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Question: 5 a rectangular plate 0 s x a o s...

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5. A rectangular plate 0 S x a, O S y < b has a steady temperature T(x, y) satisfying the Laplace equation The plate is kept at a temperature of T = 100 at the edge at x = a, and at a temperature of T = 0 at all other edges. (a) Use separation of variables, T(x,y) = X(x)Y(y) to show that with a suitable separation constant λ. Find the boundary conditions for X and Y [8 marks] (b) You are given that the solutions of the separated equations can be written as X(x) Acosh(kx) + B sinh(kx), Y(y) C, cos(ky) + D sin(ky). Determine A, C and k, such that these solutions satisfy the boundary conditions at the edges with T = 0. Hint: You can use that cosh(0) = 1 and sinh(0) = 0. [4 marks] (c) Write down the general solution for T(x, y), and determine any remaining arbitrary constants Hint: You can use the fact that the function f(x) = 1 for 0 by the Fourier series x 1 can be represented where b 4/(nza) if n is odd, and b,-0 if n is even. [8 marks]

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