Electric field in the overlap of two solid, uniformly charged spheres

[KaiserBrandon]

Homework Statement

Two spheres, each of radius R and carrying uniform charge densities +\rho
and -\rho, respectively, are placed so that they partially overlap.
Call the vector from the positive centre to the negative centre \vec{d}. Show
that the field in the region of overlap is constant and find its value. Use
Gauss’s law to find the electric field inside a uniformly charged sphere
first.

Homework Equations

law of superposition
Gauss Law

The Attempt at a Solution

I found the field inside one sphere to be
(r\rho)/(3\epsilon)
in the radial direction. Now for the overlapping spheres, I said that the vector from the centre of the positive sphere to some point P in the interlapping area is \vec{r}. And from P to the centre of the negative sphere, I denoted \vec{r'}. so \vec{r'}=\vec{d}-\vec{r}. So in order for P to be inside the spheres, |\vec{r}|<R and |\vec{d}-\vec{r}|<R. So using the law of superposition, inside the overlap, the electric is
E = (|\vec{r}|-|\vec{d}-\vec{r}|)\rho/3\epsilon
in the radial direction, with the boundaries in effect. Now I am stumped here, as I'm unsure how to reduce this to a constant. Any suggestions?

 

[granpa]

the electric field is a vector so why on Earth are you reducing r and d-r to scalars?

 

[KaiserBrandon]

yep, realized my mistake while sitting in my thermodynamics class this morning. It's funny how I usually figure stuff out while I'm not actually trying to do the question.

 

[KaiserBrandon]

k, so I changed the E function to Cartesian coordinates. So in the overlap I got:

\vec{E}=\frac{\rho*d}{3*\epsilon}*\hat{i}

where d is the magnitude of \vec{d}

And this is under the condition that \vec{d} runs along the x axis.

 

[granpa]

sometimes you just need to sleep on it and get a fresh perpective on it in the morning