# Overlapping spheres of charge: finding the E field between them

1. Jan 29, 2013

### jerro

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

You have two spheres. The first is centered at the origin and as uniform positive charge density ρ and radius R. The second is shifted up a distance d, and it has uniform negative charge density -ρ and radius R.

Find the E field in the region of overlap.

2. Relevant equations

Gauss' Law

3. The attempt at a solution

I first found an expression for the E field inside a single sphere.

∫E dA = $\frac{Q}{\epsilon}$
E(4$\pi$*$r^{2}$) = $\frac{\rho*(4/3)\pi*r^{3}}{\epsilon}$
E=$\frac{\rho*r}{3\epsilon}$

Now, for extending the case to include both spheres.

I add the E field from one to the E field of the other, giving $\frac{\rho*r}{3\epsilon}$ - $\frac{\rho*r}{3\epsilon}$, which gives zero.

I'm not sure that this is correct, I'm feeling weary of r, the radius of the Gaussian surface, and whether it is the same for both spheres.

2. Jan 29, 2013

### Staff: Mentor

r is the distance to the center of the sphere, and that is different for those spheres.
In addition, both E and r are vectors.