Physics 505 Fall 2005
Homework Assignment #1 — Due Thursday, September 15
Textbook problems: Ch. 1: 1.4, 1.5, 1.10, 1.14
1.4 Each of three charged spheres of radius a, one conducting, one having a uniform charge
density within its volume, and one having a spherically symmetric charge density that
varies radially as rn(n > −3), has a total charge Q. Use Gauss’ theorem to obtain the
electric fields both inside and outside each sphere. Sketch the behavior of the fields
as a function of radius for the first two spheres, and for the third with n=−2, +2.
1.5 The time-averagesd potential of a neutral hydrogen atom is given by
Φ = q
4π0
e−αr
r1 + αr
2
where qis the magnitude of the electronic charge, and α−1=a0/2, a0being the Bohr
radius. Find the distribution of charge (both continuous and discrete) that will give
this potential and interpret your result physically.
1.10 Prove the mean value theorem: For charge-free space the value of the electrostatic
potential at any point is equal to the average of the potential over the surface of any
sphere centered on that point.
1.14 Consider the electrostatic Green functions of Section 1.10 for Dirichlet and Neumann
boundary conditions on the surface Sbounding the volume V. Apply Green’s theorem
(1.35) with integration variable ~y and φ=G(~x, ~y ), ψ=G(~x 0, ~y), with ∇2
yG(~z, ~y ) =
−4πδ(~y −~z ). Find an expression for the difference [G(~x, ~x 0)−G(~x 0, ~x)] in terms of
an integral over the boundary surface S.
a) For Dirichlet boundary conditions on the potential and the associated boundary
condition on the Green function, show that GD(~x, ~x 0) must be symmetric in ~x
and ~x 0.
b) For Neumann boundary conditions, use the boundary condition (1.45) for GN(~x, ~x 0)
to show that GN(~x, ~x 0) is not symmetric in general, but that GN(~x, ~x 0)−F(~x )
is symmetric in ~x and ~x 0, where
F(~x ) = 1
SIS
GN(~x, ~y )day
c) Show that the addition of F(~x ) to the Green function does not affect the potential
Φ(~x ). See problem 3.26 for an example of the Neumann Green function.