Electrodynamics (I): Homework 6
Due: May 28, 2015
Exercises in Griffiths
10.3, 10.7, 10.15, 10.17, 10.20, 10.23, 10.24, 10.26(5%, hand-in), 10.28 (10%, hand-in), 10.30,
10.31, 10.32 (10%, hand-in), 10.34(15% hand-in), 12.56, 12.58, 12.70 (10%, hand-in)
Ex.1 hand-in
A point charge of q is subject to a constant force F and moves along x axis. Suppose that
the charge starts from rest at the origin at time t = 0 and moves toward +x direction.
(a) 10% If the rest mass of the particle is m, find the velocity v(t) of the charge at time t
and from v(t), compute the position x(t) at time t. If we define the proper time τ as the
time of the inertial frame that the charge particle is instantaneously at rest in the frame and
set τ = 0 when t = 0, find the velocity v(τ ) at proper time τ experienced by the charge.
(b) 10% Find the Lienard-Wiechert potentials for the position x to the right of the charge
at time t.
Ex.2 hand-in Charge moving in a medium
The Lienard-Wiechert potentials developed in the class are for charged particles moving in
vacuum. The formulation can be easily extended to the situation when a charged particle
moves in a medium. Consider a point charge q moving in a medium with permittivity and
permeability µ. The speed of light in the medium is cn = c/n, where n is the refraction
index and is equal to
q
(µ)/(0 µ0 ). Answer the following equations:
(a) 5% If the trajectory of the point charge is described by ~r0 (t) at time t, find the retarded
potentials of the particle.
(b) 5% Consider the special case when the particle moves with a constant velocity ~v =
vx̂. The fields at position ~r are determined the charge at retarded time tr which satisfies
~ = ~r − ~r0 (t). The angle
cn (t − tr ) = |~r − ~r0 (tr )|. Let the displacement at current time t be R
between R̂ and x̂ be θ. Show that if v > cn , there are regions in which for a given ~r, there
may be two solutions of tr . Find this region in terms of θ and the retarded times in terms
~ and ~v .
of R
(c) 10% Following (b), find the retarded potentials and electric and magnetic fields in terms
of θ,R and v. Note that while a charge moving with constant velocity in vacuum does not
radiate energy, this problem shows that a charge moving with constant velocity v > cn in a
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medium radiates. This is known as Cherenkov radiation.
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