Chapter 23, Problem 23.61P

CALC Coaxial Cylinders. A long metal cylinder with radius, a is supported on an insulating stand on the axis of a long, hollow, metal tube with radius b. The positive charge per unit length on the inner cylinder is λ, and there is an equal negative charge per unit length on the outer cylinder, (a) Calculate the potential V(r) for (i) r < a; (ii) a < r < b; (iii) r > b. (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take V = 0 at r = b. (b) Show that the potential of the inner cylinder with respect to the outer is

${\text{V}}_{\text{ab}}\text{ = }\frac{\lambda }{2\text{π}{ϵ}_0}\ln \frac{\text{b}}{\text{a}}$

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

$\text{E}(\text{r})\text{ = }\frac{{\text{V}}_{\text{ab}}}{\ln (\text{b}/\text{a})}\frac{1}{\text{r}}$

(d) What is the potential difference between the two cylinders the outer cylinder has no net charge?