nassp self-study review 0f electrodynamics created by dr g b tupper [email protected]

NASSP Self-studyReview 0f Electrodynamics

Created by Dr G B [email protected]

The following is intended to provide a review of classical electrodynamics at the 2nd and 3rd

year physics level, i.e. up to chapter 9 of Griffiths book, preparatory to Honours.

You will notice break points with questions. Try your best to answer them before

proceeding on – it is an important part of the process!

Basics

• Maxwell’s equations:

• Lorentz force:

0

1

0

t

0 0 0J

t

F q

Basics

• Mathematical tools:– Gauss’ Theorem

– Stokes’ Theorem

– Gradient Theorem

– Green’s function

d da

da dl

b

af dl f b f a

2 1

4r r

r r

Basics

• Mathematical tools:– Second derivatives

– Product rules

• Potentials

0f A 2A A A

,Vt

Questions

• Where is “charge conservation”?

• Where is Coulomb’s “law”?

• Where is Biot-Savart “law”?

• What about Ohm’s “law”?

Work done on charge

• Power (Lorentz)

• Now

• So

• Use Ampere-Maxwell

F v qE v

,q d v J

dWE J d

dt

0

0

E BdW EE d

dt t

Conservation of energy

• Identity

• Use Faraday

• So

E E E

BE E

t

0

0 0

1E BdW B EB E d

dt t t

Poynting’s Theorem

• Define– Mechanical energy density

– Electromagnetic energy density

– Poynting vector

EM fields carry energy

mechudWd

dt t

2 20

0

1

2 2emu E B

0

E BS

0mech emu u St

Questions

• Problem: an infinite line charge along z-axis moves with velocity :

Determine

v

, , andemE B u S

v

Waves in vacuum

• Maxwell’s equations:

• Curl of Faraday:

0

0

0t

0 0 t

2 0Et t

Waves in vacuum

• Use Gauss & Ampere-Maxwell; wave equation

• Speed of light

• Monochromatic plane-wave solutions

22

0 0 20E

t

8

0 0

1 3.00 10 msc

0, cosE x t E k x t ck

constant

0ˆ0 0k E

Transverse

Questions

• What is the meaning of the wave-number ?

• What is the meaning of angular frequency ?

• What is the associated magnetic field?

k

2k ˆ :k directionof propagation

2T

Wavelength

Period

0 0

1 ˆ, cos cosk

Faraday B x t E k x t k E k x tc

0ˆ :E directionof polarization

Monochromatic plane-wave


• Energy density

• Poynting vector

• Momentum density

2 2 2 2 2 2 20 00 0

0

1cos

2 2 2emu E B E c B E k x t

2 2 20

0 0 0

ˆ ˆˆcos em

E B k kS E E k x t kcu

c c

2 20 0 0 0

ˆ ˆcosem em

k kS E k x t u

c c


• Time average

• Intensity

2 2 12

0

cos cosT

k x t k x t dt

2002emu E

ˆ

emS kc u ˆ

em em

ku

c

2002

I S c E

Questions

0, cos ,sin ,0 cosE x t E kz t

90 o

102

, cos , sin ,0E x t E kz t kz t

A monochromatic plane-polarized wave propagating in the z-direction has Cartesian components in phase:

.

In contrast, a circularly-polarized wave propagating in the z-direction has Cartesian components out of phase:

Describe in words what such a circularly-polarized wave looks like. One of the two casess “left-handed”, and the other is “right handed” – which is which?

i

Determine the corresponding magnetic field.

Determine the instantaneous energy-density and Poynting vector.

Electrostatics in matter

• Electric field polarizes matter

– Potential in dipole approximation

– Bound charge density

Dipole moment p qd

3

0

1

4

r r P rV r d

r r

b P

Polarization: dipole moment per unit volume

Electrostatics in matter

• Rewrite Gauss’ law

– Displacement field

– For linear isotropic media

0 f b f P

Free charge density

0D

fD

0 , De

Dielectric constant 0r

Magnetostatics in matter

• Magnetic field magnetizes matter

– Vector potential

12magnetic moment m d r J r I a

034

M r r rA r d

r r

Magnetization: magnetic moment perunit volume

Magnetostatics in matter;Dipole moment proof

• Picture

• Dipole approximation

• For arbitrary constant vector

• Therefore

0J

0 034 4

J rA r d d r r J r

r r r

0 d V r r r J r d r r V J r V r r J r V r r r J r

=0

1 12 2d r r J r d r r J r r r J r d r J r r

m Q.E.D.

