At first, I thought I could just use Coulomb's law to find the force of attraction between the spheres. But then I was told that since the sphere has a uniform charge density, it isn't the same as Coulomb's law between two point charges with the same charge as the spheres. Hence, I tried using...
I believe that Gauss's law would be another way to find that electric field due to the image charge (well technically the conducting plate), but the problem is still that it seems to come to an integral that doesn't seem doable.
I tried to find the the Electric field due to the image charge. So the potential due to the image charge is V=-(pR^2)/√(4R^2-4rRcos(θ)+r^2). When I took the gradient of that in spherical coordinates, I got a mess that doesn't seem to be possible to integrate.