Potential energy of two protons

[Opus_723]

Homework Statement

Consider the electric field of two protons, b cm apart. The potential energy of the system ought to be given by 1/4[itex]\pi[/itex][itex]\int[/itex]E[itex]_{1}[/itex][itex]\cdot[/itex]E[itex]_{2}[/itex]dv

Set up the integral in some appropriate coordinates, and show, without actually evaluating it, that it must be expressible as [itex]\frac{Ce^{2}}{b}[/itex], where C is a purely numerical constant, the value of a definite integral involving dimensionless quantities only.

The Attempt at a Solution

It's obvious that the integral will end up expressible as [itex]\frac{Ce^{2}}{b}[/itex], since that's just the potential of one proton times the charge of the other. It's setting up the integral that's troubling me.

I don't see how to set up the integral over all of space so that it will converge. The electric field grows infinitely strong near either of the two point charges, so it shouldn't converge, right? I've only ever used this integral for potential energy with spheres and such, where the field doesn't go to infinity, so this is new to me.

 

[jbsteele]

I suppose it is possible to set up the integral over a sphere with radius b, centered on one of the protons, and then take the limit as b goes to infinity. However, I'm not sure how to show that this yields an expression of the form \frac{Ce^{2}}{b}, where C is a numerical constant.Any help would be appreciated!