Charge Distributions

Three Dimensional

\(\rho = \frac{dq}{dV}\)

One Dimensional

\(\lambda = \frac{dq}{d\ell}\)

Two Dimensional

\(\sigma = \frac{dq}{dA}\)

Pillbox enclosing some surface with area \(\Delta a\). \(\int \vec{E}\cdot\hat{n}da=\vec{E}_2\cdot\hat{n}_{21}\Delta a-\vec{E}_1\cdot\hat{n}_{21}\Delta a=\frac{\sigma\Delta a}{\varepsilon_0}\). Then, \((\vec{E}_2-\vec{E}_1)\cdot\hat{n}_{21}=\frac{\sigma}{\varepsilon_0}\). Hence, it must always be continous on a surface by the charge density over epsilon naught.