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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 22: Comments on Mid-Term Exam Start reading Chap. 11 Equations in cgs (Gaussian) units Special theory of relativity Lorentz transformation relations. Comments on Mid-Term Exam. F. r. E. - PowerPoint PPT Presentation
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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
PHY 712 Spring 2014 -- Lecture 22 203/24/2014
PHY 712 Spring 2014 -- Lecture 22 303/24/2014
Comments on Mid-Term Exam
PHY 712 Spring 2014 -- Lecture 22 403/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 503/24/2014
PHY 712 Spring 2014 -- Lecture 22 603/24/2014
Recall from Lecture 4: Green’s function for 3-d Poisson equation in Cartesian coordinates:
PHY 712 Spring 2014 -- Lecture 22 703/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 803/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 903/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 1003/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 1103/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 1203/24/2014
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
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
PHY 712 Spring 2014 -- Lecture 22 1303/24/2014
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
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
PHY 712 Spring 2014 -- Lecture 22 1403/24/2014
/ /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
PHY 712 Spring 2014 -- Lecture 22 1503/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 1603/24/2014
PHY 712 Spring 2014 -- Lecture 22 1703/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 1803/24/2014
PHY 712 Spring 2014 -- Lecture 22 1903/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 2003/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 2103/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 2203/24/2014
Examples of other 4-vectors applicable to the Lorentz transformation:
2222221
1
1
'' :Note
'
'
'
'
'
'
'
'
'''' :Note
/
'
'
'
/'
'
'
'
/'/
1000
0100
00
00
1000
0100
00
00
:ˆ with frame moving For the
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
PHY 712 Spring 2014 -- Lecture 22 2303/24/2014
The Doppler Effect
'''' :Note
/
'
'
'
/'
'
'
'
/'/
1000
0100
00
00
1000
0100
00
00
:ˆ with frame moving For the
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
PHY 712 Spring 2014 -- Lecture 22 2403/24/2014
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
PHY 712 Spring 2014 -- Lecture 22 2503/24/2014
Electromagnetic Doppler Effect (q=0)
c
vv detectorsource 1
1'
Sound Doppler Effect (q=0)
/1
/1'
source
detector
s
s
cv
cv
PHY 712 Spring 2014 -- Lecture 22 2603/24/2014
Lorentz transformation of the velocity
'
'
''
''
frame Moving frame Stationary
zz
yy
ctxx
xctct
For an infinitesimal increment:
dz' dz
dy' dy
βcdt'dx' γ dx
dxcdtcdt
''
frame Moving frame Stationary
PHY 712 Spring 2014 -- Lecture 22 2703/24/2014
Lorentz transformation of the velocity -- continued
dz' dz
dy' dy
βcdt'dx' γ dx
dxcdtcdt
''
frame Moving frame Stationary
'
''
'
''
'
''
: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
'
/''
PHY 712 Spring 2014 -- Lecture 22 2803/24/2014
ux/c
uy/c
Example of velocity variation with :b