Class_21new - PHYS809 Class 21 Notes Boundary conditions at...

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1 PHYS809Class21Notes 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 0 ∇× = 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 charge-free 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 semi-infinite dielectric of dielectric constant . ε The equations to be solved are ( ) ( ) ( ) 0 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 z -axis, 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 0 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 ε
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