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# Volume Charge Density and Non-Uniform Polarization

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Obtain an expression for the volume charge density ρ(r) associated with a non-uniform polarization P(r) in a dielectric material.

The polarization in a spherical region of radius a is:
P(r) = (P0)r(a-r)R, where P0 is a constant and R is a unit vector in the outward direction from the center of the sphere.

The sphere has no free charge and is placed in a region in which there is no externally applied electric field. Find the surface and volume charge densities on and within the sphere. Deduce the electric field E(r) within the sphere (in spherical coordinates div(F) = (1/(r^2))(d/dr)((r^2)Fr) + derivatives with respect to theta and phi).

https://brainmass.com/physics/polarisation/volume-charge-density-non-uniform-polarization-180943

## SOLUTION This solution is FREE courtesy of BrainMass!

Please refer to the attached files. The solution is provided in a Microsoft Word document and a pdf. file.

The polarization can be related to the volume density by:

The polarization volume density is equal to the gradient with respect to r of your polarization expression (P0)r(a-r)R which is equal to (P0)(aR-2raR), therefore the volume density is equal to:

ρ = - (P0)(aR-2raR)

On the other hand the surface polarization charge density is given by the following expression:

The polarization surface density is equal to the multiplication of -n (index of refraction) times your polarization expression (P0)r(a-r)R which is equal to - n (P0)r(a-r), therefore your surface density is given by:

σ = - n (P0)r(a-r)

In order to find the electric field just apply Gauss's Law, which is given by:

Therefore, the electric field is equal to:

E = - (P0)(aR-2raR) divided by ε

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