Post on 11-Jan-2016
Magnetic vector potential
0 x x
-.d
0
E
EE For an electrostatic field
We cannot therefore represent B by e.g. the gradient of a scalarsince
Magnetostatic field, try
B is unchanged by
)
o
o
.( always 0 . also
zero) not (x
EB
jB rhs
later) (see xx x
0 x
x
AB
AB
AB
..
0xx'x
AAA
AA
'
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 mWhen all moments have same magnitude & direction M=NmN 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 electric dipole moment of m magnetic dipole moment oflocalised charge distribution localised current distribution
r(r)rp
P(r)(r)j
n(r).jnP(r).
P
P
d
t
dt
allspace
0
ˆˆ
space all
x2
1
x
x2
1
dr j(r) rm
M(r)(r)j
j(r) r M(r)
M
Magnetic moment of current loop
2-1-
22
m Cs
2a
a)(r )a(r ˆ Iˆ I
j
a
2
2
space all
2
space all
2
space all
a Az ax
A z 4a
a4 z a
d dr ra-r4a
z a
dr d r2a
a-rx
2
1
x2
1
ˆˆ
Iˆ Iˆ
Iˆ
I ˆ
r
)(
)( r
dr j(r) rm
For a planar current loop m = I A z A m2
z unit vector perpendicular to plane
Magnetic moment and angular momentum• Magnetic moment of a group of electrons m
• Charge –e mass me
momentum angular total 2m
e-
2m
e-
momentum angularxm
xq2
1
d)(xq2
1
)(q)(
i
iei
ie
iiei
i
iii
i space all
iii
iiii
LL Lm
v r
v r m
r rrv r m
rrvrj
Ov1
r1
v4
v3v2
v5r5
r4
r3
r2
Force and torque on magnetic moment
)(x d )(x)(x Torque
)(-U c.f. )(.-U U
)(.-U suggests )(
...)(B.)(B)(B
d )(x)(
current of ondistributi continuous d )(x)( )(
force Lorentz x q
space all
pm
m
kkk
space all
space all
iii
0B mrrB rj rT
0p.E0Bm F
0Bm0m.B F
0r0r
rrB rj
rrB rvr F
BvF
Torque on magnetic moment
B m
TT
T F rT
F r T
Bv BF
x
sin BA
sin LB2
L2 x
xTorque
x q c.f. x
ˆ I
ˆI
I
F
FL/2 d
m.B
tF
cos ABdsinABU
dsin2
LLB2
dsin2
LF2d
2
L .2dU
I I
I
ˆ
L/2
IB
F
r
rT
F
mL
Origin of permanent magnetic dipole moment
non-zero net angular momentum of electronsIncludes both orbit and spin Derive general expression via circular orbit of one electron
radius: acharge: -emass: me
speed: vang. freq: ang. momentum: Ldipole moment: m
Similar expression applies for spin.
Lm
e
2e
22
2m
eamL
2
ae
2
eva
2
qvaam
-eqa2
v qa2
v
2
1q
I
I
I
I
a
-e
Origin of permanent magnetic dipole moment
Consider directions: m and L have opposite sense
In general an atom has total magnetic dipole moment:
ℓ quantised in units of h-bar, introduce Bohr magneton
Lm
e2m
e
m
L-e
eii
e 2m
e
2m
e Lm
eB 2m
e
,...,1,
,...,1,
m
mm
Bz
B
z
mm
0,1,2 1m
L
0,1,2 1L
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 momentApplied 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)
If B perp to orbit (up), extra inwards Lorentz force:Approx: radius unchanged, ang. freq increased from o to
-e+Ze
21
3eo
2
o2o
22oe am4
Zeω
a4
Zeam
aBee Bv
-eB
Larmor frequency (L)balance equation when B ≠ 0(mass) x (accel) = (electric force) + (extra force)
L is known as the Larmor frequency
Loe
o
3o
e2
e3
eo
2
2o
22
e
2m
eB
a
ZmB
2m
eB
am4
Ze
inquadratic aBea4
Zeam
21
Classical model for diamagnetism• Pair of electrons in a pz orbital
= o + L
|ℓ| = +meLa2 m = -e/2me ℓ
= o - L
|ℓ| = -meLa2 m = -e/2me ℓ
a
v-e
m
-e v x B
v-e
m
-e v x B
B
Electron pair acquires a net angular momentum/magnetic moment
Induced dipole momentIncrease in ang freq increase in ang mom (ℓ) increase in magnetic dipole moment:
Include all Z electrons to get effective total induced magneticdipole moment with sense opposite to that of B
Bme
22
e
222
ee
e
2Le
e
4m
aeB
4m
aea
2m
eBm
2m
em
am2m
em
-eB
m
224-B
27-
2o
2o
e
2
Am9.274.10 1 c.f. 1T B 12Z for 10~
orbit electron of radiussquaremean:a aZ6m
e
Bm
Critical comments on last expressionAlthough expression is correct, its derivation is not formally correct(no QM!)
