Fri. 6.2 Field of a Magnetized Object

Mon. Tues.

6.3, 6.4 Auxiliary Field & Linear Media

HW9

Dipole term for a loop Observation location

𝑥

𝑦

𝑧

r

Magnetic Dipole Moment

aIm

...

ˆ)(

24r

rmrA o

r

m

I

zRIm ˆ2

...

ˆˆ)(

2

2

4r

rzRIrA o

...ˆsin

2

2

4

r

RIo

Same direction as current AB

3

ˆˆ3

4)(

r

rr m-mrB o

dip

ˆsinˆcos24

)(3

rr

mrB o

dip

(yes, same form as E for p)

Generally (m’s not necessarily at origin or pointing up)

Magnetic Field Effects on Atomic Dipoles Proto-Quantum Derivation

Aqvmppp fieldkinsystem

1

Consider a charged particle moving in the presence of a magnetic field. The ‘momentum’ in the particle + field system:

Aqppkin

M

pH kin

nHamiltonia2

2

222

22

22A

m

epA

m

e

m

p

eee

2

2

1Aqp

M

For an electron, M = me, q = -e

2

2

1Aep

me

Magnetic field uniform and in the z direction gives 2

ˆ2

rBsBA z

222

822rB

m

eprB

m

e

m

pH

eee

Product Rule 2

222

822Bs

m

eBL

m

e

m

pH

eee

prB

LB

Product Rules 1 & 2 and define B in z direction, s in x-y plane, reduces to 22 BssBB

So if dB

dHm 2

2

42Bs

m

eL

m

e

e

z

e

Paramagnetic Diamagnetic Orbits orient to minimize energy- add to field

Opposes field What kind of atom is predominately diamagnetic?

*

*not really how you build a Hamiltonian, but happens to work in this case

Magnetization

M≡ density of magnetic dipole moments

d

mdM

i

imrA o

24

ˆ)(

i

i

r

r

Consider patch of material covered in magnetic dipoles

Observation location

r

r

r

If differentially small

24

ˆ

r

rmdo

24

ˆ

r

r

d

mdd

o

d

MrA o

24

ˆ

r

r

Each magnetic dipole is a current loop, so

with constant M, opposite currents where two loops meet cancel, leaving only edge current

nMd

nmdˆ

ˆ

side

bdl

dIK

sideside

side

adl

dIa

Consider a block of some thickness

d

nadIK

loop

b

ˆ

If there’s an M, what does that say about currents?

Magnetization

d

mdM

Consider patch of material covered in differentially-small magnetic dipoles

Looks a lot like one term in a cross product. So, if the magnetization, which here point’s z, varies across y, there’s a net current in x

MJb

If not equal magnetizations / currents, inner current crossings only partially cancel, giving current across body

x

y

yMyyM zz ˆ

The current down one stripe is the difference between that around two adjacent loops: yIyyI

x

zy

yIyyI

da

IdJ ˆ

xy

z

yI

z

yyI

ˆ

xy

zyx

yxyI

zyx

yxyyI

ˆ

xy

ayIayyI encenc

ˆ

xy

M z ˆ

Magnetization

d

mdM

It may be familiar (without the b-subscripts) that

Observation location

r

r

r

d

MrA o

24

ˆ

r

r

nMKbˆ

MJb

rr

adKdJrA bb oo

44

(Derivation almost identical to that for polarization’s scalar potential)

Magnetization

Example: An infinitely long circular cylinder carries a uniform magnetization parallel to its axis. Find the magnetic field inside and outside the cylinder.

zMM ˆ

Jb M 0 and Kb M n Mz s M

Like a solenoid (can you say “bar magnet”)

encloIldB

Amperian loop and argument for (comparatively) uniform and 0 outside

MLBL o

MB o

B 0Mz 0M s R,

0 s R.

R

nMKbˆ

MJb

d

mdM

AB

JB o

rr

adKdJrA bb oo

44

Before the math, think about the arrangement of microscopic current loops that would give this M.

Magnetization

Exercise: An infinitely long circular cylinder carries a uniform, circumferential magnetization. Find the magnetic field inside and outside the cylinder.

MM

R

nMKbˆ

MJb

d

mdM

AB

JB o

rr

adKdJrA bb oo

44

Before the math, think about the arrangement of microscopic current loops that would give this M.

Warning: the math of J is a little subtle.

Ex. 6.1: What’s the magnetic potential of a sphere with constant magnetization in the z direction?

𝑥

𝑦

𝑧

R r ’

r

Magnetization nMKbˆ

MJb

d

mdM

AB

JB o

rr

adKdJrA bb oo

44

rzMKbˆˆ

ˆsin M

Griffiths points out that this has the same form as example 5.11: uniform surface charge density, s, rotating with angular speed w.

swwss ˆsinˆsin RRvK

So you can jump to the conclusion and substitute M in place of swR

RrrRr

RrrR

rAo

o

3

3

4

3

ws

ws

becomes

rMRr

rM

o

o

3

33

3

Magnetization

d

mdM

Exercise: An iron rod of length L and square cross section (side a) is given a uniform longitudinal magnetization M, then is bent around into a circle. Find the magnetic field everywhere.

nMKbˆ

MJb

rr

adKdJrA bb oo

44

AB

MM

JB o

a a

Magnetization

d

mdM

Example: An iron rod of length L and square cross section (side a) is given a uniform longitudinal magnetization M, then is bent around into a circle. with a narrow gap (width w). Find the magnetic field at the center of the gap, assuming w<<a<<L.

nMKbˆ

MJb

rr

adKdJrA bb oo

44

AB

MM

JB o

a a w

Magnetization

d

mdM

Example: Like bound charge, total bound current must be 0 for any shaped object.

nMKbˆ

MJb

rr

adKdJrA bb oo

44

AB

JB o

dltKadJI bbbˆ

dltnMadMIbˆˆ

dltnMldMIbˆˆ

Stokes

bK

bJ

t

l

n

a b c a b c Product Rule 1

dltnMldMIbˆˆ

ltn ˆˆˆ

ldMldMIb

0bI

Fri. 6.2 Field of a Magnetized Object

Mon. Tues.

6.3, 6.4 Auxiliary Field & Linear Media

HW9