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Homework Help: Calculating the charge distribution on the surface of an assymetric conductor

  1. Jul 25, 2012 #1
    How do I calculate the charge distribution on the surface of any asymmetric closed conducting surface? Is it possible for me to calculate the surface charge density 'σ' as a function of '[itex]\bar{r}[/itex]' the position vector in a spherical co-ordinate system in space, provided I know that the conductor has been qiven a net charge 'Q' and the equation of the conductor in space is ((x/a)^2)+((y/b)^2)+((z/c)^2)=1...
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jul 26, 2012 #2
    For any conducting surface you will have to resort to an approximation.

    In brief: use a numerical method to establish the Electric field for the system. Use that E field result to establish surface charge distribution.
  4. Jul 26, 2012 #3
    But how do I calculate [itex]\bar{E}[/itex] if I don't know my σ? It's like the chicken egg problem except that it's not what comes first that matters, but I need one to know the other. All I know unfortunately is the shape of the conductor and the total charge Q which according to the uniqueness theorum has a unique way of settling on the surface in the abscence of any external electric field...
  5. Jul 26, 2012 #4
    you don't need to know anything about the charge. Field lines are normal to a conducting surface and themselves normal to equipotentials. This means that you can set up a grid with random starting values for potential and by recalculating each value in turn end up closer to the correct solution.

    a quick google gave this http://www.physics.hku.hk/~phys3231/pdf/P1%20-%20Static%20Electric%20Field%20-%20Laplaces%20equation%20in%202D.pdf [Broken] The 3d process is very similar.

    Also http://www.ece.msstate.edu/~donohoe/ece3323analytical_numerical_techniques.pdf you need the last few pages.

    The procedure looks hard but really isn't: Though it is a long time since I last did one!
    Last edited by a moderator: May 6, 2017
  6. Jul 26, 2012 #5


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    Just solve Laplace's equation for the potential. You know that the potential on the surface is a constant (which you may as well set to zero) and that the potential at large distances should look like that of a point charge, so you have your 2 boundary conditions. From the potential, you can determine the charge density.
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