Potential energy of a two-charge system

U (r) = k_{e} \frac{q Q}{r}

Work done to assemble a system of charges

W_{12 \dots N} = \frac{k_{e}}{2} \sum_{i}^{N} \sum_{j}^{N} \frac{q_{i} q_{j}}{r_{i j}} for i \neq j

Potential difference

Δ V = \frac{Δ U}{q} or Δ U = q Δ V

Electric potential

V = \frac{U}{q} = - \int_{R}^{P} \vec{E} \cdot d \vec{l}

Potential difference between two points

Δ V_{A B} = V_{B} - V_{A} = − \int_{A}^{B} \vec{E} \cdot d \vec{l}

Electric potential of a point charge

V = \frac{k_{e} q}{r}

Electric potential of a system of point charges

V_{P} = k_{e} \sum_{1}^{N} \frac{q_{i}}{r_{i}}

Electric dipole moment

\vec{p} = q \vec{d}

Electric potential due to a dipole

V_{P} = k_{e} \frac{\vec{p} \cdot \hat{r}}{r^{2}}

Electric potential of a continuous charge distribution

V_{P} = k_{e} \int \frac{d q}{r}

Electric field components

E_{x} = - \frac{\partial V}{\partial x}, E_{y} = - \frac{\partial V}{\partial y}, E_{z} = - \frac{\partial V}{\partial z}

Del operator in Cartesian coordinates

\vec{\nabla} = \hat{i} \frac{\partial}{\partial x} + \hat{j} \frac{\partial}{\partial y} + \hat{k} \frac{\partial}{\partial z}

Electric field as gradient of potential

\vec{E} = − \vec{\nabla} V

Del operator in cylindrical coordinates

\vec{\nabla} = \hat{r} \frac{\partial}{\partial r} + \hat{φ} \frac{1}{r} \frac{\partial}{\partial φ} + \hat{z} \frac{\partial}{\partial z}

Del operator in spherical coordinates

\vec{\nabla} = \hat{r} \frac{\partial}{\partial r} + \hat{θ} \frac{1}{r} \frac{\partial}{\partial θ} + \hat{φ} \frac{1}{r sin θ} \frac{\partial}{\partial φ}