Week 6

Basic Facts

Dielectrics

A dielectric material placed in an electric field acquires a polarization which leads to a bound surface charge; a charge tied to the material. The charge σ_b is related to the polarization per unit volume P by

σ_b = |P|.

We describe the total internal field with (or without) a dielectric material present by the Electric Displacement D.

D = ε₀E - P and div D = ρ_f where ρ_f is the Free charge density.

The electric field inside a material with dielectric constant ε is weaker than the field that would be there if the dielectric were absent by the factor ε.

If a capacitor of value C₀ is filled with a dielectric of constant εthen its capacitance increases by factor of ε.

At a boundary between two materials with different dielectric properties
1) The component of D perpendicular to the surface is the same on both sides of the boundary
2) The component of E parallel to the surface is the same on both sides of the boundary.

We often express this by saying that the perpendicular component of D is continuous across the boundary and the parallel component of E is continuous.

If both materials are Linear and Isotropic with dielectric constants ε₁ and ε₂ then we have
ε₁ E₁•n = ε₂ E₂•n where n is the normal to the surface.

If the electric field is not uniform then there may arise a bound volume charge. The bound volume charge density ρ_b is related to the polarization per unit volume P by
ρ_b=-∇·P=-divP

Gauss's Law in differential form : The amount of electric flux generated at each point in space is related to the charge density ρ at that point by the equation

where

and

so that