The Macroscopic Electric Field Inside a Dielectric

What’s the electric field inside matter on the microscopic level?

Suppose we want to calculate the macroscopic electric field inside a solid dielectric sphere of radius R

Outside of the region of this small imaginary sphere, (electric dipoles are far enough away from the field point)

The average electric field inside a sphere of radius δ( too close to the field point)

(from problem 3.41)

Because of the size of the imaginary sphere of radius δ<<R,electric polarization P(r) should not vary significantly over this small volume

a uniformly polarized dielectric sphere (Example 4.2)

~ this is precisely what Ein puts back in!

for the entire dielectric

For an "ideal", linear, homogeneous & isotropic dielectric

Define

χe is a scalar quantity – it is dimensionless

Total electric permittivity

Relative electric permittivity (or dielectric constant)

THE MACROSCOPIC ELECTRIC DISPLACEMENT FIELD

the effect of polarization of a dielectric is to produce bound surface andvolume charge densities

Suppose this dielectric also had embedded in it free electric charges

The TOTAL volume electric charge

Then Gauss’ Law (in differential form) becomes

(macroscopic) Electric Displacement Field:

Then we realize that Gauss’ Law (for dielectrics) becomes:

(differential form)

(integral form)

Summary:

Example 4.4

Gaussian surface

A (very) long, straight conducting wire carries a uniform, free line electric charge λ which is surrounded by rubber insulation out to radius, a. Find the electric displacement D(r)

BOUNDARY CONDITIONS ON THE ELECTRIC DISPLACEMENT D

BOUNDARY CONDITIONS ON THE ELECTRIC FIELD E

BOUNDARY CONDITIONS ON THE ELECTRIC POLARIZATIONΡ

since

Since

RELATIONSHIPS BETWEEN FREE & BOUND VOLUME CHARGE DENSITIES

Example 4.5A metal sphere of radius a carriesa charge Q, surrounding, out to radiusb, by linear dielectric material ε. Find the potential at the center .

Symmetry → find D by Gauss’s law

Note that inside the metal sphere,

Example 4.5 (conti.)Therefore, potential at the center

Polarization

Bound volume charge

Bound surface charge

@outersurface

@innersurface

negative