Embed Size (px)
Citation preview
PHY 712 Spring 2014 -- Lecture 24 103/28/2014
PHY 712 Electrodynamics10-10:50 AM MWF Olin 107
Plan for Lecture 24:
Continue reading Chap. 11 – Theory of Special Relativity
A. Transformations of the Electromagnetic Fields
B. Connection to Liénard-Wiechert potentials for constant velocity sources
PHY 712 Spring 2014 -- Lecture 24 203/28/2014
PHY 712 Spring 2014 -- Lecture 24 303/28/2014
Velocity transformation detail from Lecture 26:
222
2
22222
222
2222222
222
2
/'1
/'1
/'/'/'1/1/'1
1
/'/'//'/'1/'1
11
:proof"" of Elements
cvu
cu
cucucucvcvu
cucucvcucvucvuc
u
xv
zyxv
xv
zyxvxv
xv
22
2
2
222
/1/'1
/'1
/1
1 that Note
./'1
'
/'1
'
/'1
' :Consider
cvcu
cvu
cuγ
cvu
uu
cvu
uu
cvu
vuu
xu
xv
zz
xv
yy
x
xx
PHY 712 Spring 2014 -- Lecture 24 403/28/2014
Special theory of relativity and Maxwell’s equations
AB
AE
JAA
A
J
1
:relations Field
4
1
4 1
:equations Potential
0 1
:condition gauge Lorentz
0 :equation Continuity
22
2
2
22
2
2
tc
ctc
tc
tc
t
PHY 712 Spring 2014 -- Lecture 24 503/28/2014
A
A
A
A
J
J
J
J
c
x
z
y
x
ct
z
y
x
z
y
x
:potentialsscalar andVector
:current and Charge
:position and Time
More 4-vectors:
PHY 712 Spring 2014 -- Lecture 24 603/28/2014
Lorentz transformations
1000
0100
00
00
vvv
vvv
v
L
'' :potentialscalar andVector
'' :current and Charge
'' :space and Time
AAA
JJJ
xxx
vv
vv
vv
LL
LL
LL
PHY 712 Spring 2014 -- Lecture 24 703/28/2014
4-vector relationships
,vector -4index lower -- signs of track Keeping
0,1,2,3for vector -4index upper : ,
0
0
3
2
1
0
A
A
AA
AA
A
A
A
A
z
y
x
ct
, ,
:operators Derivative
tctc
PHY 712 Spring 2014 -- Lecture 24 803/28/2014
Special theory of relativity and Maxwell’s equations
AB
AE
JAA
A
J
1
:relations Field
4
1
4 1
:equations Potential
0 1
:condition gauge Lorentz
0 :equation Continuity
22
2
2
22
2
2
tc
ctc
tc
tc
t
0 J
0 A
J
cA
4
PHY 712 Spring 2014 -- Lecture 24 903/28/2014
Electric and Magnetic field relationships
1
tc
A
E
0330
0220
0110
AAtc
A
zE
AAtc
A
yE
AAtc
A
xE
zz
yy
xx
AB
1221
3113
2332
AAy
A
x
AB
AAx
A
z
AB
AAz
A
y
AB
xyz
zxy
yzx
PHY 712 Spring 2014 -- Lecture 24 1003/28/2014
Field strength tensor AAF
0
0
0
0
xyz
xzy
yzx
zyx
BBE
BBE
BBE
EEE
F
Transformation of field strength tensor
0'''''
'0''''
''''0'
'''''0
1000
0100
00
00
'
xzvyvyvzv
xyvzvzvyv
zvyvyvzvx
yvzvzvyvx
vvv
vvv
vvv
BEBBE
BEBBE
EBEBE
BEBEE
F
FF
LLL
PHY 712 Spring 2014 -- Lecture 24 1103/28/2014
Inverse transformation of field strength tensor
0'''''
'0''''
''''0'
'''''0
'
1000
0100
00
00
' 111
xzvyvyvzv
xyvzvzvyv
zvyvyvzvx
yvzvzvyvx
vvv
vvv
vvv
BEBBE
BEBBE
EBEBE
BEBEE
F
FF
LLL
yvzvzyvzvz
zvyvyzvyvy
xxxx
EBBBEE
EBBBEE
BBEE
'' ''
'' ''
' '
:results ofSummary
PHY 712 Spring 2014 -- Lecture 24 1203/28/2014
Example:
y
z
x
y’
z’
x’q
b
yvzvzyvzvz
