation of Bound Currents

All the internal

currents cancel out.

taMMm v

The magnetic dipole moment is

( : a unit vector directing outward)n

Thus, the surface current is

in terms of the magnetization M.

aIm in terms of the circulating current I.

M tI

Mt

IKb

nMKbˆ

There is no current on the top or bottom surface of the slab.

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6.2.2 Physical Interpretation of Bound Currents

taMMm v

When the magnetization is nonuniform along z-direction,

aIm

The corresponding volume current density is

M tI

y

zx

xb

M

dzdy

IJ

nMKbˆ

In general,

dzdyy

M dzyMdyyMI z

zzx

When the magnetization is nonuniform along y-direction,

z

M

dzdy

IJ

yx

xb

dydzz

M dyzMdzzMI

y

yyx

Thus, the total current density is z

M

y

MJ

yz

xb

MJb

For a steady-current, the conservation law should be satisfied 0 MJb

- continued

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6.2.2 The Magnetic Field Inside Matter

Microscopic magnetic field

(The averaged field over regions

containing many atoms)

The magnetization M is “smoothed out”.

Macroscopic magnetic field

(fields for a specific point or atom)

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Next Class

Chapter 6. Magnetic Fields in Matter

6.1 Magnetization




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[Appendix 1] Problems 1.13 & Its Solution

[Problem 1.13] Let s be the separation vector from a fixed point (x’, y’, z’) to the point

(x, y, z), and let s be its length. Show that

(a)

(b)

(c) What is the general formula for ?

zzzyyyxxxs ˆ 'ˆ 'ˆ '

ss

22

(a)

szzzyyyxxx

zzzyyxxz

yzzyyxxy

xzzyyxxx

s

2ˆ '2ˆ '2ˆ '2

ˆ '''ˆ '''ˆ '''2222222222

2/ˆ/1 sss

ns 222

''' zzyyxxs

(b)

23

23-222

23-222

23-22223-222

21-222

21-22221-222

ˆ1ˆ 'ˆ 'ˆ ''''

ˆ '2'''2

1

ˆ '2'''2

1ˆ '2'''

2

1

ˆ '''

ˆ '''ˆ '''1

s

ss

szzzyyyxxxzzyyxx

zzzzzyyxx

yyyzzyyxxxxxzzyyxx

zzzyyxxz

yzzyyxxy

xzzyyxxxs

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[Appendix 1] Problems 1.13 & Its Solution

[Problem 1.13]

(c) What is the general formula for ?

zzzyyyxxxs ˆ 'ˆ 'ˆ '

(c) x

snss

x

nn

1

ns 222''' zzyyxxs

snsss

ns

zzzyyyxxxs

nszz

sy

y

sx

x

snss

nn

nnn

ˆ1

ˆ 'ˆ 'ˆ '1

ˆ ˆ ˆ

11

11

s

xxxxzzyyxxzzyyxx

xx

s ''2'''

2

1'''

21-22221222

Back

- continued

For the case of , ' ns ''

1

x

snss

x

nn

s

xxxxzzyyxxzzyyxx

xx

s ''2'''

2

1'''

''

21-22221222

snsss

nszzzyyyxxxs

nszz

sy

y

sx

x

snss nnnnn ˆ

1 ˆ 'ˆ 'ˆ '

1 ˆ

'ˆ

'ˆ

'' 1111

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[Appendix 2] Problem 1.61(b) & Its Solution

[Problem 1.61] Although the gradient, divergence, and curl theorems are the fundamental

integral theorems of vector calculus, it is possible to derive a number of corollaries

from them. Show that

(b)

The divergence theorem :

S

add

vv vv

c

v

(b) S

adcdc

vv vv

[Hint: Replace v by in the divergence theorem.]

S

add

vvvv

Vector product rule #4 BAABBA

( : a constant vector)c

0 c vvvv

cccc

adcadccadadc

vvvv

S

adcdc

vv vv

S

add

vv v v

BAC

ACBCBA

From Chapter 1,

Back

Inha University 30

s

5.4 Magnetic Vector Potential

[Example 5.11]

(Solution)

Eq. (5.66b)

''

4

0 das

rKrA

(5.66b)

where v

K

'cos222 RrrRs

'''sin' 2 ddRda

The velocity of a point r’ in a rotating rigid body is 'v r

zyxRω

RRR

zyx

r

ˆ'sin'sinsinˆ'cossin'cos'sincosˆ'sin'sincos-

'cos'sin'sin'cos'sin

cos0sin

ˆˆˆ

'v

[Appendix 3]

Inha University 31

s


[Example 5.11] (Solution)

Since

'.cosu

- Continued (1)

We let

. when 3

2

when 3

2

2

2

Rr r

R

R,r R

r

'''sin

'cos2

'

4'

'

4

2

22

00

ddR

RrrR

rda

s

rKrA

and

yRr ˆ'cossin'v

,

Inha University 32

s



Since

- Continued (2)

The field inside this spherical shell is

yrr ˆ sin

when 33

sin

3

2

2

sin

when 33

sin

3

2

2

sin

3

4

0

3

4

0

2

3

0

00

2

3

0

Rrrωr

R

r

rR

r

RR

RrrωRrR

R

rR

rA

For the original coordinates with in the z-direction,

x

y

(r,,)z

ˆ sinrr

when ˆ sin

3

when ˆ sin3

2

4

0

0

Rrr

θR

RrrR

rA

ˆ v

v1

ˆ vv

sin

11

ˆ v

vinsin

1v

θ

r

rrrr

rrr

rsr

zRrR

rR

rrr

rrR

sr

rArr

rAsr

AB

ˆ 3

2ˆ sinˆ cos3

2

ˆ sin3

1ˆ sin

3 in

sin

1

ˆ 1

ˆ insin

1

00

00

r

z

RB 03

2

: uniform

'4

0 dar

KA

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Inha University 33



The vector potential outside the spherical shell is

- Continued (3)

The field outside this spherical shell is

when ˆ sin

3 2

4

0 Rrr

θRrA

ˆsinˆ

cos

ˆsinˆ

sin

sin

ˆˆsin

2

3

3

1

3 in

1

1

in1

33

4

0

2

4

0

2

4

0

rr

r

θR

r

Rr

rrr

r

Rs

r

rArr

rAsr

AB

R

z

d

R dR sin

(5.70b) ˆsinˆcos 23

3

4

0 rθr

RB

Inha University 34


[Example 5.11] (Solution) - Continued (4)

R

z

d

R dR sinThe total charge on the shaded ring is dRRdq sin2

The time for one revolution is

2dt

The current in the ring is

dR

dRR

dt

dqI sin

2

sin2 2

The cross sectional area of the ring is 2sin Rda

The magnetic moment of the ring is

dRRdRdaIdm sinsin sin 3422

The total magnetic moment of the shell is

44

0

4

0

4

0

34

3

4

4

3

12

1

4

3

12

1cos

4

33cos

12

1

sin33sin4

1 sin

RRR

dRdRdmm

4

3

4Rm (5.70c)

Eq. (5.70c) → Eq. (5.70b) :

ˆsinrθcosr

mB 2

4

3

0

(5.70d)

sin33sinsin3

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lecture note #6a chapter 6. magnetic fields in matter...

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