Dielectric Sphere in Uniform Field

[Apteronotus]

Hi,

One of the boundary conditions when solving for the potential, $\Phi$, outside a dielectric sphere placed within a uniform electric field is
$\lim_{r→0}\Phi(r,θ)<\infty$

Can anyone explain/prove why this so.

Thanks,

 

[Jano L.]

If the charges are contained in some finite volume $V$, then the Coulomb formula applies:

$$
\Phi(\mathbf x) = \int_V \frac{\rho(\mathbf r)}{4\pi |\mathbf x-\mathbf r|}\,d^3\mathbf r.
$$

If total charge in the volume $V$ is finite, the integral can be estimated by (is lower than) $Q/(4\pi |\mathbf x-\mathbf r|)$ for some $Q$. As the latter expression falls off to zero as distance increases, so does the potential.

 

[Apteronotus]

Thank you Jano!