Finding and labeling equilibrium points between multiple point charges.

[nontradstuden]

Homework Statement

How to find equilibrium points between four point charges?
How do you determine whether these equilibrium points are stable, unstable, or neutral?


I know the sum of the forces should equal zero, but that's it. How do you determine the stability of the equilibrium points by looking at the electric field lines? How do you determine the stability of equilibrium points mathematically?

 

[apelling]

One approach might be to make an equation for the electric potential. Because potential is a scalar this is easier than trying to find an equation for force or field strength, which would require vector addition.
Now field strength is equal to -potential gradient (E=-dV/dr or more generally E=-grad V).
So differentiate electric potential and put equal to zero. Solve for r. This may get very complex depending on the initial set up, so may not be the best approach.
The second differential will determine if the turning point you have found is a max or min point which tells us about stability.