- #1

Haorong Wu

- 413

- 89

- 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.