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 electrosta for 0only 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..

0..

jj

?j

B?j

or

1

o

jE

E

EE

...

..

ttt oo

oo

t

t

ooo

oo

EjB

BE

j

1

IˆI

I

SS

SV V

SnSj

EE

Ej

Sjj

D

D

DD

.dA

.d

At

Q

A

1

tt

A

Q density charge ousinstantane

t

.ddvt

dv .

oo

o

Illustration of displacement current

C

I

R

S

- +

oE

- +

o E/t

Discharging capacitor

Displacement current magnitude

Suppose E varies harmonically in time

EE

EE

t

kx)tsin(o

197

12o

oo

17

1.47.106.10

8.854.10

tyconductivi is here σ NBΩm6.10 Cu

EE

Ej

t

~ 10.19 rads-1 for oE/t to be comparable to E

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

1st form of Maxwell’s Equations

all field terms on LHS and all source terms on RHSThe sources ( and j) are multiple (free, bound, mag, pol)special status of free source suggests 2nd Form

)(

0.

)( .

PMf

bf

t(4)

0t

(3)

(2)

(1)

jjjjjE

B

BE

B

E

ooo

o

Extending Ampere’s Law to H

D/t is displacement current postulated by Maxwell (1862)to exist in the gap of a capacitor being charged

In vacuum D = oE and displacement current exists throughoutspace

tt

tt

t

1t

ff

f

PMf

DjHPEjM

B

EPMj

EjjjB

EjB

oo

o

oo

ooo

2nd form of Maxwell’s EquationsApplies only to well behaved LIH media

Focus on sources means equations (2) and (3) unchanged!Recall Gauss’ Law for D

In this version of (1), f and o Recall H version of (4)

In this version of (4), j jf , also o and o

HHBEED oror and

f

ff EED ...

ff

ff

jE

BjD

H

jD

HD

jH

tt

tt

2nd and 3rd forms

LHS: 2nd form, free sources only, other sources hidden inpermittivity and permeability constantsRHS: 3rd form (Minkowsky) free sources only, mixed fields, noconstants

ff

ff

tt(4)

tt(3)

(2)

(1)

jD

HjE

B

BE

BE

BB

DE

0 0

0. 0.

. .

Electromagnetic Wave Equation

jE

BB

E

BE

ooo

o

t(4)

t(3)

(2)(1)

0

0. .

tt(1)

ttidentity

tt(4)

t(3)

jEE

EjEE

EjE

BE

oo

oo

ooo

ooo

2

22

2 0.

0

0

Firstform

Electromagnetic Waves in Vacuum

wavetransverse implies chosen form for z

0. then z If 0. then z|| If

ck

vk

kt

expexpt

expkexpz

exp

vector constant a isexpRe

o

oo

p2

22

2

22

2

22

22

oo

kztikzti

kztikztikzti

kzti

E

EEEE

EEE

E

EEE

oo

oo

oooo

1

00

2

22

speed of lightin vacuum

= ck = 2c/

k

Dispersion relation

Relationship between E and B

j j B

j B

E

iE

ˆˆ

ˆ

ˆ

kztikzti

kzti

kzti

expc

E expE

k

expE ikt

-x

expERe

oo

o

o

E || i

B || j

k || k

Plane waves in a nutshell

2

k i.e.

r 2r k

r k

expexp )i(i(kz)

k.r

0k.r)rk.(rk.r ||

k.r

rr

r||

k

Consecutive wave fronts

EM Waves in insulating LIH medium

Less than speed of light in vacuum complex in general, real (as has been assumed) if h<<Eg

rr

rr

o

tt

pp

22

2

2

2

ff

ff

v

cnindex Refractive

c

kv

k

k

t

exp

form (2nd

kzti

1

0

0

0

)0

2

2

2

22

EE

EE

EE

j

jEE

= ck/n = 2c/n

k

Dispersion relation

Slope=±c/n

Bound Charges

2222o

2222o

22oo

o2o

t)sin(t)cos(

m

eE-x

t)cos(eExmxmxm

1 2 3 4 5 6

-1

-0.5

0.5

1

xB

vB

t

o

-/2

-

+/2

0xB

vB

Phase relative to driving field vs frequency

Bound charge displacement xB

Or velocity vB versus time

Free Charges

tyconductivi Drude m

ec.f.

m

e

Et)cos(Em

et)cos(

m

Eexj

t)cos(

m

eE-)(x

t)sin(

m

eE-)x(

t)cos(eExmxm

22

o

2o

2

o

o

o

NN

NN

0

0

For a free charge, spring constant and o tend to zero

Dielectric susceptibility

litysusceptibi Dielectric V

ex-

moment Dipole-ex Ep

onPolarisati V

pEP

oE

o

Eo

2222o

2222o

22o

o

2

E

t)sin(t)cos(

mV

e

EM waves in conducting LIH medium

0

0

2

2

2

complex iskk

exp

t

t

and 0

form) (2nd tt

2

o

2

2

f

ff

ff2

2

kzti

i

t

t

EE

EEE

Ej

conduction Ohmic Ej

jEE

EM waves in conducting LIH medium

50Hz at Cu in 1cm2

depth skin 2

exp expexp

i)(12

1ik k

t

ztizkztioo

2

EEE

EE

EEE

i

t

ti

0

0

2

2

EM wave is attenuated within ~ skin depth in conducting mediaNB Insulating materials become ‘conducting’ when radiation frequency tuned above Eg

Energy in Electromagnetic Waves

Energy density in matter for static fields

2

1kz)t(cos

4

1

dv

Ud

kz)t(cos2

12

1

dv

dU

expexp

1 vacuum In2

1

dv

dU

2oooo

2oooo

kz)ti(o

kz)ti(o

.HH.EE

.HH.EE

H.HE.E

HH EE

HB ED

B.HD.E

oo

oo

oo

rroo

Average obtained over one cycle of light wave

Energy in Electromagnetic Waves

Average energy over one cycle of light wave

oooo4

1

dv

Ud.HH.EE oo

Distance travelled by light over one cycle = 2c/= cAverage energy in volume ab c

a

b

c

Energy in Electromagnetic Waves

)( timeperiodic per (ab) area unit crossingEnergy HE2

1

ab

U

HE2c

1

2

1

2

1HE

2

1

abc

U

B

cB

H

E

E

H

2

1

H

E

2

1HE

2

1

abc

U

c

EB c

BH

abc4

1U

oo

oooo

o

o

o

o

o

o

o

ooo

oo

oo

oooo

oo

oo

o

o

o

o

oo

oo

oo

ooo

oo

1

.HH.EE

Poynting Vector

N = E x H is the Poynting vector

Equal to the instantaneous energy flow associated with an EM wave

In vacuum N || wave vector k

Example If the E amplitude of a plane wave is 0.1 Vm-1

Energy crossing unit area per second is

25

o

o2ooo Wm1.3.10E

2

1HE

2

1