Magnetic vector potential
For an electrostatic field
 E.d  0
E  -
 x E   x   0
We cannot therefore represent B by e.g. the gradient of a scalar
since
 x B   o j (rhs not zero)
Magnetostatic field, try

also .B  0 always (.E  )
o
BxA
.B  . x A   0
 x B   x  x A  (see later)
B is unchanged by
A'  A  
 x A'   x A      x A  0
5). Magnetic Phenomena
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
Definitions
• Electric polarisation P(r)

Magnetic polarisation M(r)
P(r). nˆ   jP (r). nˆ dt
1
M(r)  r x j(r)
2
P(r)
t
jM (r)   x M(r)
0
jP (r) 
p
 r  (r)dr
allspace
p electric dipole moment of
localised charge distribution
1
m
r x j(r) dr

2 all space
m magnetic dipole moment of
localised current distribution
Magnetic moment of current loop
For a planar current loop m = I A z A m2
z unit vector perpendicular to plane
1
m
r x j(r) dr

2 all space
1
 (r - a) 2
ˆ

r x I
r d dr

2 all space
2a
ˆ
a

a zˆ I
2


(
r
a
)
r
dr d

4a all space
a zˆ I 4 a 2

 zˆ I A
4a
r x ˆ  a zˆ A   a 2
 (r  a) ˆ
j  I  (r 2  a2 ) ˆ  I

2a
Cs-1 m-2
Magnetic moment and angular momentum
• Magnetic moment of a group of electrons m
• Charge –e mass me
j(r )   qi v i (r  ri )
v5
i
v4
r5
O
1
m    qi r x v i  (r  ri ) dr
2 i all space
r4
r3
v3
1
m   qi ri x v i
2 i
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
Force and torque on magnetic moment
Fi  qi v i x B
F
Lorentz force
  (r ) v(r ) x B(r ) dr
continuous distributi on of current
all space

 j(r ) x B(r ) dr
all space
Bk (r )  Bk (0)  r.Bk (0)  ...
F  m.B(0) suggests
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
Torque on magnetic moment
F
F  I  x B c.f. q v x B
Torque T  r x F
L
ˆ
T  r x F  2 ILB sin  T
2

T  IA B sin  Tˆ
mxB
/2
F
L/2 d
L/2
F
 r

m
T
r
L
I
B
F
 ˆ L
 L

dU  2 F.t d  2 F sin d
2

 2

L


 2 ILB sin  d
2


U  IAB sin d  IAB cos  m.B
Origin of permanent magnetic
dipole moment
non-zero net angular momentum of electrons
-e
Includes both orbit and spin
a
Derive general expression via
circular orbit of one electron
I
q
1 
v
I


radius: a

 2 2 a
charge: -e
qv
I
q  -e
mass: me
2 a
speed: v
2
qva
eva
e

a
ang. freq: 
m  Ia 2  


2
2
2
ang. momentum: L
dipole moment: m
 e 
2
L
L  mea  m  
 2me 
Similar expression applies for spin.
Origin of permanent magnetic
dipole moment
 e 
L
m  
2me 

m and L have opposite sense
Consider directions:
L
-e
m
In general an atom has total magnetic dipole moment:
e
e
m
 i  L  

2me i
2me
e


ℓ quantised in units of h-bar, introduce Bohr magneton B
2m e
L    1   0,1,2
L z  m 
m  ,  1,..., 
m    1B   0,1,2
m z  m  B
m  ,  1,..., 
Diamagnetic susceptibility (r < 1)
Characterised by r < 1
In previous analysis of permanent magnetic dipole moment,
m = 0 when net L = 0: now look for induced dipole moment
Applied magnetic field causes small change in electron orbit,
leading to induced L, hence induced m
Consider force balance equation when B = 0
(mass) x (accel) = (electric force)
1
2
2
2


Ze
Ze
2


meo a 
 ωo  
2
3 
4oa
 4omea 
-e
+Ze
B
If B perp to orbit (up), extra inwards Lorentz force:
Approx: radius unchanged, ang. freq increased from o to 
ev B  eaB
-e
Larmor frequency (L)
balance equation when B ≠ 0
(mass) x (accel) = (electric force) + (extra force)
2
Ze
me 2a 
 eaB
2
4oa
1
2
 Ze

eB


 

3 
2me
 4omea 
eB
   o 
 o  L
2me
2
L is known as the Larmor frequency
quadratic in 
me Z
B 
oa3
2
Classical model for diamagnetism
• 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
-e v x B

