Separation of variables, laplace equation

[merrypark3]

Homework Statement

Two coaxial pipes(radius a) of the same diameter with a small gap between them are maintained at a potential difference V.

By separation of variables and fitting coefficient, guess the potential within the pipe, near the gap.

Homework Equations

The Attempt at a Solution

Cylindrical laplace equation.
$\frac{d^2 R}{dr ^2} + \frac{1}{r} \frac{dR}{dr} =0$

Wow can I handle the boundary condition at r=a? R=V/2 or R=-V/2 or R=0?

 

[mark726]

Hello there,

Thank you for your post. It seems like you are trying to solve for the potential within the pipe near the gap using the Laplace equation. In order to handle the boundary condition at r=a, you can use the fact that the potential at the surface of a conductor is constant. Therefore, at r=a, the potential must be equal to V/2. Using this information, you can solve for the potential at different points within the pipe and near the gap. I suggest trying to solve for the potential using separation of variables and fitting coefficients, as mentioned in the homework statement. Let me know if you have any further questions. Good luck with your research!