Summary: Electrostatics in Free Space

A brief summary of the basic equations for electrostatics in empty space

Dielectric constant: $\epsilon_0=8.854*10^{-12} ~C^2N^{-1}m^{-2}$

Electron charge: $e=1.6*10^{-19}~C$

Coulomb’s law of attraction: $\vec{F}=\frac{1}{4\pi \epsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r}=k_0\frac{q_1q_2}{r_{12}^2}\hat{r}= k_0\frac{q_1q_2}{r_{12}^3}\vec{r}$

Electrostatic field: $\vec{E} = \frac{1}{q}\vec{F}$

Flux of the electrostatic field: $\Phi_S(\vec{E}) = \int \vec{E}\cdot \hat{n}dS=\int_S \vec{E}\cdot\vec{dS}$

Gauss’ theorem:  $\Phi_S(\vec{E}) = \int \vec{E}_0\cdot \hat{n}dS=\frac{Q_{int}}{\epsilon_0}=\frac{1}{\epsilon_0}\int_{V(S)} \rho dV$

Divergence theorem: $\int_S\vec{A}\cdot d\vec{S}=\int_{V(S)}\vec{\nabla}\cdot\vec{A}dV$

Maxwell’s first equation: $\vec{\nabla}\cdot \vec{E}=\frac{\rho}{\epsilon_0}$

The electrostatic field is conservative: $\oint\vec{E}\cdot d\vec{l}=0$

Electrostatic potential: $\int_{A}^{B} \vec{E}\cdot d\vec{l}= V(A)-V(B)$ and

Differential relationship between electric field and potential $\vec{E}=-\vec{\nabla} V$

Gradient of a scalar function: $\vec{\nabla}f(x,y,z)= \hat{\imath}\partial_x f + \hat{\jmath}\partial_y f + \hat{k}\partial_z f$

Electric potential due to a point charge: $V(x,y,z) = \frac{1}{4\pi\epsilon_0}\frac{q}{r}+C$

Electric potential due to a charge distribution (superposition principle): $V(x,y,z) = \frac{1}{4\pi\epsilon_0}\int_V\rho dV$

Curl of the electrostatic field in free space: $\vec{\nabla}\times\vec{E}=0$

Energy density of the electrostatic field: $u = \frac{1}{2}\epsilon_0 |E|^{2}$

Energy of the electrostatic field: $U = \int_{V} u dV = \frac{1}{2}\epsilon_{0}\int_V|E|^2 dV$

Electric dipole moment: $\vec{p}=q\vec{\delta}=\int_V\rho(r)dV$

Potential due to an electric dipole: $U = -\vec{E}\cdot\vec{p}$

Torque of an electric dipole: $\tau = -\vec{E}\wedge\vec{p}$