Transcript Feb25
Feb 25, 2011
RETARDED POTENTIALS
At point r =(x,y,z), integrate over charges at positions r’
r
rr rr d r
r
r r
j 3r
1
A( r ,t) r r d r
c r r
r
( r ,t)
3
(7)
(8)
where [ρ] evaluate ρ at retarded time:
1
r
,tr
r
c
Similar for [j]
So potentials at point
r
and time t are affected by conditions at
r
1r r
pointr at a retarded time,t r r
c
Given a charge and current density,
find retarded potentials and
Aby means of (7) and (8)
Then use (1) and (2) to derive E, B
Fourier transform spectrum
Radiation from Moving Charges
The Liénard-Wiechart Potentials
Retarded potentials of single, moving charges
Charge q
moves along trajectory
velocity at time t is
charge density
current density
r
r
(t)
0
(
u
(
t)
r
)
ot
(
r
,
t
)
q
(
r
r
(
t
))
0
j
(
r
,
t
)
q
u
(
t
)
(
r
r
(
t
))
0
delta function
Can integrate over volume d3r to get total charge and current
3
q(r,t)d r
3
quj(r,t)d r
What is the scalar potential for a moving charge?
Recall
3
(
r
,t)
r
d
r
r
where [ ] denotes
evaluation at retarded
time
r
r
3
(
r
,
t
)
(
r
,
t
)
d
r
d
t
t
t
r
r
c
Substitute
and integrate
(
r
,
t
)
q
r
r
(
t
)
0
d
r
3
light
el
time
trav
between
rand
r
r
r
q
(
r
r
(
t))
3
0
(
r
,t)
t
t
t
drd
r
r
c
r
r
(
t
t
)
c
q
d
t
r
r
(
t)
0
Now let
then
R(t)rr0(t)
R(t)R(t)
vector
scalar
R
(
t
)
1
(
r
,
t
)
q
R
(
t
)
(
t
t
)
d
t
c
Now change variables again
then
R
(
t)
t
t
t
c
1
R
d
t
d
t
(
t
)
d
t
c
u(t) r0(t)
d
R(t ) r r0(t)
dt
u(t)
2
2
R
(
t
)
R
(
t
)
2
R
(
t
)
R
(
t
)
2
R
(
t
)
u
(
t
)
Velocity
dot both sides
Also define unit vector
R
n
R
1
dt dt R(t)dt
c
1
1 R(t )dt
c
1
1 n(t ) u(t )dt
c
so
1
R
(
t
)
(
r
,
t
)
q
(
t
)
d
t
1
1
n
(
t
)
u
(
t
)
c
This means evaluate
integral at t'‘=0,
or t‘=t(retard)
So...
(r,t)
q
(tretard
)R
(tretard
)
where
1
(tretard
)
(t)1cn(t)u(t)
beaming factor
or, in the bracket notation:
q
(r,t)
R
Liénard-Wiechart
scalar potential
Similarly, one can show for the vector potential:
qu
A
c
R
Liénard-Wiechart
vector potential
Given the potentials
q
(r,t)
R
one can use
B
A
qu
A
cR
1
A
E
c
t
to derive E and B.
We’ll skip the math and just talk about the result. (see Jackson §14.1)
The Result: The E, B field at point r and time t depends on
the retarded position r(ret) and retarded time t(ret) of the charge.
Let
(
u
r
)
0t
ret
u
r
(
t
)
0
ret
u
c
velocity
of
charged
particle
accelerati
on
1
n
r r
r r r
B(r,t) n E (r,t)
r
r2
r
r
rÝ
r r
(n ) (1 ) q n
r r
E ( r ,t) q
3 ( n )
3 2
R
c1 4 4R 442 4 4 4 43
1 4 44 2 4 4 43
"VELOCITY FIELD"
1
2 Coulomb Law
R
Field of particle w/ constant velocity
"RADIATION FIELD"
1
R
Transverse field due to acceleration
Qualitative Picture:
transverse “radiation”
field propagates at velocity c