Electrostatics in Vacuum

Two Statements That Fix the Field

Electrostatics rests on two field equations. Gauss's law says the flux of $\mathbf{E}$ through any closed surface counts the charge inside it:

$$\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0}$$

with $\varepsilon_0 = 8.854 \times 10^{-12}$ F/m. Its local form follows from the divergence theorem:

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$$

The second statement is that the electrostatic field has no circulation: $\oint \mathbf{E} \cdot d\mathbf{l} = 0$, or locally $\nabla \times \mathbf{E} = \mathbf{0}$. A curl-free field is a gradient, which is exactly what licenses a scalar potential:

$$\mathbf{E} = -\nabla V$$

Combining the two gives Poisson's equation $\nabla^2 V = -\rho/\varepsilon_0$, which reduces to Laplace's equation $\nabla^2 V = 0$ wherever the charge density vanishes.

Gauss's law is always true but only useful when symmetry lets you pull $E$ out of the integral. Three cases carry most of the work:

At a boundary the field obeys continuity conditions: the tangential component of $\mathbf{E}$ is always continuous, while the normal component jumps by $\sigma/\varepsilon_0$.

Far from a neutral but polarised object, the leading term is the dipole: $\mathbf{p} = q\mathbf{d}$, with a potential falling as $1/r^2$ and a field as $1/r^3$, faster than a point charge, because the two charges nearly cancel.

Common pitfall: reading a Gaussian surface with zero net flux as a region with no field. Flux counts only the enclosed charge. A surface drawn around a dipole encloses zero net charge and has zero total flux, while the field on it is large everywhere.