Many times in EM I've seen the classic problem of solving for the fields of a uniformly polarized sphere. Moreover, electrical engineers love giving the problem "solve for the capacitance of a conducting sphere."
Yet I've never seen anyone assign a capacitance to a uniformly polarized sphere of dielectric $\epsilon$. I feel as though solving for the for the charge and voltage in: $$C = \frac{Q}V$$ would not be too difficult. $Q$ for example would be the bound surface charge. For a given uniform polarization $P = P_0\hat{z}$, the surface charge density would be given by $\sigma = P_0\cos\theta$, and the solution to the electric potential is something like $\phi=\frac{P_0r}3cos\theta$ for $r<a$. Except I think I'm missing a term to account for $\epsilon$.
Anyway, my question is: Does it even makes sense to define a capacitance for this sphere? Is the bound charge a physical thing we can actually ascribe a capacitance to?
Edit: After working it out a bit more I realize the key difference is that a conductor is an equipotential surface, so the $\Delta V$ is not ambiguous. Can one assign a capacitance to a non-conducting (or non-equipotential) structure?
