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Magnetism in Matter
Electric polarisation
(P) - electric dipole moment per unit vol.
Magnetic ‘polarisation’ (M) - magnetic dipole moment per unit vol.
M magnetisation Am-1 c.f. P polarisation Cm-2
Element magnetic dipole moment m
When all moments have same magnitude & direction M=Nm
N number density of magnetic moments
Dielectric polarisation described in terms of surface (uniform)
or volume (non-uniform) bound charge densities
By analogy, expect description in terms of surface (uniform)
or volume (non-uniform) magnetisation current densities
Magnetism in Matter
Electric polarisation P(r)

P(r ).nˆ   jpol (r ).nˆ dt
0
jpol (r ) 
p
P(r )
t
 r  (r )dr
allspace
p electric dipole moment of
localised charge distribution
Magnetisation M(r)
1
M(r )  r x j(r )
2
jM (r )   x M(r )
1
m
r x j(r ) dr

2 all space
m magnetic dipole moment of
localised current distribution
Magnetic moment and angular momentum
• Magnetic moment of a group of electrons m
• Charge –e mass me
j(r )   qi v i (r  ri )
i
v5
1
m    qi r x v i  (r  ri ) dr
2 i all space
v4
r5
O
r4
r3
1
m   qi ri x v i
2 i
v3
v1
r1
r2
v2
 i  me ri x v i angular momentum
m
-e
2me
 i 
i
-e
L
2me
L    i L total angular momentum
i
Force and torque on magnetic moment
Fi  qi v i x B
F
Lorentz force on point charges
  (r ) v(r ) x B(r ) dr
Continuous current distributi on
 j(r ) x B(r ) dr
j(r )   (r ) v(r )
all space

all space
Bk (r )  Bk (0)  r.Bk (0)  ... Taylor expansion
F
 j(r ) x B (0)  r.B (0)  ...  dr  0   j(r ) x r.B (0) dr  ...
k
k
k
all space
F  m.B(0) suggests
all space
Um  -m.B(0)
F  U Um  -m.B(0) c.f. Up  -p.E(0)
Torque T 
 r x  j(r ) x B(r ) dr  m x B(0)
all space
Diamagnetic susceptibility
Induced magnetic dipole moment when B field applied
Applied field causes small change in electron orbit, inducing L,m
Consider force balance equation when B = 0
(mass) x (accel) = (electric force)
meo2a 
 Ze




ω

o
2
3 

4oa
 4omea 
Ze
2
2
Ze 2
me a 
 eaB
2
4oa
2
1
2
 Ze

eB


 

3 
2me
 4omea 
eB
   o 
 o  L
2me
2
B
-e
1
2
quadratic in 
ev B  eaB
me Z
B 
oa3
2
L is the Larmor frequency
Diamagnetic susceptibility
Pair of electrons in a pz orbital


m
B
a
-e
m
 = o + L
|ℓ| = +meLa2
m = -e/2me ℓ
v
-e v x B

v
-e

 = o - L
|ℓ| = -meLa2
m = -e/2me ℓ
Electron pair acquires a net angular momentum/magnetic moment
-e v x B
Diamagnetic susceptibility
Increase in ang freq  increase in ang mom (ℓ)
Increase in magnetic dipole moment:
m  
B
-e
e
   2meL a 2
2me
 eB  2
e
e 2a 2
e 2a 2
a  
m  
2me 
B  m  
B
2me
2me
2me
 2me 
Include all Z electrons to get effective total induced magnetic
dipole moment with sense opposite to that of B
e2
m
Zao2 B
2me
ao2 : mean square radius of electron orbit
~ 10 -27 for Z  12 B  1T c.f. 1B  9.274.10 -24 Am 2
1B  Intrinsic ' spin' magnetic moment for one electron
m
Paramagnetism
Found in atoms, molecules with unpaired electron spins
Examples O2, haemoglobin (Fe ion)
Paramagnetic substances become weakly magnetised in an applied field
Energy of magnetic moment in B field Um = -m.B
Um = -9.27.10-24 J for a moment of 1 B aligned in a field of 1 T
Uthermal = kT = 4.14.10-21 J at 300K >> Um
Um/kT=2.24.10-3
Boltzmann factors e-Um/kT for moment parallel/anti-parallel to B differ little at
room temperature
This implies little net magnetisation at room temperature
Ferro, Ferri, Anti-ferromagnetism
Found in solids with magnetic ions (with unpaired electron spins)
Examples Fe, Fe3O4 (magnetite), La2CuO4
When interactions H = -J mi.mj between magnetic ions are (J) >= kT
Thermal energy required to flip moment is Nm.B >> m.B
N is number of ions in a cluster to be flipped and Um/kT > 1
Ferromagnet has J > 0 (moments align parallel)
Anti-ferromagnet has J < 0 (moments align anti-parallel)
Ferrimagnet has J < 0 but moments of different sizes giving net magnetisation
Uniform magnetisation
Electric polarisation
p
i
C.m
-2
P i
(
Cm
)
3
V m
I
z
x
M
y
IyΔz
I

