sample exam questions

Physics 507 Exam 3: April 29, 2009
Do 2 of the following problems. All problems worth 50 points except 1,
which has an optional 10 point bonus.
(1) NB: In 1D the Helmholtz Green function is nonsingular:
G(x, x ′ ) = 2πi
k exp(ik|x − x′ |).
This problem involves the ideas of Foldy’s method, but applied to a single
point scatterer at the origin illuminated by an incident wave u i .
Let u(x) denote the solution to the Helmholtz equation. Let A be the
scattering amplitude associated with an isotropic point scatterer situated
at x = 0 and G the 1D Green function.
• Show that
u(x) = u i (x) + G(x, 0)Au(0)
• Solve this equation for the total field at the origin u(0) and thereby
show that that the total field at any point is:
u(x) = u i (x) +
G(x, 0; k)A
1 − G(0, 0)A ui (0)
• Expand the demoninator of this equation in powers of the scattering
amplitude and interpret the terms in terms of the order of scattering
(single-scattering, double-scattering, etc).
• Bonus: why is it that the first equation for u appears to be linear in
the scattering amplitude, but the final one appears to be nonlinear?
(2) Suppose you have a current distribution J(x). In the far-field, the vector
potential associated with this is:
A(x) = µ 0 e ikr ∫
4π r
J(x ′ )e −ik·x′ d 3 x ′
• Expanding the integrand in powers of k, show that if we keep only
the first term (the zeroth power), we get
A(x) = − iµ 0ω
4π
e ikr
r p
Hint, use the divergence theorem to integrate by parts and the continuity
equation in the frequency domain.
• Compute the resulting H field.
1

(3) Consider a thick non-magnetic slab of conductivity σ, whose normal is
in the z direction illuminated by a plane EM wave propagating in the z
direction. Let the peak amplitudes of this wave be E 0 and B 0 . For large
σ the real and imaginary parts of the wavevector are equal:
√
k ′ = k ′′ ωσµ0
≈
2
• Compute the Poynting in the slab vector averaged over one period.
• Suppose the conductivity of the slab goes to infinity. What is the
time averaged Poynting vector everywhere in space?
(4) Consider the semi-classical model for an electron bound by a linear restoring
force and subject to a time-varying electric field E = E 0 e −iωt . Let K
denote the spring constant and γ the damping constant.
• Derive the equations of motion for the electron position x.
• This model applies to plasmas, in which case the electron is not
bound, so K = 0. Let ρ e be the electron density. Derive an expression
for the conductivity of plasmas. Hint: you may assume Ohm’s law
applies.
• For dilute plasmas, collisions are neglible, so you can set γ = 0. Show
that in this case the conductivity is purely imaginary.
2