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HW-6-solution

# HW-6-solution - HW-6 1 A rectangular pipe running parallel...

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HW-6 1. A rectangular pipe, running parallel to the z axis (from - ∞ to +∞), has three ground metal sides at y=0, y=a, and x=0. The fourth side, at x=b, is maintained at a specific potential V 0 (y). (a) Develop a general formula for the potential within the pipe. (b) Find the potential explicitly for the case of V 0 (y)= V 0 (constant). Solution: (a) With the method of separation of variables, the Laplace’s equation of 0 2 2 2 2 y V x V has the solution of ) ( ) ( y Y x X V . 0 ) ( ) ( 0 ) ( ) ( 2 2 2 1 2 2 y Y C dy y Y d x X C dx x X d with 0 2 1 C C . With the boundary condition of V=0 at y=0 and at y=a, we immediately obtain 2 2 sin ) ( a y n C and a y n y Y n is integer. Thus 2 1 a y n C . From V=0 at x=0, the general solution of X is a x n x X sinh ) ( So the general solution is a y n a x n A y x V n n sin sinh ) , ( 1 (b) The coefficient An can be calculated as dy a y n y b V a b n a A a n sin ) , ( sinh 2 0 Then for V(b,y)=V 0 .

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even n odd n a b n n V n a b n an aV dy a y n a b n a V A a n 0 sinh 4 cos 1 sinh 2 sin sinh 2 0 0 0 0 2. A cubical box (side length of a) consists of five metal plates (x=0, x=a, y=0, y=a, and z=0), which are welded together and grounded. The top (z=a) is made of a separate sheet of metal, insulated from the others, and held at a constant potential V 0 . Find the potential inside the box. Solution: With the method of separation of variables, the Laplace’s equation o f 0 2 2 2 2 2 2 z V y V x V has the solution of ) ( ) ( ) ( z Z y Y x X V . From V=0 at x=0 and x=a, the general solution of X is a x n x X sin ) ( n is integer.
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HW-6-solution - HW-6 1 A rectangular pipe running parallel...

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