PHY 6346 Fall 2008
Homework #5, Due Wednesday, October 1
1. A long, hollow conducting cylinder of radius bis centered on the z-axis. The right (x>0)
and left (x<0) halves of the cylinder are separated by small insulating gaps and are kept
at potentials ±V0respectively.
(a) Use separation of variables to show that the general two-dimensional solution to ∇2Φ=0
in cylindrical coordinates (ρ, φ) can be written as Φ = R(ρ)Q(φ), with R(ρ)=ρ±mand
Q(φ)=e±imφ,wheremis zero or a positive integer. (For m=0,“ρ−0”meanslnρ).
(b) Fix the coefficients to match the given potential at ρ=band write a series for Φ(ρ, φ)
inside and outside the cylinder (two different series).
(c) Sum the series and show that inside the cylinder (ρ<b)
Φ(ρ, φ)=2V0
πtan
−12bρ cos φ
b2−ρ2.
What is the solution for ρ>b?
(d) Calculate and sketch the surface charge density on the cylinder.
2. Construct the 2-d Green’s function (∂/∂z = 0) for the interior of a cylinder of radius b:
(a) Show that
δ(φ−φ0)= 1
2π
∞
X
m=−∞
eim(φ−φ0)=1
2π"1+2
∞
X
m=1
cos m(φ−φ0)#.
(b) From the differential equation
∇02G=−4πδ(ρ−ρ0)
ρδ(φ−φ0)
we know that ∇02Gvanishes almost everywhere. Expand Gin one series G<for ρ0<ρ
and another G>for ρ0>ρ. Apply boundary conditions: that G=0atρ0=b;thatGis
nonsingular at ρ0→0; and that Gis smooth at ρ0=ρ,
G=C0(ln ρ>−ln b)+
∞
X
m=1
Cmρm
<
ρm
>
−ρm
<ρm
>
b2mcos m(φ−φ0).
(c) Integrate the differential equation over the δ-function in ρand show that
G=−2lnρ>+2lnb+
∞
X
m=1
2
mρm
<
ρm
>
−ρm
<ρm
>
b2mcos m(φ−φ0).
(d) Sum the series to obtain
G=lnρ2ρ02b−2+b2−2ρρ0cos(φ−φ0)
ρ2+ρ02−2ρρ0cos(φ−φ0).