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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