It implies that ℓ is linear in B, whereas QM requires that ℓis quantised in units of h-bar
Fortunately, full QM treatment gives same answer, to which mustbe added any paramagnetic-contribution
everything is diamagnetic to some extent
EP
BM
o3kT
p
aZ6m
e
3kT
m
2
2o
e
22
N
N
Paramagnetic media (r > 1)analogous to polar dielectricalignment of permanent magnetic dipole moment in appliedmagnetic field B
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 effectof disorder etc., hence use Langevin analysis again
Bappl Bappl
Bdip
Langevin analysis of paramagnetism
As with polar dielectric media, the field B in the expressionsshould be the local field Bloc but generally find Bloc ≈ B
3kT
m
kT3
p
3kT
m
3kT
p
smallnotkTUhen wsmallkT
Uwhenionapproximat
UU
2o
Bo
2
E
BEo
22
mp
NN
NN
BM EP
BMEP
m.Bp.E
o
mB
kT
kT
mBcoth mM N
Uniform magnetisation
Electric polarisation Magnetic polarisation
)(Amm
A.m
VCm
m
C.m
V1-
3
2i
i2-
3i
i
mM
pP )(
I
z
yx
xyΔx
yΔM
I
z
zI
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 densitybut note units: Am-1
Consider a cylinder of radius r and uniform magnetisation Mwhere 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 itsdirection
M 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.
nP nM ˆ.ˆ bM c.f.
M n
M
Solenoid with magnetic coreRecap, vacuum solenoid:
With magnetic core (red), Ampere’s Law encloses two types of current, “conduction current” in the coils and“magnetisation current” on the surface of material:
r > 1: M and I in same direction (paramagnetic)r < 1: M and I in opposite directions (diamagnetic)
Substitute for M : (see later)
InB ovac
MnB o I
I
L
vacrMo
Moenclo
BnB
LnLBLB.d
I
II
Non-uniform magnetisationA rectangular slab of material in which M is directed along y-axis only but increases in magnitude along the x-axis only
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
z
x
My
zMj
I1 I2 I3
I1-I2 I2-I3
Neighbouring elemental boxes
Consider 3 identical element boxes, centres separated by dx
If the circulating current on the central box is
Then on the left and right boxes, respectively, it is
dyMy
dx dx
dy dxx
MManddydx
x
MM y
yy
y
Upward and circulating currents
The “mag current” is the difference in neighbouring circulating currents, where the half takes care of the fact thateach box is used twice! This simplifies to
dyMdxx
MMdx
x
MMM2
1y
yy
yyy
x
Mjdxdyjdxdy
x
Mdydx
x
M22
1 yMM
yy
zz
Non-uniform magnetisationA rectangular slab of material in which M is directed along -x-axis only but increases in magnitude along the y-axis only
z
x
My
I1 I2 I3
I1-I2 I2-I3z
y
-Mx
xx
Mj yMz
y
Mj xMz
y
M
x
Mj xyMz
Total magnetisation current || z
Similar analysis for x, y components yields MMj
Magnetic Field Intensity HRecall Ampere’s Law
Recognise two types of current, free and bound
jBB oenclo or.d I
f
oo
oo
ffo
foMfoo
Magnetic Electric
orwhere
jH.D
MB
H PED
MHBMB
H
jHjMB
MjjjjB
f
Ampere’s Law for HOften more useful to apply Ampere’s Law for H than for B
Bound current in magnetic moments of atomsFree current in conduction currents in external circuits ormetallic magnetic media
freeencl
ff
.d hence
.d.d
I
H
SjSHjHss
bfo
ffovac
n'MMn B core
nH c.f.nB vacuum
I I
I I
If
L
If
L
Ib
Magnetic Susceptibility B
• Two definitions of magnetic susceptibility
• First M = BB/o is analogous to P = oEE B, field due to all currents, E, field due to all charges
B r Au -3.6.10-5 0.99996Quartz -6.2.10-5 0.99994O2 STP +1.9.10-6 1.000002
In this definition the diamagnetic susceptibility is negative andthe relative permeability is less than unity
Bror
B
o
Booo
B
1
1
1 or
HHB
BHMHB BM
c.f. D = roE
Magnetic Susceptibility M
• Second definition not analogous to P = o E E
When is much less than unity (all except ferromagnets) thetwo definitions are roughly equivalent
MrorMo
MooM
11or
HHB
HHMHB HM
11
1B(T)
H Am-1
0
1.5
-1.5
-500 +500
Para-, diamagnets
Ferromagnet ~ 150-5000 for FeHysteresis and energy dissipation
Boundary conditions on B, H
21
2211
BB
0S cosBS cosB
0.d0.
