How to calculate electric potential, given some other potential?

[Haorong Wu]

Homework Statement

Below is a configuration of a Paul trap.

The problem is to find the potential near the axis, which should be ##\Psi = \frac {V_0 \cos \Omega_T t + U_r} 2 \left ( 1+ \frac {x^2-y^2} {R^2} \right )##

Relevant Equations

None

The problem can be simplified to a configuration in ##x-y## plane where two point at ##y## axis with ##y=\pm R## have potential of ##0##, and two point at ##x## axis with ##x=\pm R## have potential of ##U=V_0 \cos \Omega_T t##.

The expression of ##U## is not important, the problem is now to find the potential near the origin.

I am stuck here. I have tried to use separation of variables, but there is no useful symmetry.

I can not figure out how to configure image charges, either.

Then, should I use the superposition of potentials? If so, I can not see how the ##V=0## points at the ##y## axis come into solution.

 

[anathema]

Dear fellow scientist,

Thank you for sharing your thoughts on this problem. It seems that you have already explored some possible approaches, but have not found a satisfactory solution yet. I would like to offer some suggestions that may help you in your analysis.

Firstly, it may be helpful to consider the physical interpretation of the problem. What does the configuration described in the problem represent? Is there a physical system that can be modeled with this potential? This can provide some insights into the behavior of the potential near the origin.

Additionally, you mentioned that there is no useful symmetry in the problem. However, have you considered any approximate symmetries that may exist? Even if the problem is not exactly symmetric, there may be some approximate symmetries that can simplify the analysis.

In terms of using image charges, it may be helpful to consider the boundary conditions at the points ##y=\pm R## and ##x=\pm R##. How can these boundary conditions be satisfied by the presence of image charges? This may provide some clues on how to configure them.

Finally, using the superposition of potentials may indeed be a useful approach. In this case, you may want to consider the contributions of each point charge separately and then add them together to obtain the total potential. This may help in addressing the issue of the ##V=0## points at the ##y## axis.

I hope these suggestions are helpful in your analysis. Remember to keep an open mind and explore different approaches, as sometimes the solution may come from unexpected places. Good luck!