The steady-state temperature φ(r) between the two concentric circular conducting cylinders...

The steady-state temperature φ(r) between the two concentric circular conducting cylinders shown in FIGURE satisfies Laplace’s equation in polar coordinates in the form

(a) Show that a solution of the differential equation subject to the boundary conditions φ(a) = k0 and φ(b) = k1, where k0 and k1 are constant potentials, is given by φ(r) = A loge r + B, where

[Hint: The differential equation is known as a Cauchy-Euler equation.]

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(b) Find the complex potential Ω(z).

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(c) Sketch the isotherms and the lines of heat flux.

FIGURE Figure for Problem