Solution to Laplace's Equation in Cylindrical Coordinates 1 ...

L
z
V = Vo
V = 0
x
V = 0
a
y
Figure 2: The geometry of a cylinder with one eddcap held at potential V = V 0 (ρ) and the
other sides grounded
V = ∑ n
A n J 0 (k n ρ)sinh(k n z)
Now V = 0 for ρ = a. This means that;
J 0 (k n a) = 0
The values of k n a are the zeros of the bessel function J 0 (k n a). The first few are, α 0n =
2.4048, 5.5201, 8.6537, · · ·. Then at Z = L we find A n using the orthogonality of the Bessel
functions.
A n = 1
a 2 [J 1 (k n a)] 2 sinh(k n L)
∫ a
0
ρdρ J 0 (k n ρ) V 0 (ρ)
The graphic form of the solution is shown in figure 3.
As another example we find the potential inside a cylinder when the potential is specified
on the end caps and the cylindrical wall is at zero potential, figure 4. The boundry
conditions are that;
V = V 0 sin(φ)
z = L
V = −V 0 sin(φ) z = -L
V = 0
ρ = a
The solution must have the form;
4