Class_8new - 1 PHYS809 Class 8 Notes The Dirichlet Green function for a sphere The Dirichlet Green function for a sphere is simply the solution

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Unformatted text preview: 1 PHYS809 Class 8 Notes The Dirichlet Green function for a sphere The Dirichlet Green function for a sphere is simply the solution found earlier for a point charge in the presence of a grounded sphere. For the exterior problem, unit charge is placed at ′ x outside the sphere of radius a . Adding the contribution to the potential from the image charge, we find the Green function to be ( ) ( ) 1 2 1 2 2 2 2 2 2 2 2 2 1 1 1 , , 2 cos 2 cos D a G a x x x x xx x a xx x a γ γ ′ =- =- ′- ′ ′ ′ +- ′ ′- ′ +- ′ x x x x x x (1.1) where γ is the angle between the vectors ′ x and . x Remembering that the unit vector n points out of the volume of interest, which, in this case, is towards the center of the sphere, we have ( ) ( ) 2 2 3 2 2 2 . 2 cos D D x a x a G x a G x a x a xa γ ′= ′= ∂- ′ ⋅∇ = - = - ′ ∂ +- n (1.2) Hence given the potential on the sphere, the solution for the potential outside the sphere, in the absence of any charge other than that on the sphere, is ( ) ( )( ) ( ) ( ) ( ) ( ) 2 2 2 3 2 2 2 0 0 1 , 4 , , sin . 4 2 cos D S G dS x a a a d d x a xa π π φ φ π φ θ φ θ θ φ π γ ′ ′ ′ ′ = - ⋅∇- ′ ′ ′ ′ ′ = +- ∫ ∫ ∫ x x n x x (1.3) Note that cos γ is given in terms of the coordinates by ( ) cos cos cos sin sin cos ....
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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_8new - 1 PHYS809 Class 8 Notes The Dirichlet Green function for a sphere The Dirichlet Green function for a sphere is simply the solution

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