Gauss’s law


The Gauss law within the electric field is the law describing the static electric field that is generated by a distribution of electrical charges . It is the law that states that the electric flux through any closed surface is proportional to the total electric charge enclosed by this surface. By convention, a positive electric charge generates a positive electric field. This law was published posthumously in 1867 as part of a collection of works by the famous German mathematician Carl Friedrich Gauss .

  • Unit: Gauss
  • Symbol:  φ
  • Formula:  φ = Q / ε0

What is Gauss’s Law?

It is one of the four important Maxwell equations that is responsible for relating the electric field to its sources . It then allows us to calculate the modulus of the electric field by knowing the distribution of charges with spherical symmetry .

  • Who proposed Gauss’s law
  • History
  • Statement of Gauss’s Law
  • Formula
  • Units
  • Explanation of Gauss’s Law
  • What is it for
  • Applications
  • Curiosities
  • Conclusions
  • Examples of Gauss’s Law

Who proposed Gauss’s law

The law was formulated by Carl Friedrich Gauss in 1835, but was published until 1867. It is one of the four equations of Maxwell , which are an important part of the basis of classical electrodynamics.



Not much is known about the history of Gauss’s law but it is known that it was created by Carl Friedrich Gauss in 1865. Carl Friedrich Gauss would formulate Gauss’s law, or Gauss’s theorem , which would be one of his most important contributions within the field of electromagnetism , and from which two of Maxwell’s four equations were born .

Apparently Gauss considered three particular cases to formulate the law, a general theorem enunciated by the Russian mathematician Mikhail Ostrogradski in a work presented to the Academy of Sciences in Paris in 1826, the French mathematicians Simeon Denis Poisson and Frederic Sarrus who presented, in 1828 demonstrations of this result and the English mathematician George Green also had some connection with this interesting and useful theorem.

Statement of Gauss’s Law

The statement of Gauss’s Law tells us that the flow of electric field through any surface or closed Gaussian surface is equal to the net charge enclosed by it, between the constant ε 0 .


The total charge enclosed in a closed surface is proportional to the total flow enclosed by the surface. If φ is total flux and ε0 is an electrical constant , the total electric charge Q enclosed by the surface is calculated by the formula:

φ = Q / ε0


  • Q = total charge within the given surface
  • ε0 = the electrical constant


Gaussian units make up a metric system of physical units that is the most common of electromagnetic unit systems based on cgs or centimeter-gram-second units . Also called Gaussian unit system, Gaussian-cgs units, or often just cgs units. The term “cgs units” is ambiguous and should therefore be avoided if possible, cgs contains within it different conflicting sets of units of electromagnetism, not just Gaussian units.

Since cgs units are ambiguous, the best alternative to them is SI units.

Conversions between Gaussian units and SI units are not as simple as normal unit conversions. For example, formulas for the physical laws of electromagnetism (such as Maxwell’s equations) must be adjusted according to the system of units used.

Explanation of Gauss’s Law

Gauss’s law is the one that states that the net flux that exists in an electric field through a closed surface is proportional to the electric charge included. It relates the electric fields at points on a closed surface generally known as a ” Gaussian surface ” and the net charge enclosed by that surface. The electrical flow is defined as the electric field passing through a given area multiplied by the surface area 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 oneconstant .

What is it for

Gauss’s law helps us to be able to find the electric fields generated by an electric charge distribution, which takes advantage of the geometric properties of the system.


Gauss’s law is generally only used in certain situations where there is a lot of symmetry , since it is necessary to know where E is constant and what its direction is . In this way, it can be applied in three types of symmetries that are:

  • Translational symmetry : the system does not vary with a rectilinear displacement.
  • Rotational symmetry : where the system does not vary before a rotation.
  • Spherical symmetry : it is where there is rotational symmetry with respect to any direction.


Some of the curiosities that the law presents are:

  • It is closely related to the Divergence Theorem .
  • Its main sources are electrical charge and
  • It can be applied to the magnetostatic field .


Gauss’s law implies that the total flow through a surface that completely encloses a charge will always be proportional to the total amount of charge.

Examples of Gauss’s Law

Some examples of Gauss’s law are:

  • Coaxial cables, electromagnetic interference-free offices and dipole antennas.

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