lec14-1 - Poisson equation The Poisson equation can be...

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Poisson equation The Poisson equation can be written, 2 u ( ~ r ) = ρ ( ~ r ) For example, in two dimensions, u ( x , y ) and ρ ( x , y ) ± 2 x 2 + 2 y 2 ² u ( x , y ) = ρ ( x , y ) For the case of electrostatics, using Gaussian units we have 2 φ ( ~ r ) = - 4 πρ ( ~ r ) For a point charge, 2 φ ( ~ r ) = - 4 πδ ( ~ r - ~ r 0 ), and we find φ ( ~ r ) = 1 | ~ r - ~ r 0 | The function G ( ~ r ,~ r 0 ) = 1 | ~ r - ~ r 0 | is the Green function for the Poisson equation
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Green function for the Poisson equation The Green function G ( ~ r ,~ r 0 ) = 1 | ~ r - ~ r 0 | solves the Poisson equation for a point source, 2 G ( ~ r ,~ r 0 ) = - 4 πδ ( ~ r - ~ r 0 ) Now we notice that ρ ( ~ r ) = R R R ρ ( ~ r 0 ) δ ( ~ r - ~ r 0 ) d τ , and then with the volume integral extending over the region of nonzero charge density, φ
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.

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lec14-1 - Poisson equation The Poisson equation can be...

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