# Potential Energy

1. Sep 11, 2011

### kuahji

Consider 4 charges all of magnitude q located at the following positions:
r1 = (+1; +1; 0) ; r2 = (-1; +1; 0) ; r3 = (-1;-1; 0) ; r4 = (+1;-1; 0)

We now want to study the stability of such an equilibrium con guration against
small displacements. Show that near the origin the potential energy felt by the test particle may be written as
V (x; y; z) = V0 + Ax^2 + By^2 + Cz^2 + ...

Where a test charge e is at the origin.

I'm a bit confused about how to arrive at this derivation. In general we have E=-$\nabla$V, so if we have the electric field at the origin, we can get the potential. However it does not turn out to be the above. Any help on getting me started?

Last edited: Sep 11, 2011
2. Sep 11, 2011

### vela

Staff Emeritus
Expand V as a Taylor series about the origin. You should find the first-order terms vanish.