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Solving for electric potential using separation of variables

  1. Apr 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Given two two grounded concentric spherical shells with radii a,b (a<b) and a point charge q between them at a<r=R<b find:

    1.The surface charge density of the point charge using the delta function, assume the charge is on the z axis

    2.By using the separation of variables method find the electric potential between the shells.

    2. Relevant equations
    I have no idea how to insert mathematical equation in the forum but the most relevant equations are the jump discontinuity in the orthogonal part of the field and the general form of the solution of the Laplace equations in spherical coordinates

    3. The attempt at a solution

    In my first try I tried to solve for the whole volume at once which produced contradictions, in my second attempt I divided the volume and solved for each volume seperately but I got stuck on the orthogonality condition for the Legendre polynomials, the precise attempt is shown in the images

    HW solution attempt_1.jpg HW solution attempt_2.jpg HW solution attempt_3.jpg HW solution attempt_4.jpg HW solution attempt_5.jpg HW solution attempt_6.jpg HW solution attempt_7.jpg
     
  2. jcsd
  3. Apr 22, 2016 #2

    TSny

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    Homework Helper
    Gold Member

    Your second approach should work. However, I noticed a few errors:

    Page 1: Your expression for σ is incorrect. You want to write a surface charge density for the surface r = R. Thus, ∫σdA = q for this surface. There is no integration over r here. Think about the expression for dA for this surface.

    Page 4: Your powers of r in ##\Phi_I## and ##\Phi_{II}## don't appear to be correct. For example, see eq. 14 here http://www.luc.edu/faculty/dslavsk/courses/phys301/classnotes/laplacesequation.pdf

    Page 4: Check the signs in the equation at the bottom of the page for the boundary condition involving R+ and R-. Recall that ##\vec{E} = -\nabla \Phi##. ( I don't understand the equation in the middle of the page, just before "in Volume II".)

    Page 7: You forgot the ##\delta_{l,l'}## that occurs when doing the integral on the left at the top of the page.

    There is another boundary condition that you can use. What can you say about the continuity of ##\Phi## at r = R when ##\theta \neq 0##?
     
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