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

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Unformatted text preview: 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(y). (a)Develop a general formula for the potential within the pipe. (b)Find the potential explicitly for the case of V(y)= V0 (constant). Solution:(a)With the method of separation of variables, the Laplaces equation of 2222yVxVhas the solution of )()(yYxXV. )()()()(222122yYCdyyYdxXCdxxXdwith 21CC. With the boundary condition of V=0 at y=0 and at y=a, we immediately obtain 22sin)(aynCandaynyYn is integer. Thus 21aynC. From V=0 at x=0, the general solution of X is axnxXsinh)(So the general solution is aynaxnAyxVnnsinsinh),(1(b)The coefficient An can be calculated as dyaynybVabnaAansin),(sinh2Then for V(b,y)=V. evennoddnabnnVnabnanaVdyaynabnaVAansinh4cos1sinh2sinsinh22.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. Find the potential inside the box. Solution:With the method of separation of variables, the Laplaces equation of 222222zVyVxVhas the solution of )()()(zZyYxXV. From V=0 at x=0 and x=a, the general solution of X is axnxXsin)(n is integer....
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This note was uploaded on 02/06/2010 for the course PHYSICS 11 taught by Professor Qiu during the Fall '09 term at University of California, Berkeley.

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

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