- 54

- 0

**1. Homework Statement**

Two spheres, each of radius R and carrying uniform charge densities +[tex]\rho[/tex]

and [tex]-\rho[/tex], respectively, are placed so that they partially overlap.

Call the vector from the positive centre to the negative centre [tex]\vec{d}[/tex]. 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.

**2. Homework Equations**

law of superposition

Gauss Law

**3. The Attempt at a Solution**

I found the field inside one sphere to be

[tex](r\rho)/(3\epsilon)[/tex]

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 [tex]\vec{r}[/tex]. And from P to the centre of the negative sphere, I denoted [tex]\vec{r'}[/tex]. so [tex]\vec{r'}=\vec{d}-\vec{r}[/tex]. So in order for P to be inside the spheres, [tex]|\vec{r}|<R[/tex] and [tex]|\vec{d}-\vec{r}|<R[/tex]. So using the law of superposition, inside the overlap, the electric is

[tex]E = (|\vec{r}|-|\vec{d}-\vec{r}|)\rho/3\epsilon[/tex]

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?