Chapter 23, Problem 47E

 (a)

To determine

The potential for the region $r<r_a$.

The potential for the region $r_a<r<r_b$.

The potential for the region $r>r_b$.

(b)

To determine

To show: The potential of the inner sphere with respect to the outer sphere $\left(V_{ab}\right)$ is $\frac{q}{4\pi {\epsilon }_0}\left(\frac{1}{r_a}-\frac{1}{r_b}\right)$.

(c)

To determine

To show: The magnitude of the electric field $E\left(r\right)$ at any point between the sphere is $\frac{V_{ab}}{\left(\frac{1}{r_a}-\frac{1}{r_b}\right)}\left(\frac{1}{r^2}\right)$.

(d)

To determine

The electric field at a point outside the larger sphere at a distance $r$ from the centre, where $r>r_b$.

(e)

To determine

To show: The answers of part (b) and part (c) are the same but answer for part (d) is different, if the charge on the outer sphere is not $-q$ but a negative charge of different magnitude is $-Q$.