c251_7 - 1 ELECTROSTATIC BOUNDARY-VALUE PROBLEMS(Chapter 6...

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Unformatted text preview: 1 ELECTROSTATIC BOUNDARY-VALUE PROBLEMS (Chapter 6) ¡ DEAL WITH THE EFFECT OF ELECTRIC CHARGES AT REST ¡ DEAL WITH DETRMINATION OF E , V AND/OR Q THROUGHOUT A REGION WHEN V AND/OR Q ARE SPECIFIED ON THE BOUNDARY V = ? V = v O Dielectric region V is known on the boundary Find V inside 2 POISSON’S & LAPLACE’S EQUATION ∇ 2 is the Laplacian operator = ∇ . ∇ i.e., in Cartesian Coord.: ∇ 2 V = ∂ 2 V/ ∂ x 2 + ∂ 2 V/ ∂ y 2 + ∂ 2 V/ ∂ z 2 V = ? V = V O ρ v , ε 9 Given a closed dielectric region where the potential V is specified on the boundary. Find V everywhere. Special case: LAPLACE’S EQUATION Simple medium with no free charge ( ρ V =0): ∇ 2 V = 0 ¡ For a homogneous & linear medium: ∇ . D = ρ V , E = - ∇ V & D = ε E ∴ ∇ . ( ε E ) = ∇ . (- ε ∇ V ) = ρ V ε ρ v V − = ∇ 2 Poisson’s equation: 3 Uniqueness Theorem STATEMENT : A SOLUTION TO POISSON’S (LAPLACE’S) EQUATION THAT SATISFIES THE GIVEN BOUNDARY CONDITIONS IS A UNIQUE SOLUTION.SOLUTION....
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c251_7 - 1 ELECTROSTATIC BOUNDARY-VALUE PROBLEMS(Chapter 6...

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