For the infinite rectangular pipe in Ex. 3.4, suppose the potential on the bottom (y = 0...

For the infinite rectangular pipe in Ex. 3.4, suppose the potential on the bottom (y = 0) and the two sides (x = ±b) is zero, but the potential on the top (y = a) is a nonzero constant V₀. Find the potential inside the pipe. [Note: This is a rotated version of Prob. 3.15(b), but set it up as in Ex. 3.4, using sinusoidal functions in y and hyperbolics in x. It is an unusual case in which k = 0 must be included. Begin by finding the general solution to Eq. 3.26 when k = 0.]²⁶

Reference prob 3.15

rectangular pipe, running parallel to the z-axis (from −∞ to + ∞), has three grounded metal sides, at y = 0, y = a, and x = 0. The fourth side, at x = b, is maintained at a specified potential V₀(y).

(a) Develop a general formula for the potential inside the pipe.

(b) Find the potential explicitly, for the case V0(y) = V₀ (a constant).

Reference equation 3.26