PHY 3323
October 26, 2011
Exam #2
. . . such names as Laplace, Maxwell and Einstein — do they mean anything to
you? . . . Physicists, according to our not-so-reliable historians. Responsible for
the rapid rise of the European-American culture . . .
A Canticle for Leibowitz
(1) A point charge q of mass m is released from rest at a distance d from an infinite
grounded conducting plane. How long ∆t will it take for the charge to hit the plane?
a) Dimensional analysis gives you how the time ∆t depends upon the parameters q,
m, d and ǫ0 , up to an overall constant. What is this form? (30 points)
b) Suppose the charge q is at height z. Use the Method of Images to find the force
exerted upon it. (30 points)
c) What is the exact formula for ∆t? (20 points)
(2) A charge +Q is distributed uniformly along the z axis from z = −a to z = +a.
a) Find the general expansion of the potential V (r, θ) for r > a, up to dimensionless
constants, using the symmetry of the charge distribution, dimensional analysis and
∇2 V (r, θ) = 0. (30 points)
b) What are the first three multipole moments? (30 points)
c) What is the leading term in the expansion of the potential for large r? (20 points)
(3) A point dipole p~ = pb
z is embedded at the center of a sphere of linear dielectric material
with radius R and dielectric constant ǫr .
a) The general form of the potential V (r, θ), both inside and outside the sphere can be
guessed, up to dimensionless constants, using dimensional analysis, the presence of
~ 3 (~r). What is this form?
the dipole source and the fact that ∇2 V (r, θ) = ǫr ǫ0 ~p · ∇δ
(30 points)
b) What are the boundary conditions which apply at the edge of the sphere? (30
points)
c) What is V (r, θ) both inside and outside the sphere? (20 points)