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Solution to Laplace's Equation

  1. May 27, 2017 #1
    • Moved from technical forum so the template is missing
    A point charge q is situated at a distance d from a grounded conducting plane of infinite extent. Find the potential at different points in space.

    I want to solve this problem without using the image charge idea.

    I assumed azimuthal symmetry and took the zonal harmonics. And we know that as r tends to infinite the potential in the opposite side of the conducting plane goes to a constant. So we are left with the Pn(theta)r^{-(n+1)} terms.

    $$V(r,\theta)=A_1 + \frac{C_1}{r} + \frac{C_2}{r^2}\cos(\theta)) + \frac{C_3}{2}(3\cos^2(\theta) - 1) + .....$$

    When cos(theta)=d/r, the potential vanishes. So we get the condition:

    $$A_1 + \frac{C_1}{r} + \frac{C_2}{r^2}\frac{d}{r} + \frac{C_3}{2}(3\frac{d^2}{r^2} - 1) + ..... =0$$

    for all r

    Now how to progress further? How do I get the coefficients
     

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    Last edited: May 27, 2017
  2. jcsd
  3. May 27, 2017 #2

    jasonRF

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    I don't think you are answering the same question in the image - it just asks for the total charge on the plane (which you can compute from the image solution). What is the real question you are trying to answer?

    Jason
     
  4. May 27, 2017 #3

    jasonRF

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    By the way, if you want to solve without images, you will have better luck I think if you use Cartesian or cylindrical coordinates since your boundary is a plane. It will be fairly involved - more difficult than the kinds of problems in a typical undergrad (in US) textbook like Griffiths. For fun I solved it using integral transform techniques and contour integration, but other approaches could be used. I'm pretty sure that whatever way you solve it, instead of a discrete sum of functions you will find a continuous sum (an integral). EDIT: the integral can be evaluated to yield the image solution, of course.

    Jason
     
    Last edited: May 27, 2017
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