# Electric potential in cylindrical coordinates using separation of variables

1. Feb 23, 2009

### itwong

Hi, I am new to the forum. I've encountered a couple problems with separation of variables in cylindrical coordinates.

Problem #1: a clindrical surface of radius R is oriented along the z-axis, and is split into two
conducting half-cylinders. The potential satisfies the boundary conditions
V(R,phi) = V0 , for 0<phi<pi; V(R,phi) = -V0, for pi<phi<2*pi. Find the potential
inside and outside the cylinder and the capacitance per unit length.

For this problem, I know the general solution is independent of z, namely V(r,phi) and it contains sin(n*phi), cos(n*phi) and r^n terms. But how to find the coefficients using the boundary condition confuses me. Is it necessary to assume V->0 as r -> infinity? Please
help!

Problem #2: an empty can haas the shape of a sector of a cylinder. The volume of the can
is defined by the cylindrical coordinates: 0<= r <= R, 0<= phi <= pi/2, -inf<z<inf
Basically one quarter of a cylinder in the first quadrant with boundary conditions
(i) V= 0 when phi = 0; (ii) V = 0 when phi = pi/2; (iii) V= V0 when r = R. Find the
potential everywhere inside the can.

For this problem, I know it's similar to the first problem except the boundary conditions.
maybe even simpler because of the zero potential. but I still have no idea how to approach this problem. Thank you for your help.