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