Chapter 13.5, Problem 12P

Separate Laplace’s equation in two dimensions in polar coordinates [equation (5.1) without the $z$ term] and solve the $r$ and $\theta$ equations. (See Problem 11.) Remember that for the $\theta$ equation, only periodic solutions are of interest. Use your results to solve the problem of thesteady-state temperature in a circular plate if the upper semicircular is ${100}^{°}$ and the lower atComment: Another physical problem whose mathematical solution is identical with this temperature problem is this: Find the electrostatic potential inside a capacitor formed by two half-cylinders, insulated from each other and maintained at potentials 0 and 100.