# PHYS 598 hw9 - Physics 498/MMA Handout 9 Oct 10th 2002...

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Physics 498/MMA Handout 9 Oct 10th 2002 Mathematical Methods in Physics I m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Conducting Strip : A thin insulated conducting strip of width 2 a extends infinitely far in the z direction. It lies in the plane y = 0 between x = - a and x = + a and is held at potential V 0 , The rest of the y = 0 plane consists of two semi-infinite conducting sheets, one occupying all x > a , and one all x < - a . Both these sheets are grounded i.e. held at potential V = 0. a) Represent the potential V ( x ) in the y = 0 plane as a Fourier integral V ( x ) = -∞ dk 2 π A ( k ) e ikx , and explicitly compute A ( k ). b) There is no charge or conductors elsewhere, so the electrostatic potential in the rest of three dimensional space satisfies Laplace’s equation with the boundary condition V 0 as | y | → ∞ . Write down the Fourier integral giving the function V ( x, y, z ) at all points in space. c) Compute the electric field E = -∇ V near y = 0 on either side of the conducting planes, and hence find the Fourier integral which gives the charge distribution σ ( x ) on the three conducting sheets.

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