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