Chapter 23, Problem 73P

A nonconducting solid sphere has a uniform volume charge density ρ. Let $\stackrel{\to }{r}$ be the vector from the center of the sphere to a general point P within the sphere. (a) Show that the electric field at P is given by $\stackrel{\to }{E}=\rho \stackrel{\to }{r}/3{\epsilon }_0$. (Note that the result is independent of the radius of the sphere.) (b) A spherical cavity is hollowed out of the sphere, as shown in Fig. 23-60. Using superposition concepts, show that the electric field at all points within the cavity is uniform and equal to $\stackrel{\to }{E}=\rho \stackrel{\to }{a}/3{\epsilon }_0$, where $\stackrel{\to }{a}$ is the position vector from the center of the sphere to the center of the cavity.

Figure 23-60 Problem 73.