How to Solve for the Electric Field Between Angular Plates?

[kargak]

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 couldn't find the answer.


http://img85.imageshack.us/i/adsztw.jpg/

 

[jdwood983]

Looks to me like cylindrical Laplacian is necessary for this problem. What are your boundary conditions??

 

[kargak]

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 couldn't get a meaning

 

[jdwood983]

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 $z$ dependence so from separation of variables we can write $V(r,\theta)=R(r)\Theta(\theta)$ (where $0\leq\theta\leq\beta$) so that we get

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

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