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 , r0 ) =-4 πδ ( r-r0 ) and a sine series, expand a one dimensional delta function in the range (0,a) as below, δ ( x-x0 ) = Σ j [ A j sin jπx/a sin jπx0 /a ] , (3) and ﬁnd the coe±cients A j . This is called the completeness relation for the sine series,
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 12/24/2011 for the course PHYS 5406 taught by Professor Blecher during the Spring '10 term at Virginia Tech.