Solving Laplace's Eqn: Setting Zero of Potential

  1. 1. The problem statement, all variables and given/known data
    I have two quick related questions that I think will help clear something up for me.

    (1) An uncharged metal sphere of radius R is placed in an otherwise uniform
    electric field E = E0 zhat. The field induces charge. Find the potential in the region outside the sphere.

    (2) Find the potential outside an infitely long metal pipe of radius R placed at right angle to an otherwise uniform electric field E0.

    2. Relevant equations

    3. The attempt at a solution

    Okay, so I am mostly comfortable with the solutions to these problems, with the exception of one key concept.

    Problem 1: If E=E0 zhat (for r>>R, then V for r>>R is -E0z + C)
    The one hang-up I have is that for problem (1), you set V=0 on the xy-plane "by symmetry." Thus we can say C=0. I'm not sure I understand the reasoning behind setting V=0 for all z=0.

    Similarly, from problem (2), if we suppose that the electric field is pointing along z direction, so again, the form of the potential is -E0z + C. Again, one should set V=0 on the xy-plane, so C=0. I don't have a clear picture in my head of why you set V=0 on the xy-plane.

    Any explanations of this concept would be greatly appreciated
  2. jcsd
  3. fzero

    fzero 3,120
    Science Advisor
    Homework Helper
    Gold Member

    Since the electric field is the gradient of the potential, [tex]\vec{E} = - \nabla V[/tex] (for electrostatics), a constant shift in the potential [tex]V' = V + c[/tex] doesn't change the electric field. Therefore we are usually free to define the location of [tex]V=0[/tex] by adding a constant term to the potential. In your examples, choosing [tex]V(z=0) = 0[/tex] is just the most convenient choice, but it was not the only one.
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