Question

Find the electric field produced by a uniformly polarized sphere of radius R

Answer 1

The potential outside:

$$V(r)=\frac{P}{3\epsilon_0}\frac{R^3}{r^2}\cos\theta=\frac{1}{4\pi \epsilon_0}\frac{\frac{4\pi PR^3}{3}}{r^2}\cos\theta$$

$$V(r)=\frac{1}{4\pi \epsilon_0}\frac{\bold{p\cdot \hat{r}}}{r^2}$$

where

$$p=\frac{4\pi PR^3}{3}$$

The potential inside:

$$V(r)=\frac{p}{3\epsilon_0}r\cos\theta=\frac{p}{3\epsilon_0}z$$

The  electric field:

$$\bold{E}=-\nabla V=-\frac{P}{3\epsilon_0}\hat{\bold{z}}$$