F07_Practice_Midterm - Physics 505 Fall 2007 Practice...

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Unformatted text preview: Physics 505 Fall 2007 Practice Midterm The midterm will be a two hour open book, open notes exam. Do all three problems. 1. A rectangular box has sides of lengths a , b and c c x y z a b a ) For the Dirichlet problem in the interior of the box, the Greens function may be expanded as G ( x,y,z ; x ,y ,z ) = X m =1 X n =1 g mn ( z,z ) sin mx a sin mx a sin ny b sin ny b Write down the appropriate differential equation that g mn ( z,z ) must satisfy. b ) Solve the Greens function equation for g mn ( z,z ) subject to Dirichlet boundary conditions and write down the result for G ( x,y,z ; x ,y ,z ). c ) Consider the boundary value problem where the potential on top of the box is ( x,y,c ) = V ( x,y ) while the potential on the other five sides vanish. Using the Greens function obtained above, show that the potential may be written as ( x,y,z ) = X m =1 X n =1 A mn sin mx a sin ny b sinh mn z where mn = p ( m/a ) 2 + ( n/b ) 2 and A mn = 4 ab...
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This note was uploaded on 04/11/2010 for the course PHYSICS PHYS taught by Professor Tony during the Spring '07 term at University of Michigan-Dearborn.

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F07_Practice_Midterm - Physics 505 Fall 2007 Practice...

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