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F07_Practice_Midterm

# F07_Practice_Midterm - Physics 505 Practice Midterm Fall...

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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 Green’s function may be expanded as G ( x, y, z ; x , y , z ) = m =1 n =1 g mn ( z, z ) sin mπx a sin mπx a sin nπy b sin nπy b Write down the appropriate differential equation that g mn ( z, z ) must satisfy. b ) Solve the Green’s 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 Φ( 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 Φ( x, y, z ) = m =1 n =1 A mn sin mπx a sin nπy b sinh γ mn z where γ mn = π ( m/a ) 2 + ( n/b ) 2 and A mn = 4 ab sinh γ mn c a 0 dx b 0 dy V ( x, y ) sin mπx a sin nπy b Over -→

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2. The potential on the surface of a sphere of radius
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F07_Practice_Midterm - Physics 505 Practice Midterm Fall...

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