PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 22:

PHY 712 Spring 2014 -- Lecture 22 103/24/2014

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 22:

• Comments on Mid-Term Exam

• Start reading Chap. 11

A. Equations in cgs (Gaussian) units

B. Special theory of relativity

C. Lorentz transformation relations



Comments on Mid-Term Exam


0 0

0 0

2

20

0 for

1( ) sin sin 1 for

Green's function solution:

2

a

a

a

a

x a

x x xx x dx a x a

a a a

0

for x a

0

0

2

0 for

( ) cos 1 for

0 for

x a

xE x a x a

a

x

a

a

2 320 0

20

3( ) ( )

2 2

a a

a a

W w x dxa

dx E x

E

F

r



Recall from Lecture 4: Green’s function for 3-d Poisson equation in Cartesian coordinates:


2 22

02 2

( ) ( )( ) ( ) /

( ) ( ) ( ) ( )( ,

In 2-dimen

,

sions:

, 4

) l l m m

lm l m

x y

u x u x v y v yG x x y y

r rr r ò

2

2

In our case:

1(0) (4 ) ( ) sin

2 4 4

2(0) (3 ) ( ) sin

3 3

3m

l l l l

m m m

l x lu u h u x

h h h

m x mv v h v x

h h h

4 3

0 0 0

20

20

1( ) ' ' , ', , ' ( , )

4

144 = sin sin

25 4 3

h h

dx dy G x x y y x y

h x y

h h

r


654

321

2 2

20

20

20

1 1 1 20

20

220

144( ) sin sin

25 4 3

, 3.527265230

4.98830632, 7

h x y

h h

hx y

hx y

r


654

321

2 42

0 0

1 1 1 3

2 5

4 6

1 1 3( , ) ( ,

Finite difference equation for 2-dim cartesian case:

and

) ( , )5 20 10 40

In our case: ( ,

)

A B

h hx y S S x y x y

x y

2 42

1 1 1 10 0

2 4 5 2 1 2

2 1 3 5 4 6 2 1 2

2 42

01 2

02

1 1 1 1 3

5 20 5 20 10 40

U

1 1 1 1 32 2

5 20 5 2

nique equations:

0 10 40

h h

h h

2 4 2 2

2 22

0 0 0

2 22 4 2 42 20 0

2 20 0 0

1 20

10

20

3 3 1 25, , ,

10 40 10 40 144

3 3

10 401.554261748 2.19805

1 40

08 45

0

h h hx y x y x y

h hh h h h


654

321

2 42

0 0

1 1 1 3

2 5

4 6

1 1 3( , ) ( ,

Finite difference equation for 2-dim cartesian case:

and

) ( , )5 20 10 40

In our case: ( ,

)

A B

h hx y S S x y x y

x y

21 0

20

21 0

2

2

2 0

Linear equations:

4 / 5 1/ 4 1.554261748

1/ 2 4 / 5 2.198058045

Finite difference: Analytic:

3.481406

4.923451

h

h

21 0

202

3.527265

4.988306

h


654

321

,

,

0

, where

Finite element equation for 2-dim cartesian case:

( , ) ( , ) and

( , ) ( , ) /

kl ij ij klij

kl ij kl ij

kl kl

M G

M dx d y x y x y

G dx d y x y x y

,

8for and

31

for 1 and/or 13

0 otherwise

kl ij

k i l j

M k i l j

0 0

20

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

( , ) /

kl kl k l kl

k l

G dx d y x y x y x y dx d y x y

h x y


654

321

,

,

0

, where


( , ) ( , ) and

( , ) ( , ) /

kl ij ij klij

kl ij kl ij

kl kl

M G

M dx d y x y x y

G dx d y x y x y

2 4 5 2 1

2 1 4 5 6 3 2 1

2

1 1 1

02 2

0

2

For our case:

8 1 8 1

3 3 3 3

8 1 8 1

3 3 3 3

2

4

h

h

21 0

22 0

21 0

22 0

7 / 3 2 / 3 6.043873688

4 / 3 7 / 3 8.547328140

4.346471 (rather poor result)

6.146838

h

h


654

321

,

,

0

, where


( , ) ( , ) and

( , ) ( , ) /

kl ij ij klij

kl ij kl ij

kl kl

M G

M dx d y x y x y

G dx d y x y x y

2 20 0

1 2 20 0

2

Better treatment of :

5.233806864 7.401720668

klG

G Gh h

21 0

22 0

21 0

22 0

7 / 3 2 / 3 5.233806864

4 / 3 7 / 3 7.401720668

3.763909 (pretty good result)