xyΔz x
Magnetisation
M
m
i
i
V
A.m2
-1
(Am
)
3
m
Magnetisation is a current per unit length
For uniform magnetisation, all current localised
on surface of magnetised body
(c.f. induced charge in uniform polarisation)
Surface Magnetisation Current Density
Symbol: aM a vector current density
Units: A m-1
Consider a cylinder of radius r
and uniform magnetisation M
where M is parallel to cylinder axis
Since M arises from individual m,
(which in turn arise in current loops)
draw these loops on the end face
Current loops cancel in interior,
leaving only net (macroscopic) surface current
m
M
Surface Magnetisation Current Density
magnitude aM = M but for a vector must also determine its direction
aM
M
n̂
aM is perpendicular to both M and the surface normal
Normally, current density is “current per unit area”
in this case it is “current per unit length”, length along the
Cylinder - analogous to current in a solenoid.
aM  M nˆ
c.f. d
pol
 P.dS
Surface Magnetisation Current Density
Solenoid in vacuum
Bv ac  oNI
With magnetic core (red), Ampere’s Law integration contour encloses
two types of current, “conduction current” in the coils and
“magnetisation current” on the surface of the core
 B.d   I
o encl
 BL  o NLI  a ML 
 B  o NI  a M   B v ac
B
 > 1: aM and I in same direction (paramagnetic)
 < 1: aM and I in opposite directions (diamagnetic)
 is the relative permeability, c.f.  the relative permittivity
Substitute for aM
B  o NI  M
L
I
Magnetisation
Macroscopic electric field
EMac= EApplied + EDep = E - P/o
Macroscopic magnetic field
BMac= BApplied + BMagnetisation
BMagnetisation is the contribution to BMac from the magnetisation
BMac= BApplied + BMagnetisation = B + oM
Define magnetic susceptibility via M = cBBMac/o
BMac= B + cBBMac
EMac= E - P/o = E - EMac
BMac(1-cB) = B
EMac(1+c) = E
Diamagnets
Para, Ferromagnets
Au
Quartz
O2 STP
BMagnetisation opposes BApplied
BMagnetisation enhances BApplied
cB
-3.6.10-5
-6.2.10-5
+1.9.10-6
cB < 0
cB > 0

0.99996
0.99994
1.000002
Magnetisation
Rewrite BMac= B + oM as
BMac - oM = B
LHS contains only fields inside matter, RHS fields outside
Magnetic field intensity, H = BMac/o - M = B/o
= BMac/o - cBBMac/o
= BMac (1- cB) /o
= BMac/o
c.f. D = oEMac + P = o EMac
 = 1/(1- cB)
Relative permeability
=1+c
Relative permittivity
Non-uniform Magnetisation
Rectangular slab of material with M directed along y-axis
M increases in magnitude along x-axis
z
I1-I2 I2-I3
My
x
I1
Individual loop currents increase from left to right
There is a net current along the z-axis
Magnetisation current density jM z
I2
I3
Non-uniform Magnetisation
dx
dx
Consider 3 identical element boxes, centres separated by dx
If the circulating current on the central box is My dy
Then on the left and right boxes, respectively, it is
My 
My 


 My 
dx dy and  My 
dx  dy
x
x




Non-uniform Magnetisation


My  
My 
1 M   M 
dx    My 
dx   My  dy
2 y  y x
x

 



Magnetisation current is the difference in neighbouring
circulating currents, where the half takes care of the fact that
each box is used twice! This simplifies to
M
My
My
1 2 y dx dy 
dxdy  jMz dxdy  jMz 

2  x
x
x


Non-uniform Magnetisation
Rectangular slab of material with M directed along x-axis
M increases in magnitude along y-axis
My
z
-Mx
z
y
x
jMz 
My
x
I1-I2 I2-I3
x
jMz 
 Mx
y
I1
I2
I3
My Mx

x
y
Total magnetisation current || z
jMz 
Similar analysis for x, y components yields
jM    M
Types of Current j
  B o j   o o
jP 
E
t
j  jf  jM  jP
Total current
P
t
k
jM   x M
M = sin(ay) k
j
i
jM = curl M = a cos(ay) i
Polarisation current density from oscillation of charges as electric dipoles
Magnetisation current density from space/time variation of magnetic dipoles
Magnetic Field Intensity H
Recall Ampere’s Law
 B.d   I
o encl
or   B  o j
Recognise two types of current, free and bound
  B  o j  o  jf  jM   o  jf    M 
B

     M   jf    H  jf
 o

B
where H 
 M or B  o H  M 
o
Electric
D   oE  P
Magnetic
B
H
M
.D   f
  H  jf
o
Magnetic Field Intensity H
  B o j   o  o

1
o
E
t
  B  jf  jM  jP   o
 jf    M 
E
t
P
E
 o
t
t
B


D
    M   jf   oE  P     H  jf 
t
t
 o

D/t is displacement current postulated by Maxwell (1862) to exist in the
gap of a charging capacitor
In vacuum D = oE and displacement current exists throughout space
Boundary conditions on B, H
For LIH magnetic media B = oH
(diamagnets, paramagnets, not ferromagnets for which B = B(H))
.B  0   B.d S  0
S
B1cos1 S  B 2cos 2 S  0
 B1  B 2
 H.d  I
enclf ree
H1sin1 L  H2sin 2 L  I encl f ree  0
H1||  H2||
B
 H .d
1
1
1
2
B2
S

 - H1 sin 1  1
A
B1
2
1
1B
2
1
dℓ1
C
A
 H .d
H1
2
A
2
B
I enclfree
H2
dℓ2
2
 H2 sin  2  2
Boundary conditions on B, H
H||1  H||2
H1sin1  H2sin 2
B 1  B 2  B1cos1  B2cos 2
 r1 oH1cos1  r2 oH2cos 2
H1sin1
H2sin 2

r1 oH1cos1 r2 oH2cos 2
tan 1 r1
tan 1  r1

c.f.

tan  2 r2
tan  2  r2