A spherical shell of radius $R$ carries uniform surface charge density $\sigma$. Find the electric field just outside and just inside the shell.

Outside ($r > R$): Using Gauss's law with $Q = 4\pi R^2 \sigma$:

$$E_{out} = \frac{\sigma R^2}{\epsilon_0 r^2}$$ , at $r = R^+$: $E = \dfrac{\sigma}{\epsilon_0}$

Inside ($r < R$): No enclosed charge:

$$E_{in} = 0$$

$$E_{out} = \frac{\sigma}{\epsilon_0}$$