Divergentie en krul van elektrisch veld

Gauss's law states that the flux out of a closed surface equals the total charge enclosed within it.

Expressing the total charge in terms of volume charge density gives the integral form of Gauss's law.

According to the divergence theorem, the closed surface integral of an electric field equals the volume integral of its divergence. This means that, for any arbitrary volume, equating the integrands gives the divergence of the electric field.

The divergence of an electric field indicates the presence of a source for the field.

The electric field lines diverge from the positive charge and converge at the negative charge. It follows that, field lines cannot circulate back on themselves. So, the electric field has no curl.

Now, consider a closed path in an electric field. The line integral of the electric field over the closed path, on simplifying, equals zero.

Stokes' theorem states that the surface integral of the curl of the electric field equals the line integral over the closed path. Comparing both equations shows that the curl of the electric field is zero for a static charge distribution.