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## Homework Statement

Consider the electric field of two protons b meters apart.

The potential energy of the system is equal to:

[tex] U = \frac{\epsilon_0}{2}\int {\bf E}^2dv = \int({\bf E}_1+ {\bf E}_2)^2dv [/tex]

[tex] = \frac{\epsilon_0}{2}\int {\bf E}_1^2dv + \frac{\epsilon_0}{2}\int {\bf E}_2^2dv + \epsilon_0 \int{\bf E}_1\cdot {\bf E}_2dv [/tex]

The third integral is not hard to evaluate if you set it up in spherical coordinates with

one proton at the origin and the other along the polar axis (z axis) and perform the integration

over r first. Show that it integrates to

[tex] e^2/4\pi\epsilon_0b [/tex]

**2 The attempt at a solution**

I have been trying to solve this problem for hours, but cannot find an expression for E1 dot E2 in spherical coordinates in a way that would make this integral easy. Any guidance would be appreciated.