SBB
1
2
B1
B22
1
S
||2||1
freeencl2211
freeencl
HH
0L sinHL sinH
.d
I
I
H
For LIH magnetic media B = oH(diamagnets, paramagnets, not ferromagnets for which B = B(H))
222
A
B
22
111
B
A
11
sin H .d
sin H- .d
H
H
1
2 H2
H1
2
1dℓ1
dℓ2
C ABI enclfree
Boundary conditions on B, H
||||
2
1
2
1
21
21
21
21
r
r
2
1
r
r
2
1
22or
22
11or
11
22or11or
2211
2211
tan
tanc.f.
tan
tan
cosH
sinH
cosH
sinH
cosHcosH
cosBcosBBB
sinHsinH
HH
Faraday’s Law
field varying-time tic electrosta t
tx
t
x
tx
.dt
dx
.dLaw sFaraday'.dtdt
d
fieldtic electrosta 0
AE
AABE
SB
SEE.d
SBSB
E.d
EE.d
.
SS
SS
S
BE
dℓ
SB.dd
Faraday’s Law
I
B(r)
To establish steady current, cell must do work against Ohmic losses and to create magnetic field
Energy density in magnetic fields
space allo
space all
space all
dv t
.
dv t
.dt
dW
fieldmagnetic establish to workt
heating Joule
tdv .
tdt
dW power Total
dv .da d . dv to supplied Power
d difference Potential
AB x
Aj
Aj.
Ejj.E
AEj
AE
jj
1
.
dℓ
daj
Energy density in magnetic fields
media dielectric ormagnetic in dv .Uc.f. dv .U 21
E21
M EDHB
vacuum in dv2
c.f. dv2
1 W
dvdt
d
2
1
dt
1dv
t
1
dvt
-t
1
xb)a.(-xa)b.(.(axb)dv t
. dt
dW
space all
o
space allo
space allo
ospace allo
space allo
space allo
E.E B.B
B.B
SB x AB
B.
B x AA
x B.
AB x
S
.
.
1
Time variationCombining 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 Inductionemf induced in a circuit equals the rate of change of magneticflux through the circuit
t
t
t
t
BE
SB
SE
SBE
ESB
Theorem Stokes'dd
.d.d
.d .d
..
EE
E E
by simply blerepresenti longer no so x
whichfor fieldstic electrostaonly general, in .d
0
0C
dS
B
dℓ
Displacement currentAmpere’s Law
currentssteady -non for t
1
1
0.
0..
j
Bj
BjjB
o
oo
Problem!
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
Continuity equation
Extending Ampere’s Lawadd term to LHS such that taking Div makes LHS also identically equal to zero:
The extra term is in the bracket
extended Ampere’s Law
0..
?j
B?jo
1
jE
E
EE
...
..
ttt oo
oo
t
t
ooo
oo
EjB
BE
j
1
Types of current j
• Polarisation current density from oscillation of charges inelectric dipoles• Magnetisation current density variation in magnitude ofmagnetic dipoles in space/time
PMf jjjj
tP
jP
tooo E
jB
M = sin(ay) k
k
i
j
jM = curl M = a cos(ay) i
Total current
MjM x