12. Separate Laplace's equation in two dimensions in polar coordinates [equation (5.1) without the z term] and solve the r and 0 equations. (See Problem 11.) Remember that for the 0 equation, only periodic solutions are of interest. Use your results to solve the problem of the steady-state temperature in a circular plate if the upper semicircular boundary is held at 100° and the lower at 0°. u = 100 u=0 Comment: 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. n sin ne 200 Answer: u=50+4 π Σ()" odd n n