The first of Maxwell’s four equations:

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

In words: the divergence of the electric field equals the charge density divided by $\varepsilon_0$.

This is the formal statement of something intuitive: electric charges are sources (and sinks) of electric field.

What it says

• Where there’s positive charge ($\rho > 0$), the electric field diverges outward — it’s a source.
• Where there’s negative charge ($\rho < 0$), the electric field converges inward — it’s a sink.
• Where there’s no charge ($\rho = 0$), the electric field has zero divergence — whatever flows in, flows out.

The constant $\varepsilon_0 \approx 8.854 \times 10^{-12}$ F/m (the permittivity of free space) sets the strength: how much field per unit charge.

The integral form

Gauss’s law has a beautiful equivalent form. Draw any closed surface (a “Gaussian surface”). The total electric flux through that surface — the net amount of field passing outward — equals the total charge enclosed, divided by $\varepsilon_0$:

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

Why it’s powerful

The beauty of Gauss’s law is that you don’t need to know the details of the field to use it. Choose a surface that exploits the symmetry:

• Spherical symmetry (point charge): use a spherical surface → recover Coulomb’s law
• Cylindrical symmetry (infinite wire): use a cylindrical surface → $E = \lambda / (2\pi \varepsilon_0 r)$
• Planar symmetry (infinite sheet): use a pillbox → $E = \sigma / (2\varepsilon_0)$, constant everywhere!

In each case, the symmetry makes the integral trivial. One equation, infinite situations.

Equation #1 of 4

Gauss’s law tells us where electric fields originate: at charges. It says nothing about whether those fields curl (that’s Faraday’s law) or what magnetic fields do (that’s the other two equations). Each equation captures one aspect of how fields behave.