Class_6new - PHYS809 Class 6 Notes Green function methods...

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PHYS809 Class 6 Notes Green function methods in electrostatics Green’s first theorem is 2 , V S S dV d dS φ ψ φ ψ + ∇ ⋅∇ = = ⋅∇ S n (1.1) where and are two scalar fields and S is the bounding surface of a volume V . Also the unit vector n is normal to the surface S and points out of V . This theorem follows from applying the divergence theorem to the vector identity ( ) 2 . ∇⋅ = ∇ + ∇ ⋅∇ (1.2) Interchanging and in Green’s first theorem and subtracting the result from (1.1) gives Green’s second theorem ( ) ( ) 2 2 . V S dV dS - ∇ = ⋅∇ - ⋅∇ n n (1.3) This theorem is useful for finding solutions to Poisson’s equation for the electrostatic potential by choosing an appropriate scalar field . For example, if we take 1 1 , R = = - x x (1.4) where x is the integration variable and x is the observation point, Poisson’s equation 2 0 , ρ ε = - (1.5) is converted into an integral equation for the potential ( ) ( ) ( ) 3 0 1 1 1 4 , V S d x dS R R R πδ - - + = ⋅∇ - ⋅∇ x x x x n n (1.6) where indicates that differentiation is with respect to primed coordinates. Here we have made use of ( ) 2 1 4 , R = - - x x (1.7) i.e. 1 R is the potential at x due to a point charge of strength 0 4 πε placed at . x For a point inside V , ( ) ( ) 3 0 1 1 1 1 , 4 4 V S d x dS R R R + ⋅∇ - ⋅∇ x x n n (1.8) whereas for a point outside V , the left hand side of (1.8) is replaced by zero. [This allows us to interpret the terms in the surface integral as being the potential from a surface charge density 0 , σ = ⋅∇ n and a dipole layer of density 0 . D ε φ = - The discontinuities in potential and electric field then lead to zero potential and electric field outside V .]
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.

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Class_6new - PHYS809 Class 6 Notes Green function methods...

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