PHGN462: Advanced E&M
Group Quiz 7 – November 9, 2009
NAMES:
z
P
ρ
p
r
s/2
x
The electric potential for a single oscillating dipole located at the origin is given by
V (~r, t) = −
p~ · r̂ sin[ω(t − r/c)]
,
4π0 −
λ
r
where −
λ = c/ω is the reduced wavelength. As described in problem 11.11, an electric quadrupole can be thought of as two
oppositely oriented oscillating electric dipoles. Shown in the figure is one of the electric dipoles, p~ = pẑ, which, at arbitrary
time, t, is shifted from the origin to ~r 0 (t) = + 2s cos ωt ẑ. You are to evaluate some of the expressions needed to calculate
the retarded electric potential of this dipole in the radiation zone (r s) in the long wavelength limit ( −
λ s). Define the
relative coordinate, ρ
~ = ~r − ~r 0 (the same as script-r in the text).
s
a. Show that |~
ρ| ' r(1 − 2r
ẑ · r̂ cos ωt) to first order in rs .
b. Find p~ · ρ̂ = pẑ · ρ̂ to first order in rs .
c. Find sin[ω(t − |~
ρ|/c)] to first order in
s
−.
λ
1