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Semi-Infinite planes at an angle

  1. Jun 28, 2015 #1
    This problem was on my exam last week, and i've been having some troubles coming up with a solution.

    1. The problem statement, all variables and given/known data

    Two semi-infinite conducting planes are connected by insulating glue, they make a 30º angle with each other. One of them is at ground potential, and the other one at V0. Find the potential between the conductors.
    Note: There might be solutions, in which the potential has logarithmic dependence on the distance to the axis, or even independent of this distance.
    2. Relevant equations
    Laplace's equation in cylindrical coordinates.
    3. The attempt at a solution

    IMG_1212.jpg

    IMG_1216.jpg
    The boundary conditions i came up with are:
    1- V=0 at phi=0
    2- V=V0 at phi=alpha

    But i'm note sure if this is right.
    Can you guys please help me out?
     
  2. jcsd
  3. Jun 28, 2015 #2

    TSny

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    Gold Member

    Your work looks correct to me. But you might be required to justify ignoring the parts of the Laplacian that involve derivatives of the radial variable ##s## and derivatives of the variable ##z##.

    Of course, you can make your answer more specific by substituting the given value for ##\alpha##.
     
    Last edited: Jun 28, 2015
  4. Jul 2, 2015 #3
    I'm assuming i can justify ignoring the z part by saying we have cylindrical symmetry. But what would the justification be for the s radial variable?
     
  5. Jul 2, 2015 #4

    TSny

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    Use the fact that the potential for ##\alpha = 0## or ##\alpha = 30^0## is independent of ##s## along with the general form of the solution for ##V## using separation of variables: ##V(s, \alpha, z) = f(s)g(\alpha)h(z)##.
     
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