# Potential inside sphere with empty cavity

## Homework Statement

An insulating sphere of radius R , centered at point A, has uniform chagre density ρ. A spherical cavity of radius R / 2 , centered at point C, is then cut out and left empty, see Fig.

(a) Find magnitude and direction of the electric field at points A and B.

(b) Find the potential at points A and B. Set V(r → ∞) = 0.

(c) Write down an algebraic solution (no integrals!) for E(r) and V (r) for the space outside the larger sphere, r > R. Choose r = 0 at point A, and the radius vector of point C as r = RC .

## The Attempt at a Solution

(a)First I want to figure out the volume charge distribution, which I wanted to do by finding the the volume of the whole sphere minus the volume of the cavity.
$$\int_V \rho \cdot dVolume = \frac{4}{3}\pi \rho (R^3-\frac{R^3}{8})=\frac{7}{6}\pi R^3 \rho$$

Then to get $E_B$ I wanted to use Gauss' law, but Im not sure how i would set that up because the E field isnt isotropic, so instead Im trying to use the equation for E-field of a volume charge distribution:
$$E(r)=k\int \frac{dq \hat{s}}{s^2}$$
Where i just calculated dq. $s^2=R$, but im not sure what $\hat{s}$ is

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thinking about it though, im pretty sure that what i got for dq is wrong, because i dont think what i did takes into account that the cavity isn't centered at the origin

haruspex