What is Gauss's law?
Gauss's law states that the net flux of an electric field
through a closed surface is proportional to the enclosed electric charge
It was first formulated by Carl Friedrich Gauss
and relates the electric fields at points on a closed surface (known as a "Gaussian surface
") and the net charge enclosed by that surface. The electric flux
is defined as the electric field passing through a given area multiplied by the area of the surface in a plane perpendicular to the field.
Another statement of Gauss's law is that the net flux of an electric field through a surface divided by the enclosed charge is equal to a constant.
The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity
Fig .1: Gauss law
Gauss's Law is a general law applying to any closed surface. It is an important tool since it permits the assessment of the amount of enclosed charge by mapping the field on a surface outside the charge distribution. For geometries of sufficient symmetry, it simplifies the calculation of the electric field.
Another way of visualizing this is to consider a probe of area A which can measure the electric field perpendicular to that area. If it picks any closed surface and steps over that surface, measuring the perpendicular field times its area, it will obtain a measure of the net electric charge
within the surface, no matter how that internal charge is configured.
Gauss' Law, Integral Form
The area integral of the electric field over any closed surface is equal to the net charge enclosed in the surface divided by the permittivity of space. Gauss' law is a form of one of Maxwell's equations, the four fundamental equations for electricity and magnetism
Fig.2: Gauss law integral form.
Gauss's law permits the evaluation of the electric field in many practical situations by forming a symmetric Gaussian surface surrounding a charge distribution and evaluating the electric flux through that surface.