Overlapping spheres of charge: finding the E field between them

[jerro]

Homework Statement

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.

Homework Equations

Gauss' Law

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.

 

[mfb]

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.