zvyvyzvyvy
xxxx
EBBBEE
EBBBEE
BBEE
'' ''
'' ''
' '
:frame stationaryin Fields
0'
'
ˆˆ'ˆ'ˆ'
' '
:frame movingin Fields
2/3223
B
yxyxE
bvt
bvtqyx
r
q
v
PHY 712 Spring 2014 -- Lecture 24 1303/28/2014
Example:
b
2/322
2/322
2/322
''
''
'
' '
:frame stationaryin Fields
bvt
bqEB
bvt
bqEE
bvt
vtqEE
vvyvvz
vyvy
xx
0'
'
ˆˆ'ˆ'ˆ'
' '
:frame movingin Fields
2/3223
B
yxyxE
bvt
bvtqyx
r
q
y
z
x
y’
z’
x’q
v
PHY 712 Spring 2014 -- Lecture 24 1403/28/2014
Example:
b
2/322
2/322
2/322
'
'
'
:frame stationaryin Fields
btv
bqEB
btv
bqEE
btv
tvqEE
v
vvyvvz
v
vyvy
v
vxx
0'
'
ˆˆ'ˆ'ˆ'
' '
:frame movingin Fields
2/3223
B
yxyxE
bvt
bvtqyx
r
q
y
z
x
y’
z’
x’q
v
Expression in terms of consistent coordinates
PHY 712 Spring 2014 -- Lecture 24 1503/28/2014
2/322 btv
bqE
v
vy
gv=10
gv=2
PHY 712 Spring 2014 -- Lecture 24 1603/28/2014
Examination of this system from the viewpoint of thethe Liènard-Wiechert potentials (temporarily keeping SI units
3
0
32
0
1 ( , ')( , ) ' ( | | / )
4 | |
1 ( , )( , ) ( | | / )
4 | |
Evaluating integral over ' :
( )' ( ') ' ( | ( ') | / )
''
'
( ) ( ( )
'
1
''
|
'
)r
qq r q r
tt d r dt t t c
t d r dt t cc
t
f tdt f t t t t
t
ct t
t
c
rr r r
r r
rr r r
r r
r RR r R
r
JA
ò
ò
,
( ) |q rtR
3 3 ( )( , ) ( ( )) ( , ) ( ) ( ( )) ( ) q
q q q q
d tt q t t q t t t
dt
Rr r R r R r R RJ
PHY 712 Spring 2014 -- Lecture 24 1703/28/2014
Examination of this system from the viewpoint of thethe Liènard-Wiechert potentials – continued (SI units)
0
20
1( , )
4
( , )4
( )where ( ) q r
q rr
qt
Rc
qt
c R
t
td
c
td
A
Rr
rv R
vr
v R
R vR
ò
ò
( , )( , ) ( , )
( , ) ( , )
tt t
tt t
AE
B A
rr r
r r
PHY 712 Spring 2014 -- Lecture 24 1803/28/2014
Examination of this system from the viewpoint of thethe Liènard-Wiechert potentials – continued (SI units)
2
3 2 20
2
3 22 2 20
1( , ) 1
4
/( , ) 1
4
( , )( , ) .
q R v Rt
c c c cR
c
q v ct
c c cR R
c c
tt
cR
v v vr R R R
v R
R v v R R vr
v R v R
R rr
E
B
EB
ò
ò
PHY 712 Spring 2014 -- Lecture 24 1903/28/2014
Examination of this system from the viewpoint of thethe Liènard-Wiechert potentials – continued (Gaussian units)
2
3 2 2
2
3 22 2
( , ) 1
/( , ) 1
( , )( , ) .
q R v Rt
c c c cR
c
q v ct
c c cR R
c c
tt
R
v v vr R R R
v R
R v v R R
E
B
E
vr
v R v R
BR r
r
PHY 712 Spring 2014 -- Lecture 24 2003/28/2014
Examination of this system from the viewpoint of thethe Liènard-Wiechert potentials – continued (Gaussian units)
2
3 2
2
3 2
( , ) 1
( , ) 1
q R v
tc c
Rc
q vt
c cR
c
vr R
v R
R vr
v
E
RB
3/22 2
3/22 2
ˆ( ,
This should be
, , ) (0, ,0
equivalent to the result given in Jack
, )( )
( , , , ) (0, ,0, )( )
son (11.152):
ˆ
ˆ
v t bx y z t b t q
b v t
bx y z t b t q
b v t
y
z
xE E
B B
2 2 2
For our example:
ˆ ˆ
ˆ ˆ
ˆ
( )q r r
r r
r
b
b R v t b
t vt
Rt
v
tc
t
v
R x r y
R y x
v x
