Chapter 22, Problem 22.57P

(a) An insulating sphere with radius a has a uniform charge density ρ. The sphere is not centered at the origin but at $\stackrel{\to }{\text{r}}=\stackrel{\to }{\text{b}}$. Show that the electric field inside the sphere is given by $\stackrel{\to }{\text{E}}\text{ = }\rho (\stackrel{\to }{\text{r}}-\stackrel{\to }{\text{b}})/3{\in }_0.$ (b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a < b < R (a cross section of the sphere is shown in Fig. P22.57). The solid part of the sphere has a uniform volume charge density ρ. Find the magnitude and direction of the electric field $\stackrel{\to }{\text{E}}$ inside the hole, and show that $\stackrel{\to }{\text{E}}$ is uniform over the entire hole. [Hint: Use the principle of superposition and the result of part (a).]

Figure P22.57