F07_hw03

# F07_hw03 - Physics 505 Fall 2007 Homework Assignment#3 Due...

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Physics 505 Fall 2007 Homework Assignment #3 — Due Thursday, September 27 Textbook problems: Ch. 2: 2.14, 2.15, 2.22, 2.23 2.14 A variant of the preceeding two-dimensional problem is a long hollow conducting cylinder of radius b that is divided into equal quarters, alternate segments being held at potential + V and - V . a ) Solve by means of the series solution (2.71) and show that the potential inside the cylinder is Φ( ρ,φ ) = 4 V π X n =0 ± ρ b ² 4 n +2 sin[(4 n + 2) φ ] 2 n + 1 b ) Sum the series and show that Φ( ρ,φ ) = 2 V π tan - 1 ³ 2 ρ 2 b 2 sin2 φ b 4 - ρ 4 ´ c ) Sketch the ﬁeld lines and equipotentials. 2.15 a ) Show that the Green function G ( x,y ; x 0 ,y 0 ) appropriate for Dirichlet boundary conditions for a square two-dimensional region, 0 x 1, 0 y 1, has an expansion G ( x,y ; x 0 ,y 0 ) = 2 X n =1 g n ( y,y 0 )sin( nπx )sin( nπx 0 ) where g n ( y,y 0 ) satisﬁes ³ 2 ∂y 0 2 - n 2 π 2 ´ g n ( y,y 0 ) = - 4 πδ ( y 0 - y ) and g n ( y, 0) = g n ( y, 1) = 0 b

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F07_hw03 - Physics 505 Fall 2007 Homework Assignment#3 Due...

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