Class_21new - 1 PHYS809 Class 21 Notes Boundary conditions...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 ( ) ( ) ( ) 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 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....
View Full Document

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.

Page1 / 5

Class_21new - 1 PHYS809 Class 21 Notes Boundary conditions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online