Guess what the equipotential lines u(r, ) = const in Probs. 5 and 7 may look like. Then graph some of them, using partial sums

Guess what the equipotential lines u(r, θ) = const in Probs. 5 and 7 may look like. Then graph some of them, using partial sums of the series.

Data from Prob. 5

The electrostatic potential satisfies Laplace’s equation ∇²u = 0 in any region free of charges. Also the heat equation ut = c² ∇²u (Sec. 12.5) reduces to Laplace’s equation if the temperature u is time-independent (“steady-state case”). Using (20), find the potential (equivalently: the steady-state temperature) in the disk r < 1 if the boundary values are (sketch them, to see what is going on).

u (1, θ) = 220 if -1/2π < θ < 1/2π and 0 otherwise