# Electric field inside a cavity within a uniformly charged sphere problem

## Homework Statement

A uniformly charged sphere (center O1) with radius a and charge +Q has an off-center cavity within it (center O2) with radius b. Show that the electric field within the cavity is uniform and is directed along the line of centers, according to the equation:

$$\vec{E} = \frac{Q}{4\pi\epsilon{o}(a^{3}-b^{3})}\vec{S}$$

where $$\vec{S}$$ is the vector directed from O1 to O2 along the line of centers. HINT: Use the superposition principle.

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## The Attempt at a Solution

I was pretty stumped by this one. My biggest problem here is that the lack of symmetry makes it difficult to think of an appropriate equation/integrand. I thought about dividing the larger sphere along the radius of the smaller sphere to produce two equal hemispheres, but then I still wasn't sure how to find an appropriate integral, because the change in the radius isn't symmetric in any way. Gauss' law is out of the question/irrelevant, too. So where do I even begin?

Start by calculating the charge density $\rho$ within the charged portion of the sphere. What would the electric field $\vec{E}_{bigsphere}$ be if the entire sphere was filled with the charge density $\rho$? What would the field $\vec{E}_{smallsphere}$ due to just the small sphere be if it was filled with a charge density $-\rho$? What does the superposition principle tell you about the sum of these two fields?To make your calculations easier, I recommend you center that small sphere at the origin, so that the bigshphere is centered at $\vec{S}$