# Computing field inside spherical dielectric by geometric series

It's somewhat tedious but trivial to show (e.g. by considering the potential) that the electric field $\vec{E}$ inside a spherical linear (homogeneous and isotropic) dielectric with susceptibility $\chi_e$ placed in a uniform electric field $\vec{E}_0$ satisfies $$\vec{E} = \frac{3}{3+\chi_e}\vec{E}_0.$$

Another way to compute the electric field (as suggested by Griffiths) is to consider the polarization $\vec{P}_0$ due to $\vec{E}_0$, the contribution $\vec{E}_1$ due to this polarization, the polarization $\vec{P}_1$ due to $\vec{E}_1$, the contribution $\vec{E}_2$ due to $\vec{P}_1$ etc.

Evidently (the electric field inside a spherical dielectric due to a homogeneous polarization $\vec{P}'$ is simply $\vec{E}' = -1/(3\epsilon_0)\vec{P}'$) we have $$\vec{P}_n = \epsilon_0\chi_e\vec{E}_n$$ and $$\vec{E}_{n+1} = -\frac{1}{3\epsilon_0}\vec{P}_n = -\frac{\chi_e}{3}\vec{E}_n$$ whence $$\vec{E}_n = (-\chi_e/3)^n\vec{E}_0.$$

Adding all contributions, we get a geometric series: $$\vec{E} = \vec{E}_0 \sum_{n=0}^\infty (-\chi_e/3)^n = \vec{E}_0\frac{1}{1+\chi_e/3} = \frac{3}{3+\chi_e}\vec{E}_0,$$ as desired.

However, this obviously only holds for $|\chi_e| < 3$, whereas the result in general holds for any $\chi_e$. What does this mean physically? Why does it go wrong?

You already know the mathematical reason. Physically, the perturbative argument goes wrong because when $|\chi_e|$ is sufficiently large, it no longer makes sense to think about the physical electric field as coming from a direct sequence of corrections to contributions from increasingly higher order. Instead, you can think of first computing the field $\vec E$ associated with a few auxiliary values of $\chi_e$, and 'resum' the perturbative approximation about each point. In other words, when a closed analytic form isn't obvious from terms in the series, you need to take the problem in smaller steps. Handling error is a little more subtle then, but it is doable. Physically, the resummation procedure essentially involves imagining corrections from the dialectric with respect to a system whose behavior is closer to the system you wish to approximate.