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Chapter%207

# Chapter%207

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Dr. Ray Chen© Chapter 7 Poisson’s and Laplace’s Equations

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Dr. Ray Chen© 7.1 Poisson’s and Laplace’s Equations ( ) 0 region s homogeneou a For = = = = −∇ = = −∇ = = = z y x V ) E ( D V E E D D v v ε ε ε ρ ε ε ε ρ v v v v v Equation s Poisson' ε ρ v V = ε ρ v z y x z V y V x V ) z V ( z ) y V ( y ) x V ( x V a ˆ z V a ˆ y V a ˆ x V V z z y y x x = + + = + + = + + = + + = 2 2 2 2 2 2 A A A A v
Dr. Ray Chen© 7.1 Poisson’s and Laplace’s Equations V. of Laplacian the called is Equation. s Laplace' the is which 0 have we 0 If 2 2 = = , V , v ρ true, always is 0 0 If 2 = = V , v ρ l cylindrica spherical, cartesian, as such system, coordinate different in d represente be can 2

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Dr. Ray Chen© 7.1 Poisson’s and Laplace’s Equations 2 2 2 2 2 2 2 is equation s Laplace' , coordinate cartesian In z V y V x V V + + = 2 2 2 2 2 2 1 1 coordinate l cylindrica In z V V V V + + = φ ρ ρ ρ ρ ρ 2 2 2 2 2 2 2 2 1 1 1 coordinate spherical In φ θ θ θ θ θ + + = V sin r V sin sin r r V r r r V
Dr. Ray Chen© 7.2 Uniqueness Theory have we condition, boundary same the under 0 V Equation s Laplace’ the satisfying and functions potential two are there If 2 2 1 = V V 2 1 V V =

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Dr. Ray Chen© 7.2 Uniqueness Theory Proof: ( ) 0 0 0 Equation s Laplace' to According 2 1 2 2 2 1 2 = = = V V V V Each solution must also satisfy the boundary conditions, and if we represent the given potential values on the boundaries, then 0 V V or V V V boundary) the at V boundary) the at 2b 1b 2b 1b 2b 2 1b 1 = = = = = b V ( V ( V Next page