Gauss’s Law: Statement, Mathematical Expression And Applications of Gauss’s Law

Gauss’s Law

It is among the four equations of Maxwell’s laws of electromagnetism. It was initially formulated by Carl Friedrich Gauss in the year 1835 and relates the electric fields at the points on a closed surface area and the net charge enclosed by that surface. The law was released in 1867 as part of a collection of work by the popular German mathematician, Carl Friedrich Gauss.

Statement of Gauss’s Law

Gauss Law states that the net electric flux out of a closed surface area is equal to the charge enclosed divided by the permittivity. The electric flux in an area is defined as the electric field multiplied by the area of the surface projected in a plane and perpendicular to the field.

Electric Flux

The electrical flux is specified as the electrical field traveling through a given area multiplied by the area of the surface in a plane perpendicular to the field. Yet another statement of Gauss’s law states that the net flux of an offered electrical field through an offered surface, divided by the enclosed charge must be equal to a constant.

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Normally, a positive electric charge is expected to generate a positive electrical field.

Gauss’s Law Mathematical Expression

According to the Gauss law, the overall flux linked with a closed surface area is 1/ ε0 times the charge confined by the closed surface.


For instance, a point charge q is positioned inside a cube of edge ‘a’. Now as per Gauss law, the flux through each face of the cube is q/6 ε0.

The electrical field is the fundamental principle to know about electrical flux. Generally, the electric field of the surface area is determined by using Coulomb’s law, but to calculate the electric field distribution in a closed surface, we need to understand the principle of Gauss law. It explains the electrical charge enclosed in a closed or the electric charge present in the enclosed closed surface.


Based on the Gauss theorem, the total charge confined in a closed surface area is proportional to the overall flux enclosed by the surface area. Therefore, if ϕ is overall flux and ϵ0 is electric constant, the total electric charge Q confined by the surface area is;

Q = ϕ ϵ0

The Gauss law formula is expressed by;

ϕ = Q/ ϵ0


  • Q = overall charge within the provided surface,
  • ε0 = the electrical constant.
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Applications of Gauss’s Law
  • Gauss’s law is extremely practical in figuring out expressions for the electrical field, even though the law is not directly about the electrical field; it is about the electrical flux.
  • Gauss’s law can be used to derive Coulomb’s law and vice versa. Keep in mind that since Coulomb’s law only applies to point stationary charges, there is no factor to expect Gauss’s law to hold for moving charges based upon this derivation alone. In fact, Gauss’s law does hold for moving charges, and in this respect, Gauss’s law is more basic than Coulomb’s law.