Chapter 23, Problem 23.41E

CALC A metal sphere with radius r_a is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius r_b. There is charge +q on the inner sphere and charge -q on the outer spherical shell, (a) Calculate the potential V(r) for (i) r < r_a (ii) r_a < r < r_b; (iii) r > r_b. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take V to be zero when r is infinite, (b) Show that the potential of the inner sphere with respect to the outer is

${\text{V}}_{\text{ab}}\text{ = }\frac{\text{q}}{4\text{π}{\text{ε}}_0}\left(\frac{1}{{\text{r}}_{\text{a}}}-\frac{1}{{\text{r}}_{\text{b}}}\right)$

(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the spheres has magnitude

$\text{E}(\text{r})\text{ = }\frac{V_{\text{ab}}}{(1/{\text{r}}_{\text{a}}-1/{\text{r}}_{\text{b}})}\frac{1}{{\text{r}}^2}$

(d)    Use Eq. (23.23) and the result from part (a) to find the electric field at a point outside the larger sphere at a distance r from the center, where r > r_b (e)    Suppose the charge on the outer sphere is not −q but a negative charge of different magnitude, say −Q. Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different.