BOUNDARY VALUE PROBLEMS WITH LINEAR DIELECTRICS

We have shown that the bound volume charge density bound is proportional to the free charge volume density

What happens at the boundary / interface between two Linear dielectrics?

We have the boundary condition

boundary condition

for linear dielectrics

And (eq. 2.34)

Example Consider a hemispherical linear dielectric of radius R placed in between two infinite conducting parallel plates

far away from the hemisphere (r>>R)

We want to know/determine the following quantities:

Inside the dielectric (r < R):

Outside the dielectric (r > R):

Since there is NO volume free charge density inside the dielectric, therefore

However, at r=R:i.e. a bound surface charge density will exist on/at the surface of the hemispherical dielectric.

1. Since , then

2. Note also that this problem has azimuthal / axial symmetry, therefore V, E, D, P have NO ϕ

-dependence

Therefore, the general solution can be represented in terms of Legendré polynomials

boundary conditions

since

Now, we can solve this problem directly.

Example 4.7

A dielectric sphere is placed in a Uniform electric field, find theelectric field inside the sphere.

Boundary conditions:

Example 4.7 (conti.)

Solution of Laplace’s equation

BC3

Therefore

Example 4.7 (conti.)

BC1

BC2

Example 4.7 (conti.)

Therefore

Energy in dielectric systemsAs ρf is increased by an amount Δρf, the work done is

Since

integrating by parts

By divergence theorem, vanishes as →∞

Energy in dielectric systems (conti.)

Therefore

For linear dielectric material

compare

Forces on dielectrics

wL

x

dielectric

Assume Q=constant

Forces on dielectrics (conti.)Therefore

In this case, ( Check it by yourself)