Chapter 6.10, Problem 12P

A cylindrical capacitor consists of two long concentric metal cylinders. If there is a charge of $k$ coulombs per meter on the inside cylinder of radius $R_1,$ and $-k$ coulombs per meter on the outside cylinder of radius $R_2$, find the electric field $E$ between the cylinders. Hint: Use Gauss’s law and the method indicated in Figure 10.7. What is $E$ inside the inner cylinder? Outside the outer cylinder? (Again use Gauss’s law.) Find, either by inspection or by direct integration, the potential $\varphi$ such that $E=-\nabla \varphi$ for each of the three regions above. In each case $E$ is not affected by adding an arbitrary constant to $\varphi .$ Adjust the additive constant to make $\varphi$ a continuous function for all space.