10—Partial Differential Equations 26610.35 For the preceding equations show that there are solutions of the formf(r) =Arn, and recall
the analysis in section4.11for thegsolutions. What values of the separation constantC will allow
solutions that are finite asx→ ± 1 (θ→ 0 , π)? What are the corresponding functions ofr? Don’t
forget that there are two solutions to the second order differential equation forf — two roots to a
quadratic equation.
10.36 Write out the separated solutions to the preceding problem (the ones that are are finite asθ
approaches 0 orπ) for the two smallest allowed values of the separation constantC: 0 and 2. In each
of the four cases, interpret and sketch the potential and its corresponding electric field,−∇V. How do
you sketch a potential? Draw equipotentials.
10.37 From the preceding problem you can have a potential, a solution of Laplace’s equation, in the
form
(
Ar+B/r^2
)
cosθ. Show that by an appropriate choice ofAandB, this has an electric field that
for large distances from the origin looks likeE 0 zˆ, and that on the spherer=Rthe total potential is
zero — a grounded, conducting sphere. What does the total electric field look like forr > R; sketch
some field lines. Start by asking what the electric field is asr→R.
10.38 Similar to problem10.16, but the potential on the cylinder is
V(R,φ) =
{
V 0 ( 0 < φ < π/ 2 andπ < φ < 3 π/ 2 )
−V 0 (π/ 2 < φ < πand 3 π/ 2 < φ < 2 π)
Draw the electric field in the region nearr= 0.
10.39 What is the area charge density on the surface of the cylinder where the potential is given by
Eq. (10.53)?