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Electric potential in cylindrical coordinates using separation of variables

  1. Feb 23, 2009 #1
    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

    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.
  2. jcsd
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