Magnetostatics in matter

– Bound current density

• Rewrite Ampere’s law

– Induction

– For linear isotropic media

bJ M

0f b f

BJ J J J M

0

1

fJ

1,m

Free current density

Electrodynamics in matter

• New feature

• Rewrite Ampere-Maxwell

b t P t

ppolarization current density J P t

0 0 00

f b p f

PJ J J J J M

t t t t


• Maxwell’s equations

• Constitutive relations

• Linear isotropic media

D

0

f

f

t

DJ

t

00

1D

0 , D

1,

e

m


• Boundary conditions

, 0

ˆ0 ,

above below f above below

above below above below f

D D B B

E E H H K n

n̂


• Energy density

• Poynting vector

2 20

0

1 1

2 2 2em emu E B u D H

0

E BS S E H

Electromagnetic waves in matter

• Assume electrical neutrality• In general there may be mobile charges; use

– Resistivity

0f

' : fOhm s law J E

Conductivity

1

Ionized plasma


• Maxwell’s equations

– Curl of Faraday

– For constant use Ampere-Maxwell

D 0

0

t

DE

t

2

2 0

Et t

D E Ht

,

2 2 0D

E H E Et t t


• Wave equation

• In an ideal insulator– Phase velocity– Plane wave solution

22

20E

t t

New

0

1v

0

0

, cos

ˆ 0

E x t E k x t

vk k E

0

0

, cos

1 ˆ cos

kB x t E k x t

k E k x tv

0 0 cc n

Index of refraction

Questions

1. What do you expect happens in real matter where the conductivity doesn’t vanish?

2. Which is more basic: wavelength or frequency?


• Take propagation along z-axis

– Complex ‘ansatz’

– Substitution gives

– Solution

2 2

2 20E

t t z

0, Re expE z t E ikz i t

2 20exp 0k i E ikz i t

2k i k i 2 2: ,2 2

r x r xNB x iy i r x y


Thus general solution is

0 0, cos , cos , 0x yE z t exp z E kz t E kz t

2

1 12

k

2

1 12

: cPhase velocity v k n 1:Skin depth

Transverse

PhaseAttenuation!

Frequency dependant: dispersion


• Limiting cases– High frequency

– Low frequency

2k

2

2k v

Good insulator

Good conductor

Note: at very high frequencies conductivity is frequency dependant


• Magnetic field – take for simplicity0 0, cosE z t E exp z kz t

0

0

ˆ, cos sin

ˆ cos sin

B z t dt z E exp z kz t k kz t

kz E exp z kz t kz t

2

0ˆ, 1 cos

tan

B z t z E exp z kz t

k


Good conductor

Questions

What one calls a “good conductor” or “good insulator” is actually frequency dependant; i.e. is or ?

Find the value of for pure water and for copper metal. Where does it lie in the electromagnetic spectrum in each case?

For each determine the high-frequency skin depth.

For each determine the skin depth of infrared radiation ( ).

In the case of copper, what is the phase velocity of infrared radiation?

In the case of copper, what is the ratio for infrared radiation?

1410 Hz

Frequency dependence

• Electric field polarizes matter

• Model

Dipole moment p qd

m x K x q E m x

“Restoring force” Driving force

Damping (radiation)

…dynamically


• Electromagnetic wave

– Rewrite in complex form

– Steady state solution

20 0 cosm x m x m x q E t

20 0

i tm x m x m x q E e

Natural frequency

0i tx t x e


– Substitution of steady state solution

– Dipole moment

2 20 0 0

i t i tm i m x e q E e

0

0 2 20

q Ex

m i m

2

0

2 20

i tq E ep q x

m i m


• Polarization

• Complex permittivity

2

02 20

i tqP E e

m i m

N

Number of atoms/molecules per unit volume

2

0 2 20

q

m i m

N


• Even for a “good insulator”

• Low density (gases)


0 2k k i

Ignore paramagnetism/diamagnetism

2 220

22 2 2 20

0

12

iqk

c m

N

Absorption coefficient


• Low density

2 220

22 2 2 20

0

12

c k qn

m

N

2 2

22 2 2 20

0

q

mc

N

Frequency dependent: dispersion


!v c

Anomalous dispersion

QuestionsIt was obtained that the frequency dependant index of refraction is given by:

2 220

22 2 2 20 0

12

c k Nqn

m

.

This implies that for 0 the phase velocity v k exceeds c ! That’s not the whole story,

however; to send information one cannot use infinite plane waves. Instead one must use a wave-packet – e.g.

0,

2

i k z td kE z t E f k e

.

Here we take k and assume for simplicity that the wave-number distribution is a

“gaussian”:

2 221 k kf k e

.

This distribution is peaked at k with width .

By approximating k k k k k , carry out the wave-number integral and then

take the real part to explicitly find ,E z t

. [Hint: I’ve normalized things such that

12

d kf k

; try shifting integration variable.] Sketch the electric field (take 0 ˆoE E x

).

Show that while the phase velocity is k

k

, the group velocity of the wave-packet is

gv k .

Electromagnetic waves in Plasma

• Electrons free to move; inertia keeps positive ions almost stationary

• Model

– Solution

m x eE m x

Electron mass No restoring force!

iev t E t

m i


• Current density

• Conductivity

eJ e v N

Electron number density

2

eie

m i

N

Drude model


• Electron collisions rare, so dissipation small

Recall for conductor

2

eie

m

NPurely imaginary!!


2k i


• As

– Above the plasma frequency: waves propagate with negligible loss

– Below the plasma frequency: no propagation, only exponential damping

0 0&

2 21pk

c 2

0

ep

e

m

N

Dispersion relation Plasma frequency

!gv c but v c

F&F 2013 L46

Plasma - Ionosphere

nassp self-study review 0f electrodynamics created by dr g b tupper [email protected]

Documents

energy slide

solution slide

simplicity slide

dispersion slide

frequency dependant

matter good conductor

unit volume slide

dielectric constant