02/18/2015PHY 712 Spring 2015 -- Lecture 151 PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 15: Finish reading Chapter 6 1.Some details

PHY 712 Spring 2015 -- Lecture 15 102/18/2015

PHY 712 Electrodynamics9-9:50 AM MWF Olin 103

Plan for Lecture 15:

Finish reading Chapter 61. Some details of Liénard-Wiechert results

2. Energy density and flux associated with electromagnetic fields

3. Time harmonic fields





Solution of Maxwell’s equations in the Lorentz gauge -- continued

Liénard-Wiechert potentials and fields --Determination of the scalar and vector potentials for a moving point particle (also see Landau and Lifshitz The Classical Theory of Fields, Chapter 8.)

Consider the fields produced by the following source: a point charge q moving on a trajectory Rq(t).

3(Charge density: , ) ( ( )) qt q t r r R

3 ( )Current density: ( , ) ( ) ( ( )), where ( ) .q

q q q

d tt q t t t

dt J r r

RR R R

q

Rq(t)



3

0

1 ( ,' ')( , ) ' ' ( | | / )

4 |'

|'

']t

td r dt t t c

r

r r rr rò

32

0

1 ( ', t')( , ) ' ' ' ( | ' | / ) .

4 | ' |t d r dt t t c

c

J r

A r r rr rò

We performing the integrations over first d3r’ and then dt’ making use of the fact that for any function of t’,

( )( ) ( | ( ) | / ) ,

( ) ( ( ))1

'

(

'

| ) |

' ' rq

q r q r

q r

f td f t c

t t

c t

t t t t

r RR r R

r R

where the ``retarded time'' is defined to be

| ( ) |.q r

r

tt t

c

Rr



Resulting scalar and vector potentials:

0

1( , ) ,

4

qt

Rc

v Rr

ò

20

( ,

) ,4

qt

c Rc

vr

v RA

ò

Notation: ( )q rt r RR

( ),q rtv R | ( ) |

.q rr

tt t

c

Rr



In order to find the electric and magnetic fields, we need to evaluate ( , )

( , ) ( , )t

t tt

r

r rA

E

( , ) ( , )t tA rB r

The trick of evaluating these derivatives is that the retarded time tr depends on position r and on itself. We can show the following results using the shorthand notation:

andrt

c Rc

Rv R

.rt R

tR

c

v R



2

3 2 20

1( ,

v ) 1 ,

4

q vt R

c c c cR

c

v R Rr R R

v

v

R

ò

2

3 2 2 20

( , ) 1.

4

t q R v RR

t c c Rc c c cR

c

R

A v vr v v R v R

v R

ò

2

3 2 20

1( , ) 1 .

4

q R v Rt

c c c cR

c

v v vr R R R

vE

R

ò

2

3 22 2 20

/ ( , )(

, ) 1

4

q v c tt

c c c cRR R

c c

R v R R R rr

v

v v EB

R v R

ò


Energy analysis of electromagnetic fields and sources

trd

dt

dW

rddt

dE

dt

dW

t)(

mech

DHE

JE

rJ

:producesit fields of in termscurrent source Expressing

:field neticelectromagby , sourceon done work of Rate

3

3

0 :monopoles magnetic No

0 :law sFaraday'

:law sMaxwell'-Ampere

:law sCoulomb'

