Charge Distributions

Types

Point Charge

Zero Dimensional

Charge Densities

Volume

Three Dimensional

$ρ = \frac{d q}{d V}$

Linear

One Dimensional

$λ = \frac{d q}{d ℓ}$

Areal - Surface

Two Dimensional

$σ = \frac{d q}{d A}$

Pillbox enclosing some surface with area $Δ a$ . $\int \vec{E} \cdot \hat{n} d a = {\vec{E}}_{2} \cdot {\hat{n}}_{21} Δ a - {\vec{E}}_{1} \cdot {\hat{n}}_{21} Δ a = \frac{σ Δ a}{ε_{0}}$ . Then, $({\vec{E}}_{2} - {\vec{E}}_{1}) \cdot {\hat{n}}_{21} = \frac{σ}{ε_{0}}$ . Hence, it must always be continous on a surface by the charge density over epsilon naught.