Homework_3 - Homework 3 solutions 1.3 (a) The charge...

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Unformatted text preview: Homework 3 solutions 1.3 (a) The charge density will be of form ( ) , k r R - where k is a constant. Integrating over a spherical volume with radius greater than R , we find that ( ) 2 4 . k Q R = Hence ( ) ( ) 2 . 4 Q r R R =- x (b) Similarly to part (a) by integrating over a cylinder of unit length and radius greater than b , we find ( ) ( ) . 2 r b b =- x (c) Assume the disk is centered on the origin of coordinates and lies in the z = 0 plane. For , r R < the surface charge density is ( ) 2 . Q R = The charge density is zero everywhere else. By integrating the charge density over the volume between two concentric cylinders of radii r and r r + where r < R , we find ( ) ( ) 2 , , . 0, . Q z r R R r R < = > x (d) For , r R < the charge density will be proportional to ( ) 2 . - By integrating the charge density over the volume between two spheres of radii r and r r + where r < R , we find ( ) 2 , , . 2 0, . Q r R R r r R - < = > x 2.3 (a) The line charge will have 3 image line charges. Images of charge density - need to be placed at ( ) , x y- and ( ) , , x y- and an image of charge density needs to be placed at ( ) , . x y-- The potential from the 4 line charges is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 , ln . 4 x x y y x x y y x y x x y y x x y y + +-- + + = - +- + + + The potential on y = 0 is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 ,0 ln 0. 4 x x y x x y x x x y x x y + + -- + = = - + - + + Similarly ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 0, ln 0. 4 x y y x y y y x y y x y y +-- + + = = - +- + + We conclude that the potential is zero on the conducting surfaces. The electric field has components ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ln 4 2 2 2 2 , 4 ln 4 x y x x y y x x y y E x x x y y x x y y x x x x x x x x x x y y x x y y x x y y x x y y x x y y x x y y E y x x y y x x + +-- + + = - - +- + + + +-- + = -- +- + +-- +-- + + + + + + +-- + + = - - +- + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 . 4 y y y y y y y y y y x x y y x x y y x x y y x x y y + + -- + + = -- +- + +-- +-- + + + + + On y = 0, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 0....
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.

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Homework_3 - Homework 3 solutions 1.3 (a) The charge...

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