 = o - L
|ℓ| = -meLa2
m = -e/2me ℓ
Electron pair acquires a net angular momentum/magnetic moment
Induced dipole moment
Increase in ang freq  increase in ang mom (ℓ)
 increase in magnetic dipole moment:
e
m  
   meLa2
2me
B
-e
m
 eB  2
e
e 2a 2
e 2a 2
a  
m  
me 
B  m  
B
2me
4me
4me
 2me 
Include all Z electrons to get effective total induced magnetic
dipole moment with sense opposite to that of B
e2
m
Zao2 B
6me
ao2 : mean square radius of electron orbit
~ 10-27 for Z  12 B  1T c.f. 1B  9.274.10 -24 Am2
Critical comments on last expression
Although expression is correct, its derivation is not formally correct
(no QM!)
It implies that ℓ is linear in B, whereas QM requires that ℓ
is quantised in units of h-bar
Fortunately, full QM treatment gives same answer, to which must
be added any paramagnetic-contribution
 m2
e2
2
M  N 

Zao B
 3kT 6me

 p2

P  N 
  o E
 3kT

everything is diamagnetic to some extent
Paramagnetic media (r > 1)
analogous to polar dielectric
alignment of permanent magnetic dipole moment in applied
magnetic field B
Bappl
Bdip
Bappl
An aligned electric dipole opposes the applied electric field;
But here the dipole field adds to the applied field!
Other than that, it is completely analogous in thermal effect
of disorder etc., hence use Langevin analysis again
Langevin analysis of paramagnetism
Up  p.E  Um  m.B
approximat ion when U
kT
Np 2
Nm2
P
E M 
B
3kT
3kT
P   o  EE
B
M
B
o
small
w hen U
not small
kT

 mB  kT 
M  Nm coth


kT
mB




oNm2
Np 2
E 
 B 
3 okT
3kT
As with polar dielectric media, the field B in the expressions
should be the local field Bloc but generally find Bloc ≈ B
Uniform magnetisation
Electric polarisation
p
i
C.m
-2
i
P
(
Cm
)
3
V m
I
z
x
y
IyΔz
I
M

xyΔz x
Magnetic polarisation
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: M ; a vector current density
but note 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 volume,
leaving net surface current.
M
m
Surface Magnetisation Current Density
magnitude M = M but for a vector must also determine its
direction

M
M
n̂
M is perpendicular to both M and the surface normal
 M  M  nˆ
c.f. 
b
 P.nˆ 
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.
Solenoid with magnetic core
Recap, vacuum solenoid:
Bv ac  onI
I
With magnetic core (red), Ampere’s Law
L
encloses two types of current, “conduction current” in the
coils and“magnetisation current” on the surface of material:
 B.d   I
o encl
 BL  o nL I   ML 
 B  o nI   M   rB v ac
r > 1: M and I in same direction (paramagnetic)
r < 1: M and I in opposite directions (diamagnetic)
Substitute for M :
B  o nI  M
(see later)
Non-uniform magnetisation
A rectangular slab of material in which M is directed along
y-axis only but increases in magnitude along the x-axis only
z
I1-I2 I2-I3
My
x
I1
I2
I3
As individual loop currents increase from left to right,
there is a net “mag current” along the z-axis,
implying a “mag current density” which we will call jM z
Neighbouring elemental boxes
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




Upward and circulating currents


My  
My 
1 M   M 
dx    My 
dx   My  dy
y
y

2
x
x

 



The “mag 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
A rectangular slab of material in which M is directed along
-x-axis only but increases in magnitude along the y-axis only
My
z
-Mx
z
y
x
jMz 
My
x
I1-I2 I2-I3
x
jMz
 Mx

y
Total magnetisation current || z
Similar analysis for x, y components yields
I1
I2
I3
My Mx
jM 

x
y
jM    M
z
Magnetic Field Intensity H
Recall Ampere’s Law
 B.d  o Iencl 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
Magnetic
B
D   oE  P
H
.D   f
  H  jf
o
M
Ampere’s Law for H
Often more useful to apply Ampere’s Law for H than for B
  H  jf     H.d S   jf .d S
s
s
hence  H.d   I enclf ree
Bound current in magnetic moments of atoms
Free current in conduction currents in external circuits or
metallic magnetic media
If Ib
If
L
L
vacuum B v ac  onI f c.f. H  nI f
core
 o nI f  M M  n' Ib
B