5.322971

h

h


/ /0

2 1

0 0 10 1

1 00

1 1 1' ' ' ' ' '

2 1

For o

44

ur case

( ) ( ) 3

r l l llm lm lml r

m

r a r b

l

Y θ,φ r dr r r r dr rl r

rr Q r Qe e

b

r


2 11 0

0

3

3 4 2 2/ /0 1

2 20

1 1 1' ' ' ' ' '

2 1

Evaluating the expressions with the help of Maple:

2 2411 1 cos 1 1

2 82

r l l llm lm lml

m

r

rl

r a b

Y θ,φ r dr r r

Q r r re

r dr rl r

a Q bre

r br ba b

r

r

2 2 30 1

0

Also with the help of Maple, can evaluate the

)

limit 0:

1( c )os (

r

a O rQ Q br O r

r

3

0 12

30

4

4

0

1

For the limit r :

2 241cos

Identify monopole charge: 8

Identify dipole: 96

Q

a Q

a Q b

r r

p Q

q

b

r



0 0

320 0

3 30 0

) :

for 02

) 3 2 for 6

fo

Find solution fo

6

r (

(

r

f

J

a

b a a

Jbf

b a

a b

Jb

0 0

0 0

( )1ˆ: =

ˆ for 0

ˆ) for

0

Find

(

for

d f

d

J b a a

J b a b

b

B B A z

z

B z



Notions of special relativity

The basic laws of physics are the same in all frames of reference (at rest or moving at constant velocity).

The speed of light in vacuum c is the same in all frames of reference.

x

y y’

x’

v

x’y’

y

x


x

y y’

x’

v

x’y’

y

x

Lorentz transformations

21

1

:notation Convenient

c

v

'

'

''

''

frame Moving frame Stationary

zz

yy

ctxx

xctct


Lorentz transformations -- continued

2222222222

1

1

''''

:Notice

'

'

'

'

'

'

'

'

1000

0100

00

00

1000

0100

00

00

:ˆ with frame moving For the

zyxtczyxtc

z

y

x

ct

z

y

x

ct

z

y

x

ct

z

y

x

ct

v

-

-

LL

LL

xv


Examples of other 4-vectors applicable to the Lorentz transformation:

2222221

1

1

'' :Note

'

'

'

'

'

'

'

'

'''' :Note

/

'

'

'

/'

'

'

'

/'/

1000

0100

00

00

1000

0100

00

00


cpEcpE

cp

cp

cp

E

cp

cp

cp

E

cp

cp

cp

E

cp

cp

cp

E

tωωt

k

k

k

c

k

k

k

c

k

k

k

c

k

k

k

c

v

z

y

x-

z

y

x

z

y

x

z

y

x

z

y

x-

z

y

x

z

y

x

z

y

x

-

LL

LL

LL

rkrk

xv


The Doppler Effect

'''' :Note

/

'

'

'

/'

'

'

'

/'/

1000

0100

00

00

1000

0100

00

00


1

1

rkrk

xv

tωωt

k

k

k

c

k

k

k

c

k

k

k

c

k

k

k

c

v

z

y

x-

z

y

x

z

y

x

z

y

x

-

LL

LL

' '

/' //'

zzyy

xxx

kkkk

ckkkcc


The Doppler Effect -- continued

' '

/' //'

zzyy

xxx

kkkk

ckkkcc

x

y

x’

v

y’

k’

k

βkβk

βkβk

kβ

ˆˆ'

/cos'cos'/ˆˆ'

cos/ //'

:generally More

ckkc

βkccc

cos

sin'tan

)( :0For

1

1' 1'

)( :0For

ω/ck

ω/ck


Electromagnetic Doppler Effect (q=0)

c

vv detectorsource 1

1'

Sound Doppler Effect (q=0)

/1

/1'

source

detector

s

s

cv

cv


Lorentz transformation of the velocity

'

'

''

''


zz

yy

ctxx

xctct

For an infinitesimal increment:

dz' dz

dy' dy

βcdt'dx' γ dx

dxcdtcdt

''



Lorentz transformation of the velocity -- continued

dz' dz

dy' dy

βcdt'dx' γ dx

dxcdtcdt

''


'

''

'

''

'

''

:Define

dt

dzu

dt

dyu

dt

dxu

dt

dzu

dt

dyu

dt

dxu

zyx

zyx

yx

y

xx

x

ucvu

u

cdxdt

dy

dt

dy

ucvu

vu

cdxdt

βcdt'dx'γ

dt

dx

2

2

/'1

'

/''

'

/'1

'

/''


ux/c

uy/c

Example of velocity variation with :b

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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 22:

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