See "Technical note on computing radiation fields'
Problem 1.
(Optional) Electric field in the far field
If you get stuck check the notes online. The scalar and vector potential in the far field are
drop(T,ro)
(1)
A(t,r) d3r.J(T,ro)/c 4Tr
(2)
where the retarded time T = t - |r - ro|/c in the far field is
n.ro T =t-r/c+ c
(3)
The goal is to compute the electric field
(4)
(a) (Optional) Consider the change of variable t,r. T,r.. Show that
a OT t0-
(5)
n o c Ot
(6)
dro
(b) (Optional) Compute
- cn dt or
=0
(7)
You should find a simple result. Interpret the answer using the definition of T
T = the time when the light should be emitted from r. to arrive at the observation point (t,r).
How do you interpert the derivative:
It is the derivative moving with the light. a a If you change the observation point (t,r) + cn to (t + dt, r + c n dr) the light th(8) Qt ar emitted at (T,r0) will reach both of these points without changing T or r0
(c) (Optional) Show that
1
OJ(T,r.) n 1 OT c 4Tr J
Op(T,r.)
E :
(9)
4Trc2
(d) (Optional) Use current conservation to express
n OJ
(10)
OT
where (Vr. : J)t denotes the divergence at fixed observation time
1
(e) (Optional) Conclude that only the transverse piece of the current to n contributes to the radiation field
11
E=
[(re.u)u -re]
(11)
the part of O J transverse to n r 1oJ(T,r.) ar
(12) (13)
2
Problem 2. Dipole Fields
Consider a small ectric dipole with harmonic time dependence, p(t) = p.e-iot. Recall that in homework 6 we determined the electric field through order w2 in frequency using a quasi- static near field expansion 3n(n:p(t))-p(t)n(np)+p E(t)= (14) 4Tr2 8Trc2 The purpose of the problem is to examine the transition to the far field, by computing the exact electric field as a function of radius.
(a) Define near and far field. Express your results in terms of the wave number k = w/c
(b) Start from the exact expressions
P(T,r.) ro4r|r-rol J(T,ro)/c '4T|r -rol
et,r)
(15)
A(t,r)
(16)
where
-=I
n.ro
(17)
c
c
and assume harmonic time dependence of p(to,r.) = p(r.)e-ioto and J(to,r.) = J(r.)e-iot, without making a far field expansion show that
Y(t,r)
(18)
e~ot+ikr A(t,r)=-ik -Po 4Tr
(19)
(c) Show by direct differentiation of the potentials A and that in the far field you recover the result given in class k2e-iot+ikr E = [-n X n x p.] (20) 4TT Comment on all qualitative features. When differentiating, note carefully the contribution from A and , and how they conspire to make a field E which is transverse to n.
(d) Show that in general
-ik
(21)
4Tr3
4Tr2
4TT
Comment on all qualitative features. Write Eq. (21) in coordinate space expressed in terms of p(te), p(te), and p(te) with te = t -- r/c (compare to the typed course no