Magnetic Susceptibility B
•
Two definitions of magnetic susceptibility
•
First M = BB/o is analogous to P = oEE
B, field due to all currents, E, field due to all charges
B
M
B  B  o H  M  oH   BB
o
o
1
or B 
H  r oH  r 
c.f. D = roE
1  B
1  B
B
r
Au
-3.6.10-5
0.99996
Quartz
-6.2.10-5
0.99994
O2 STP
+1.9.10-6
1.000002
In this definition the diamagnetic susceptibility is negative and
the relative permeability is less
than unity

Magnetic Susceptibility M
•
Second definition not analogous to P = o E E
M  MH  B  o H  M  o H  MH
or B  o 1 M H  r oH  r  1 M
When  is much less than unity (all except ferromagnets) the
two definitions are roughly equivalent
1.5
B(T)
-500
0
+500
-1.5
H Am-1
Ferromagnet  ~ 150-5000 for Fe

Hysteresis and energy dissipation
1
 1 
1 
Para-, diamagnets
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
B1cos1 S  B2cos 2 S  0
 B1  B2
 H.d  I
H1sin1 L  H2sin 2 L  I encl f ree  0
H1||  H2||
2
B2
S

 H .d
1
 - H1 sin 1  1
A
B1
2
B
1
1
1
enclf ree
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
Faraday’s Law
 E.d  0
E  
electrosta tic field
d
B
 E.d   dt  S t .dS Faraday' s Law   S B.dS
E
B
B


E.d



x
E
.
d
S


.d
S
d  B.d S

S
S t
B
  x A 
 A 
xE

  x 

t
t
 t 
A
E   
electrosta tic  time - varying field
t
dℓ
S
Faraday’s Law
B(r)
I
To establish steady current, cell must do work against
Ohmic losses and to create magnetic field
Energy density in magnetic fields
Potential difference   .d
Power supplied to dv  . j d da  . j dv
Total power
j.E
dW
A 

  E 
. j dv
dt all space 
t 
Joule heating
j  E
A
j.
work to establish magnetic field
t
dW
A
  j.
dv
dt all space t

1
A



x
B
.
dv

t
o all space
A
E   
t
j
da
dℓ
Energy density in magnetic fields
dW
1

dt
o
A
 x B. dv

t
all space
.(axb)  b.( xa) - a.( xb)
 
A 
 A


B.  x
x B  dv
 - .


o all space  
t 
 t

1
B
1  A


B.
dv

x
B

.dS


o all space t
o S  t

1
1 d

B.B dv

2o dt all space
W 
UM 
1
2
1
2o
 B.B dv
c.f.
all space
 B.H dv c.f. U
E

1
2
o
2
 E.E dv
in vacuum
all space
 D.E dv
in magnetic or dielectric media
Time variation
Combining electrostatics and magnetostatics:
(1) .E = /o
where  = f + b
(2) .B = 0
“no magnetic monopoles”
(3)  x E = 0
“conservative”
(4)  x B = oj
where j = jf + jM
Under time-variation:
(1) and (2) are unchanged,
(3) becomes Faraday’s Law
(4) acquires an extra term, plus 3rd component of j
Faraday’s Law of Induction
emf x induced in a circuit equals the rate of change of magnetic
flux through the circuit

x 
   B.d S x   E.d 
t
B

 E.d   t  B.dS
B
   E.dS   t .dS Stokes' Theorem
B
  E  
t
dS
dℓ
 E.d  0 in general, only electrosta tic fields for which E  
C
x  0 so E no longer representi ble simply by E  
Displacement current
  B o j  j 
 .j 
1
1
o
 B
Ampere’s Law
.  B  0
Problem!
o

Continuity equation
 .j  
 0 for non - steady currents
t
Steady current implies constant charge density so Ampere’s
law consistent with the Continuity equation for steady currents
Ampere’s law inconsistent with the continuity equation
(conservation of charge) when charge density time dependent
Extending Ampere’s Law
add term to LHS such that
taking Div makes LHS also
identically equal to zero:
.E 

  o .E  
o
j  ?  
1
o
 B
. j  .?   0

 o.E    .  o E   .j
t
t
 t 
E  1
 
j   o
 B

The extra term is in the bracket
 t  o

extended Ampere’s Law
E
  B o j   o o
t
Types of current j
E
  B  o j   o  o
t
j  jf  jM  jP Total current
P
jP 
t
jM   x M
k
M = sin(ay) k
j
i
jM = curl M = a cos(ay) i
• Polarisation current density from oscillation of charges in
electric dipoles
• Magnetisation current density variation in magnitude of
magnetic dipoles in space/time