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).