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 of magnetisation is 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 bulk (non-uniform) bound charge densities
Magnetisation described in terms of surface (uniform)
or bulk (non-uniform) magnetisation current densities
Magnetism in Matter
Paramagnetism
Found in atoms, molecules with unpaired electron spins (magnetic moments)
Examples O2, haemoglobin (Fe ion)
Paramagnetic substances become weakly magnetised in an applied field
Magnetic moments align parallel to applied magnetic field to lower energy
Paramagnetic susceptibility is therefore positive
Moments fluctuate because system is at finite temperature
Energy of magnetic moment in B field Um = -m.B
Um = -9.27.10-24 J for a moment of 1 mB aligned in a field of 1 T
Uthermal = kT = 4.14.10-21 J at 300K >> Um
Um/kT=2.24.10-3
This implies little net magnetisation at room temperature
Magnetism in Matter
Diamagnetism
Found in atoms, molecules, solids with paired electron spins
Examples H2O, N2
Induced electric currents shield interior of a body from applied magnetic field
Magnetic field of induced current opposes the applied field (Lenz’s Law)
Diamagnetic susceptibilty is therefore negative
Generally small except for type I superconductor where interior is completely
shielded from magnetic fields by surface currents in superconducting state
Strong, non-uniform magnetic fields can be used to levitate bodies
via diamagnetism
Magnetism in Matter
Ferromagnetism, Ferrimagnetism, Antiferromagnetism
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
Magnetic susceptibilities non-linear because of domain formation
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
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)
Magnetisation
Uniform magnetisation and surface current density
Symbol: aM current density (vector )
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
Magnetisation
magnitude aM = M but for a vector must also determine its direction
aM
M
n
aM is perpendicular to both M and the surface normal n
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  n
c.f. 
pol
 P. n
Magnetisation
Non-uniform magnetisation and bulk current density
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-direction
Magnetisation current density jM z
I2
I3
Magnetisation
dx
dx
Consider 3 identical element boxes, centres separated by dx
If the circulating current on the central box is My dy, on the left and right
boxes, respectively, it is
My 
My 


 My 


dx dy and  My 
dx  dy
x
x




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


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
Magnetic Susceptibility
Solenoid in vacuum
Bv ac  moNI
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  m I
o encl
 BL  mo NLI  a ML 
 B  mo NI  a M   mB v ac
B
m > 1: aM and I in same direction (paramagnetic)
m < 1: aM and I in opposite directions (diamagnetic)
m is the relative permeability, c.f. e the relative permittivity
Substitute for aM
B  mo NI  M
L
I
Magnetic Susceptibility
Macroscopic electric field
EMac= EApplied + EDep = E - P/eo
Macroscopic magnetic field
BMac= BApplied + BMagnetisation
BMagnetisation is the contribution to BMac from the magnetisation
BMac= BApplied + BMagnetisation = B + moM
Define magnetic susceptibility via M = cBBMac/mo
BMac= B + cBBMac
EMac= E - P/eo = 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
m
0.99996
0.99994
1.000002
Magnetic Susceptibility
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
m
v4
r5
O
r4
r3
1
qi ri x v i

2 i
v3
v1
r1
r2
v2
 i  me ri x v i angular momentum
-e
m
2me
-e
  i  2m L
e
i
L    i L total angular momentum
i
Magnetic Susceptibility
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eoa
 4eomea 
Ze
2
2
Ze 2
me a 
 eaB
2
4eoa
2
-e
1
2
quadratic in 
1
2
 Ze

eB

   

3 
2me
 4eomea 
eB
   o 
 o  L
2me
2
B
B2 
ev B  eaB
me Z
eoa3
L is the Larmor frequency
Magnetic 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
Magnetic Susceptibility
Increase in ang freq  increase in ang mom (ℓ)
Increase in magnetic dipole moment:
m  
B
-e
e
   2m eL a 2
2m e
 eB  2
e
e 2a 2
e 2a 2
a  
m  
2m e 
B  m  
B
2m e
2m e
2m e
 2m e 
Include all Z electrons to get effective total induced magnetic
dipole moment with sense opposite to that of B
e2
m
Zao2 B
2m e
ao2 : mean square radius of electron orbit
~ 10 -27 for Z  12 B  1T c.f. 1mB  9.274.10 -24 Am 2
1mB  Intrinsic ' spin' magnetic moment for one electron
m
Magnetic Field
Rewrite BMac= B + moM as
BMac - moM = B
LHS contains only fields inside matter, RHS fields outside
Magnetic field intensity, H = BMac/mo - M = B/mo
= BMac/mo - cBBMac/mo
= BMac (1- cB) /mo
H = BMac/mmo
c.f. D = eoEMac + P = eoe EMac
The two constitutive relations
m = 1/(1- cB)
Relative permeability
e=1+c
Relative permittivity
Boundary conditions on B, H
For LIH magnetic media B = mmoH
(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  H2 sin 2
B 1  B 2  B1cos1  B 2cos 2
 mr1 moH1cos1  mr2 moH2cos 2
H1sin1
H2 sin 2

mr1 moH1cos1 mr2 moH2cos 2
tan 1 mr1
tan 1 e r1

c.f.

tan  2 mr2
tan  2 e r2