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ANTENNAS , WAVE ANTENNAS , WAVE PROPAGATION &TV ENGGPROPAGATION &TV ENGG
Topics to be coveredTopics to be covered Retarded Potential
RETARDED POTENTIALS
At point r =(x,y,z), integrate over charges at positions r’
(r r ,t)
r r r r d3r r
r A (r r ,t)
1c
r j
r r r r d3r r
(7)
(8)
where [ρ] evaluate ρ at retarded time:
rr
ctr 1, Similar for [j]
So potentials at point r and time t are affected by conditions at
point r r at a retarded time, t 1
cr r r r
Given a charge and current density,
find retarded potentials by means of (7) and (8)
Then use (1) and (2) to derive E, B
Fourier transform spectrum
A and
Radiation from Moving ChargesRadiation from Moving Charges
)(0 trr
)()( trtu o
The Liénard-Wiechart PotentialsRetarded potentials of single, moving charges
Charge q
moves along trajectory
velocity at time t is
charge density
current density
))((),( 0 trrqtr
))(()(),( 0 trrtuqtrj
delta function
Can integrate over volume d3r to get total charge and current
rdtrjuq
rdtrq
3
3
),(
),(
What is the scalar potential for a moving charge?
rdrr
tr
3),( Recall where [ ] denotes
evaluation at retardedtime
c
rrtt
rrtrtdrdtr
),(),( 3
rr and between timeellight trav
Substitute )(),( 0 trrqtr
and integrate rd 3
)(
)(
))((),(
0
03
trrc
rrtt
tdq
crr
ttrr
trrqtdrdtr
Now let
)()(
)()( 0
tRtR
trrtR
vector
scalar
then
tdctRtttRqtr ))(()(),( 1
Now change variables againctRttt )(
then tdtRc
tdtd )(1
Velocity
)(
)()(
)()(
0
0
tu
trrtd
dtR
trtu
)()(2)()(2
)()( 22
tutRtRtR
tRtR
dot both sides
Also define unit vector RRn
tdtutnc
tdtRc
tdtRc
tdtd
)()(11
)(11
)(1
so
tdttutn
tRqtrc
)()()(1
)(),(1
1
This means evaluateintegral at t'‘=0, or t‘=t(retard)
)()(1)()(
)()(),(
1 tutnttwhere
tRtqtr
cretard
retardretard
So...
beaming factor
or, in the bracket notation:
Rqtr
),(
RcuqA
Liénard-Wiechartscalar potential
Similarly, one can show for the vector potential:
Liénard-Wiechartvector potential
Rqtr
),(
RcuqA
Given the potentials
one can useAB
tA
cE
1
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 onthe retarded position r(ret) and retarded time t(ret) of the charge.
Let
ncu
tru
tru
ret
ret
1
onaccelerati )(
particle charged of velocity )(
0
0
r B (r r , t) r n
r E (r r , t)
r E (r r ,t) q
( r n r )(1
r 2)
3R2
"VELOCITY FIELD"
1
R 2 Coulomb Law
1 2 4 4 4 3 4 4 4
qc
r n 3R
( r n r )
r Ý
"RADIATION FIELD"
1R
1 2 4 4 4 4 3 4 4 4 4
Field of particle w/ constant velocity Transverse field due to acceleration
Qualitative Picture:
transverse “radiation”field propagates at velocity c