B

BE

JD

H

D

t

t free

free

Maxwell’s equations


Energy analysis of electromagnetic fields and sources -- continued

JES

BHDE

HES

BH

DEHE

DHEJE

t

u

u

ttrd

trdrd

dt

dW

densityenergy 2

1

vector"Poynting" Let

3

33


Energy analysis of electromagnetic fields and sources -- continued

trdtrddt

dE

dt

dE

t

u

turdE

u

rddt

dE

fieldmech

field

mech

,ˆ ,

:analysisenergy previous theFrom

: vectorPoynting

,

2

1 :densityenergy neticElectromag

23

3

3

rSrrS

JES

HES

r

BHDE

JE


Momentum analysis of electromagnetic fields and sources

BBEE

PP

BEP

BvE

BJEP

220

3

30

3

2

1

: tensorstress Maxwell

:fields for vacuum Expression

:force Lorentzth analogy wiby Follows

cBBcEET

r

Tr d

dt

d

dt

d

rd

qF

rddt

d

ijjijiij

j j

ij

i

fieldmech

field

mech


Comment on treatment of time-harmonic fields

ti

ti

e,t)(dt) ,(

e),(d ,t) (

~

~

2

1

:domain in timeon ansformatiFourier tr

rErE

rErE

),(),(,t)( rErErE *~~ real is that Note

tititi )e,()e,()e,(,t)( rErErErE *~~

2

1~

:atmentcommon tre the tolead

principle,ion superposit theofnotion theand relations These


Comment on treatment of time-harmonic fields -- continued


2

1~

:fields harmonic for time Equations

0~


0~~

0 :law sFaraday'

~~~ :law sMaxwell'-Ampere

~~ :law sCoulomb'

BB

BEB

E

JDHJD

H

DD

it

it freefree

freefree

Note -- in all of these, the real part is taken at the end of the calculation.

Equations in time domain in frequency domain


Comment on treatment of time-harmonic fields -- continued


2

1~

:fields harmonic for time Equations

),(),(,t

)e,(),()e,(),(

),(),(),(),(

)e,()e,()e,()e,(,t

,t)(,t)(,t

t

titi

titititi

rHrErS

rHrErHrE

rHrErHrE

rHrHrErErS

rHrErS

*

avg

2**2

**

**

~~

2

1

~~~~

4

1

~~~~

4

1

~~~~

4

1

: vectorPoynting



0 :law sFaraday'

:law sMaxwell'-Ampere

:law sCoulomb'

B

BE

JD

H

D

t

t free

free


Summary and review



0 :law sFaraday'

0 :law sMaxwell'-Ampere

0 :law sCoulomb'

:sources no and

; -- media isotropiclinear For

B

BE

EB

E

HBED

t

t



0 2

22

2

t

tt

BB

EB

EB

Analysis of Maxwell’s equations without sources -- continued:


0 :law sFaraday'

0 :law sMaxwell'-Ampere

0 :law sCoulomb'

B

BE

EB

E

t

t

0 2

22

2

t

tt

EE

BE

BE


2

20022

2

2

22

2

2

22

where

01

01

n

ccv

tv

tv

EE

BB

Analysis of Maxwell’s equations without sources -- continued: Both E and B fields are solutions to a wave equation:

tiitii etet rkrk ErEBrB 00 , ,

:equation wave tosolutions wavePlane



00

222

00

where

, ,

:equation wave tosolutions wavePlane

nc

n

v

etet tiitii

k

ErEBrB rkrk

Note: , e m, n, k can all be complex; for the moment we will assume that they are all real (no dissipation).

0ˆ and 0ˆ :note also

ˆ

0 :law sFaraday' from

t;independennot are and that Note

00

000

00

BkEk

EkEkB

BE

BE

c

n

t



0ˆ and where

, ˆ

,

: wavesneticelectromag plane ofSummary

000

222

00

Ekk

ErEEk

rB rkrk

nc

n

v

etec

nt tiitii

kEkE

EkES rkrk

ˆ2

1ˆ2

ˆ1

2

1

: wavesneticelectromag planefor vector Poynting

2

0

2

0

*

00

c

n

ec

ne tiitii

avg



0ˆ and where

, ˆ

,

: wavesneticelectromag plane ofSummary

000

222

00

Ekk

ErEEk

rB rkrk

nc

n

v

etec

nt tiitii

2

0

*

00

*

00

2

1

ˆˆ1

4

1

4

1

: wavesneticelectromag planefor density Energy

E

EkEk

EE

rkrk

rkrk

tiitii

tiitii

avg

ec

ne

c

n

eeu

Transverse Electric and Magnetic (TEM) waves


More general result (for homework)

Documents

02/18/2015PHY 712 Spring 2015 -- Lecture 151 PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 15: Finish reading Chapter 6 1.Some details