How Do You Calculate Charge and Potential in a Polarized Dielectric Sphere?

[deedsy]

Homework Statement

A dielectric sphere radius R is injected with free charge so that the resultant polarization can be described by $\vec P = \frac{K}{r} \hat r_1$ where $\hat r_1$ is the unit radial vector.
a) Calculate the volume and the surface density of bound charge
b) Calculate the volume density of free charge
c) Calculate the potential inside and outside the sphere

Homework Equations

The Attempt at a Solution

Well, this is the first problem I've ever attempted involving dielectrics, polarization, bound charges, etc...

a) I know that $\rho_b = - \nabla \cdot \vec P = -[\frac{2K}{r^2}-\frac{K}{r^2}] = \frac{-K}{r^2}$
$\sigma_b = \vec P \cdot \hat n$ which is equal to $\frac{K}{R}$

b) Next, they want the volume density of free charge.
I know $\rho_f = \nabla \cdot \vec D$ and you can use $\oint \vec D \cdot d\vec a = Q_{free inside}$ to find the electric displacement, however this isn't much use here since I am trying to find the density of the free charge.
Then I see you can use $\vec D = \epsilon_0 \vec E + \vec P$ but this requires knowing the electric field inside the dielectric... I don't believe I can find that unless I know the free charge density first. I know $\rho_{tot} = \rho_b + \rho_f$ but I don't think that can be helpful here... Any other attempt to find an equation relating my knowns (polarization/bound charge densities) to relevant unknowns required unknown constants like the relative permativity, or electric susceptibility - things they don't give in the initial question. So right now, I'm stuck!

c) I'm pretty confident I can do this part once I have my answer for (b). I'll just find the electric field inside and outside the sphere by using Gauss Law for the free and bound charges, then use $V = \int_r^{\infty} \vec E \cdot d \vec r$ to get the potentials

 

[deedsy]

problem is solved - had to assume we knew the relative permittivity, making it very straight forward