The potential energy between charges q and q' at distance r is given by:
A regular plane arrangement of alternate positive and negative charges of the same magnitude is obtained by placing the charges at the centre of squares of side a.
Calculate the potential energy of charge A in the electric field created by the others charges for the finite grid (see the attached image).
The Attempt at a Solution
So, we know that the distance between two charges is going to be 2a, as described in the problem. So then, if we work out this equation:
according to our problem, we get:
and so on, alternating the signs.
And at this point, I don't know what should I do, it looks rather like a geometrical problem involving a finite series (i.e. converge). It looks like most of the distances are going to cancel, but it's not trivially seen how they cancel, because it's not that we're going to sum the 48 different distances anyway, neither that it looks to give 0 as a result. I guess it should have something to do with Pythagoras' theorem, but I don't see how to apply it here. Any help would be appreciated.
Note: For some reason the equations are shown up as if I used the slash /, but in fact I used \, I don't know why the LaTeX is not working... Hope you can see the equations correctly.