10—Partial Differential Equations 31010.13 Fill in the missing steps in finding the solution, Eq. ( 31 ).
10.14 A variation on the problem of the alternating potential strips in section10.6. Place a grounded conducting
sheet parallel to thex-yplane at a heightz=dabove it. The potential there is thenV(x,y,z=d) = 0. Solve
for the potential in the gap betweenz= 0andz=d. A suggestion: you may find it easier to turn the coordinate
system over so that the grounded sheet is atz = 0and the alternating strips are at z =d. This switch of
coordinates is in no way essential, but it is a bit easier. Also, I want to point out that you will need to consider
the case for which the separation constant in Eq. ( 34 ) is zero.
10.15 The equation ( 33 ) is in rectangular coordinates. In cylindrical coordinates it is
∇^2 V =
∂^2 V
∂r^2+
1
r∂V
∂r+
1
r^2∂^2 V
∂θ^2+
∂^2 V
∂z^2= 0
Take the special case of a potential function that is independent ofzand try a solutionV(r,θ) =rnf(θ). Show
that this works and gives a simple differential equation forf. Solve that equation. Mustnbe positive? Mustn
be an integer?
10.16 A very long conducting cylindrical shell of radiusRis split in two along lines parallel to its axis. The two
halves are wired to a circuit that places one half at potentialV 0 and the other half at potential−V 0. What is the
potential everywhere inside the cylinder? Use the results of the preceding problem and assume a solution of the
form
V(r,θ) =∑∞
0rn(
ancosnθ+bnsinnθ)
V 0
−V 0
Match the boundary condition that
V(R,θ) ={
V 0 ( 0 < θ < π)
−V 0 (π < θ < 2 π)