Electrodynamics (I): Homework 7
Due: June 11, 2015
Exercises in Griffiths
11.3, 11.4(10%, hand-in), 115, 11.6, 11.8, 11.11, 11.13 (5%, hand-in), 11.16, 11.17, 11.22,
11.25, 11.27, 11.28(10%, hand-in)
Ex.1 hand-in
A plane electromagnetic wave of wavelength λ is incident on an insulating dielectric sphere
of radius a. The dielectric constant of sphere is κ and λ a. Suppose that the incident
wave is linearly polarized in z direction and the differential cross section
dσ
dΩ
in the direction
Ω̂ is defined as
dσ
power of EM fields scattered into dΩ around Ω̂
≡
dΩ
incident power per unit area
(a) 10% Find the differential cross section as a function of scattering angles θ and φ in
spherical coordinates.
(b) 10% Following (a), if the sphere is replaced by a perfectly conducting sphere with the
total magnetic field being kept zero inside the sphere, find the the differential cross section
as a function of scattering angles.
Ex.2 hand-in
(a) 5% A non-relativistic positron of charge e is incidents with velocity v1 (v1 c) from
infinite distance and head- collides on a fixed nucleus of charge Ze. The positron is decelerated until it comes to rest and then is accelerated again in the opposite direction until it
reaches a terminal velocity v2 . Taking radiation loss into consideration but assuming that
it is small, find v2 as a function of v1 and Z.
(b) 5 % A relativistic particle of charge q and rest mass m moves on a circular orbit in a
constant and uniform magnetic field B. The particle loses energy by radiation. Assuming
that the energy of the particle is E0 at t = 0 and the particle moves in a quasi-circular spiral,
find the energy of the particle E(t) and the radius of its orbit as a function of time t.
Ex.3 hand-in Charges moving on a circle
Consider charges in uniform circular motions (counterclockwise) on a ring of radius a. Let
ω be the angular frequency for the circular motion. As shown in the following figure, the
center of the ring is set to be the origin O of coordinate, the angular coordinates for charges
are denoted by φn , and the distance and the relative angle to the point for radiation fields
1
z
θ
R
O
x
ϕ
y
are denoted by R and θ.
(a) 7% Suppose that there are only two charges q and −q on the ring with angular coordinates being φ+ and φ− respectively q. Two charges move in the way that they orbit
around each other with distance being 2a, i.e., φ+ − φ− = π. Find the angular distribution
of average radiation power in the regime: R a and c/ω a.
(b) 8% Because motions of charges are periodic, the corresponding current density J(r, t)
and the radiation field Arad (r, t) are periodic functions of t with period T being
2π
.
ω
Hence
J and Arad can be expressed as
J(R, t) =
X
Arad (R, t) =
X
Jm (R)e−imωt ,
m
Am (R)e−imωt ,
m
with Jm (R) =
1
T
RT
0
dtJ(R, t)eimωt is the mth Fourier component of J. When there is only
one charge of q moving on the ring, the corresponding Jm and Am are denoted by jm and
am . Suppose that there are N equally-spaced charges moving on the ring and each charge
has the same charge q.
(i) Find the corresponding Am in terms of am .
(ii) Using the following mathematical facts for the Bessel function Jm (z) (m = integer):
im Z 2π imφ−izcosφ
e
dφ,
Jm (z) =
2π 0
(ex/2)m
Jm (mx) ≈ √
,
2πm
show that for large N , Am ∼ (aω/c)N → 0 and hence the radiation field gets cancelled due
to destructive interference.
Ex.4 hand-in Radiation and index of refraction
(a) 5% Consider a charge distribution and a current distribution that are oscillating with
angular frequency ω so that ρ(r, t) = ρ(r, ω)e−iωt and J(r, t) = J(r, ω)e−iωt / Express the
2
potentials: vector potential A(r, ω) and scalar potential V (r, ω) in terms of integrals over
ρ(r, ω) and J(r, ω).
(b) 10 % Consider a think layer of dielectric material placed at z = 0. The dielectric layer is
parallel to xy plane and occupies 0 ≤ z ≤ d. Suppose that there are n0 molecules per unit
volume in the dielectric layer and there are fj electrons with natural frequency ωj in each
molecule. Furthermore, for the electron with natural frequency ωj , it can be described by a
damped harmonic oscillator only driven by the electric field:
e
dx
d2 x
+ ωj2 x = E(t),
+ γj
2
dt
dt
m
where γj is the damping coefficient and E(t) is the electric field. Let Einc = E0 ei(kz−ωt) x̂
be the electric field of a plane wave incident upon the dielectric layer. Assuming that
kd 1 and n0 is small so that each molecule is radiating independently, find the electric
field Erad (z, t) due to the dipole radiation of the dielectric layer. By combing Erad and Einc
at z = d, show that E = Erad + Einc = E0 ei(nz−ωt) x̂ with n being the refraction index. Find
n.
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