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E Field between angular plates

  1. Jul 20, 2010 #1
    the electric field between two plates . plates have an angle with each other. first plate has zero potential and the second plate has a V potential.

    i guess i must use laplace equation on cylndric coordinates but i couldnt find the answer.

  2. jcsd
  3. Jul 20, 2010 #2
    Looks to me like cylindrical Laplacian is necessary for this problem. What are your boundary conditions??
  4. Jul 20, 2010 #3
    V=0 at angle=0
    V=V(0) at angle=beta

    all givens are : http://img85.imageshack.us/i/adsztw.jpg/
    sylindrical laplacian's only teta part is not equal zero. i wrote it and couldnt get a meaning
  5. Jul 20, 2010 #4
    You are correct about the boundary conditions, but there still can be a contribution from the radial component (you'll see why/how soon).

    Clearly there is no [itex]z[/itex] dependence so from separation of variables we can write [itex]V(r,\theta)=R(r)\Theta(\theta)[/itex] (where [itex]0\leq\theta\leq\beta[/itex]) so that we get

    [tex]\frac{\nabla^2V}{V}\rightarrow-\frac{r}{R}\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r}\right)=\lambda_\theta=\frac{1}{\Theta}\frac{\partial^2\Theta}{\partial\theta^2}[/tex]

    So with the boundary conditions such that [itex]V(r,0)=0[/itex] and [itex]V(r,\beta)=V[/itex], what can you determine about the angular function, [itex]\Theta[/itex] and the eigenvalue [itex]\lambda_\theta[/itex]?
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