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Unformatted text preview: 1 PHYS809 Class 21 Notes Boundary conditions at the interface between two dielectric media Let n 21 be the unit normal to the interface with direction from medium 1 into medium 2. Since ∇× = E on both sides of the interface, the tangential component of electric field is continuous across the interface, ( ) 2 1 21 0. × = E E n (21.1) By considering a Gaussian pillbox, the normal components of electric displacement are related by ( ) 2 1 21 , σ ⋅ = D D n (21.2) where σ is the density of free charges on the interface (i.e. not including polarization charges). For a chargefree surface, the normal component of the electric displacement is continuous at the interface. Boundary value problems in dielectrics Consider a point charge q in vacuum a distance d from a plane interface with a semiinfinite dielectric of dielectric constant . ε The equations to be solved are ( ) ( ) ( ) in 0, 0 in 0, 0 everywhere. q z d x y z z ε δ δ δ ε ∇⋅ = > ∇⋅ = < ∇× = E E E (21.3) Due to azimuthal symmetry about the zaxis, we can restrict attention to the xz – plane. To find the potential for z > 0 (which will be region 2), we try placing an image charge q ′ at z =  d . The potential in region 2 is then ( ) ( ) 2 2 2 2 2 1 . 4 q q x d z x d z πε ′ Φ = + + + + (21.4) Since there are no charges in region 1 ( z < 0 ), the potential is a solution of Laplace’s equation there. If ), the potential is a solution of Laplace’s equation there....
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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.
 Fall '11
 MacDonald

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