325_Sp2011_8_Laplace's_and_Poisson's_equations

# 325_Sp2011_8_Laplace's_and_Poisson's_equations - 8 Laplaces...

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 1 8. Laplace’s and Poisson’s equations EE325 Mikhail Belkin

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 2 What we have so far Gauss’s Law (in differential form) we also have the electric field as the gradient of potential so combining v D ρ ∇• = r r E V = -∇ r r D E ε = r r ( 29 ( 29 ( 29 v E V = ∇• -∇ = r r r r v V ∇•∇ = - r r if ε is constant wrt space
© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 3 The Laplacian operator we’ve written things symbolically, but what are the actual derivatives? let’s try things in rectangular coordinates 2 V V ∇•∇ = ∇ r r ˆ ˆ ˆ ˆ ˆ ˆ V V V V x y z x y z V x y z x y z = + + = + + ÷  ÷ r ˆ ˆ ˆ grad x y z x y z = ∇ = + + r ( 29 y x z F F F divergence F F x y z = ∇ • = + + r r r { { { 2 ˆ ˆ ˆ ˆ ˆ ˆ x z y F F F V V V V V V V V x y z x y z x y z x y z ÷ = ∇•∇ = ∇• + + + + ÷ ÷ ÷ ÷ r r r r r 2 2 2 2 2 2 V V V V V V x x y y z z x y z = + + = + + ÷ ÷ ÷

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 4 Laplace and Poisson’s Equations for a uniform dielectric , the relation between the potential and the charge density is given by Poisson’s equation if the charge density is zero we get Laplace’s equation (a special case of Poisson) note we can now change our method of solution from an integral approach to one involving the solution of a differential equation for the potential: once we have V(r ), we have everything since we can get E from V 2 v V V ρ ε ∇•∇ = ∇ = - r r 2 0 V = 2 2 2 2 2 2 2 V V V V x y z = + + if ε is constant wrt space
© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 5 various coordinates rectangular cylindrical: spherical: ( 29 ( 29 ( 29 1 1 z F F F F z ρ φ = + + r r g ( 29 ( 29 ( 29 2 2 sin 1 1 1 sin sin r r F F F F r r r r θ = + + r r g 1 ˆ ˆ ˆ V V V V z z = + + r 2 2 2 2 2 2 2 V V V V x y z = + + ˆ ˆ ˆ V V V V x y z x y z = + + r 1 1 ˆ ˆ ˆ sin V V V V r r r r = + + r y x z F F F F x y z ∇• = + + r r 2 2 2 2 2 2 1 1 V V V V z = + + ÷ 2 2 2 2 2 2 2 2 1 1 1 sin

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325_Sp2011_8_Laplace's_and_Poisson's_equations - 8 Laplaces...

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