2_3 - PHY 5346 HW Set 3 Solutions Kimel 2. 2.3 The system...

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Unformatted text preview: PHY 5346 HW Set 3 Solutions Kimel 2. 2.3 The system is described by a) Given the potenial for a line charge in the problem, we write down the solution from the figure, R2 R2 R2 R2 ln ln ln 2 2 2 0 o o1 o2 o3 2 x x x x x x x x 2 2 2 x x x x x x Looking at the figure when y 0, o o1 , o2 o3 2 , so T |y0 0 x x o 2 o2 2 , o1 2 o3 2 , so T |x0 0 x x x x x x Similarly, when x 0, x x On the surface T 0, so T 0, however, T T T x 0 0 Et 0 xt t xt b) We remember y0 y0 0 T 2 2 y x x0 2 y2 x x0 y0 0 where I've applied the symmetries derived in a). Let y0 y0 1 / 2 2 x x0 2 y2 x x0 y0 0 This is an easy function to plot for various combinations of the position of the original line charge (x 0 , y 0 . T ln c) If we integrate over a strip of width z, we find, where we use the integral x0 0 x x 1 2 y 2 dx 1 2 arctan y0 y0 2 0 0 Q 2 dx tan z 0 and the total charge induced on the plane is , as expected. d) Expanding 4 Q 0 dx z 1 x0 y0 ln R2 R2 R2 ln ln x x0 2 y y0 2 x x0 2 y y0 2 x x0 2 y to lowest non-vanishing order in x 0 , y 0 gives xy 16 2 y x 2 0 0 x y2 so xy 4 y x asym 0 2 2 0 0 x y2 This is the quadrupole contribution. y0 2 ln R2 x x0 2 y y ...
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This note was uploaded on 01/10/2012 for the course PHYSIS 506 taught by Professor Smith during the Spring '08 term at University of Arkansas Community College at Morrilton.

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