hw3_p5406_s03 - HW 3 Phys5406 S03 due 2/10/03 1)Recall the...

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HW 3 Phys5406 S03 due 2/10/03 1)Recall the solution for the potential, Φ( r ), inside a conducting box of sides (a,b,c) in the (x,y,z) directions when all sides are grounded except for the side at z = c that is maintained at constant potential V, Φ( r ) = 16 V π 2 Σ j,k =1 , 3 , 5 ... [ sin jπx/a j sin kπy/b k sinh γ jk z sinh γ jk c ] , (1) where γ jk = [( jπ/a ) 2 +( kπ/b ) 2 ] 1 / 2 . We will use this solution to get the Green’s function G D for all problems of this type. a) First expand the sinh function in the range (0,c) in a sine series, sinh γ jk z sinh γ jk c = Σ m [ A m sin mπz/c ] , (2) and using the orthogonality relation for such a series obtain the coe±cients A m . b) To see the connection between 0 2 G D ( r , r 0 ) = - 4 πδ ( r - r 0 ) and a sine series, expand a one dimensional delta function in the range (0,a) as below, δ ( x - x 0 ) = Σ j [ A j sin jπx/a sin jπx 0 /a ] , (3) and find the coe±cients A j . This is called the completeness relation for the sine series,
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