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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?

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If my memory serves, the capacitance of an isolated spherical conductor depends solely on its radius. I'm not certain of the following expressions, but I'd definitely apply Gauss' Law here.

Something like C = 4πε0R

V = Q / 4πε0R

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  • $\begingroup$ Sorry if I was unclear, I'm asking about a dielectric sphere, not a conducting one. I believe your point about solving it with Gauss's law is correct though. $\endgroup$
    – kthaxt
    Commented Feb 6, 2017 at 2:13
  • $\begingroup$ Ah. That makes sense. Must've overlooked that. It's been too long since I've done this type of work, but thinking through what I remember, I can't recall enough to definitively answer your questions. I am, however, now very curious to see the answer. $\endgroup$ Commented Feb 6, 2017 at 2:43
  • $\begingroup$ Actually I might have a more streamlined version of my question: Does it make sense to assign a capacitance for a non-conducting structure? $\endgroup$
    – kthaxt
    Commented Feb 6, 2017 at 18:38

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