Electric field through the intersecting part of two spheres

[kopinator]

Homework Statement

Consider a pair of spheres of radius R with uniform charge densities ρ > 0 and −ρ located respectively at ~r+ = (0, 0, d/2) and ~r− = (0, 0, −d/2), with d < R.
a)Find the electric field at all points in the region of overlap of the spheres for arbitrary d < R.

Homework Equations

Coulomb's law
Gauss's law (integral and differential form) ∇E = (1/∈_0)ρ is the differential form.

The Attempt at a Solution

I haven't made it very far into the problem. I'm still trying to set it up. I believe at some point I will have to integrate from -d/2 to d/2, but that can't happen until I set up the proper equation. Now, I have to find the flux through the overlapping region while there is a uniform charge density, ρ and -ρ. One idea I had was to integrate the divergence of the electric field over the volume, each sphere. This would put me into a triple integral in spherical coordinates and I want to say integrate R^2*dr from -d/2 to d/2 and proceed with integrating sinΘdΘ from 0 to π and dΦ from 0 to 2π.

P.S. I kind of thought all of this out while I was posting.

 

[kopinator]

My final answer is (πρd^3)/(3∈_o). This answer seems too simple though. Maybe this is only part of the answer?

 

[TSny]

My final answer is (πρd^3)/(3∈_o).

Your answer does not have the right dimensions for electric field. Also, you'll need to specify the direction of the field.

Use the principle of superposition for electric field. The net field at a point is the vector sum of the fields from each sphere alone.