Electric field inside a cavity within a uniformly charged sphere problem

[mr.hood]

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.

Homework Equations

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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?

 

[gabbagabbahey]

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 $\vec{S}$