Electric field through the intersecting part of two spheres

1. Sep 15, 2015

kopinator

1. The problem statement, all variables and given/known data
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.

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

3. 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.

2. Sep 15, 2015

kopinator

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

3. Sep 15, 2015

TSny

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.

Last edited: Sep 16, 2015