Electric Charges and Fields

Gauss’s Law

The study of science is incredibly interesting and filled with fun facts. The deeper one dives into the concepts of science and its related fields, the greater amount of knowledge and information there is to learn in there. One such topic of study is the Gauss Law, which studies electric charge along with a surface and the topic of electric flux. Let us get to know more about the law and the manner of its operation so that we can understand the equation of the law.

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Gauss’s law


(Source: Study)

Gauss’s law states that the net flux of an electric field in a closed surface is directly proportional to the enclosed electric charge. It is one of 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 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. Yet another statement of Gauss’s law states that the net flux of a given electric field through a given surface, divided by the enclosed charge should be equal to a constant.

Usually, a positive electric charge is supposed to generate a positive electric field. The law was released in 1867 as part of a collection of work by the famous German mathematician, Carl Friedrich Gauss.

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Gauss Law Equation

Let us now study Gauss’s law through an integral equation. Gauss’s law in integral form is given below:

E⋅dA=Q/ε0      ….. (1)


  • E is the electric field vector
  • Q is the enclosed electric charge
  • ε0 is the electric permittivity of free space
  • A is the outward pointing normal area vector

Flux is a measure of the strength of a field passing through a surface. Electric flux is defined as

Φ=∫E⋅dA    …. (2)

We can understand the electric field as flux density. Gauss’s law implies that the net electric flux through any given closed surface is zero unless the volume bounded by that surface contains a net charge.

Gauss’s law for electric fields is most easily understood by neglecting electric displacement (d). In matters, the dielectric permittivity may not be equal to the permittivity of free-space (i.e. ε≠ε0). In the matter, the density of electric charges can be separated into a “free” charge density (ρf) and a “bounded” charge density (ρb), such that:

Ρ = ρf + ρb

Solved Examples for You

Question: There are three charges q1, q2, and q3 having charge 6 C, 5 C and 3 C enclosed in a surface. Find the total flux enclosed by the surface.

Answer:  Total charge Q,

Q = q1 + q2 + q3
= 6 C + 5 C + 3 C
= 14 C

The total flux, ϕ = Q/ϵ0
ϕ = 14C / (8.854×10−12 F/m)
ϕ = 1.584 Nm2/C

Therefore, the total flux enclosed by the surface is 1.584 Nm2/C.

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