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In fact, Gauss’s Law, as well known as Gauss’s flux theorem or Gauss’s theorem, is the law that describes the relationship between electric charge distribution and the resulting electric field. As per Gauss’s law, the total amount of electric flux passing through any closed surface is proportional to the enclosed electric charge. The electrical field of a surface is calculated using Coulomb’s law, but the Gauss’s law is required to calculate the distribution of the electrical field on a closed surface. This describes the electrical charge contained within the closed surface or the electrical charge present within the enclosed closed surface. Gauss’s Law is being used to solve complex electrostatic problems with unique symmetries like cylindrical, spherical, or planar symmetry. This even aids in the calculation of the electrical field, which is quite complicated and requires difficult integration. The Gauss’s law can also be used to evaluate the electrical field in a straightforward manner.
Overview
As per Gauss’s Law, the total electric flux out of a closed surface equals the charge enclosed divided by the permittivity. According to this, the electric flux inside an area is described as the electric field multiplied by the surface area projected in a plane perpendicular to the field.
The total flux linked with a closed surface is 1/ε0 times the charge enclosed by the closed surface, according to Gauss’ law.
The electric field is by far the most fundamental concept in electricity. In general, the electric field of a surface is calculated using Coulomb’s law, but understanding the concept of Gauss law is required to calculate the electric field distribution in a closed surface. This describes the electric charge enclosed in a closed surface or the electric charge present in a closed surface enclosed in a closed surface.
Gauss’s Law Formula
The total charge enclosed in a closed surface is proportional to the total flux enclosed by the surface, according to the Gauss’s theorem. As a result, it is total flux ϕ and ε0 is the electric constant, the total electric charge Q enclosed by the surface is as follows:
Q = ϕ ε0
Now, the Gauss’s law formula is;
ϕ = Q/ ε0
Here,
Q stands for total charge within the given surface,
ε0 is said to be the electric constant.
The Gauss’s Theorem
The total or net flux through a closed surface is proportional to the net charge in the volume surrounded by the closed surface.
That is, Φ = → E.d → A = qnet/ ε0
In basic terms, the Gauss’s theorem connects the ‘flow’ of electric field lines (flux) to the charges on the enclosed surface. When no charges are enclosed by a surface, the net electric flux is zero.
It thus means that the number of electric field lines entering the surface equals the number of field lines leaving it.
The statement of the Gauss’s theorem also includes an important corollary:
The electric flux out of any closed surface is caused solely by the sources (positive charges) and sinks (negative charges) of the electric fields enclosed by the surface. Any charges outside the surface have no effect on the electric flux. Furthermore, only electric charges can act as electric field sources or sinks. Switching magnetic fields, for example, cannot act as electric field sources or sinks.
Because the surface on the left encloses a net charge, its net flux is non-zero. Since the surface on the right does not enclose any charge, the net flux is zero.
Electric Field due to Infinite Wire – Gauss’s Law Application
First, consider an infinitely long charge line with a charge per unit length of λ. We could indeed take advantage of the situation’s cylindrical symmetry. The electric fields all point radially away from the line of charge due to symmetry; there is no component parallel to the line of charge.
As our Gaussian surface, we can use a cylinder (with an arbitrary radius (r) and length (l)) centred on the line of charge.
The electric field is perpendicular to the curved surface of the cylinder, as shown in the diagram above. As a result, the angle formed by the electric field and the area vector is zero, and cos θ= 1.
The cylinder’s top and bottom surfaces are parallel to the electric field. As a result, the angle formed by the area vector and the electric field is 90 degrees, and cos θ = 0.
So, the electric flux is solely due to the curved surface.
As per Gauss’s Law,
Φ = → E.d → A
Φ = Φcurved + Φtop + Φbottom
Φ = → E . d → A = ∫E . dA cos 0 + ∫E . dA cos 90° + ∫E . dA cos 90°
Φ = ∫E . dA × 1
The curved surface is equidistant from the line of charge due to radial symmetry, and the electric field in the surface has a constant magnitude throughout.
Φ = ∫E . dA = E ∫dA = E . 2πrl
The overall charge enclosed by the surface is as follows:
qnet = λ.l
By Gauss’s theorem,
Φ = E × 2πrl = qnet/ ε0 = λl/ ε0
E × 2πrl = λl/ ε0
E = λ/2πrε0
Also read: Kirchhoff’s Laws and their Applications
FAQs
Q. What is the differential form of Gauss’s theorem?
Ans: Gauss’s law, in its differential form, relates the electric field to the charge distribution at a specific point in space. To explain, according to the law, the divergence of the electric field (E) at a given point is equal to the volume charge density (ρ). It can be written as:
ΔE = ρ/ ε0
Q. What are the three applications of Gauss law?
Ans: There really are three distinct cases to consider:
Spherical: The charge distribution has spherical symmetry.
Cylindrical: The charge distribution has cylindrical symmetry.
Planar: Charge distribution has translational symmetry along a plane.
What is called Gaussian surface?
In general, a Gaussian surface (abbreviated G.S.) is a three-dimensional closed surface through which the flux of a vector field is calculated; typically, the gravitational field, electric field, or magnetic field.
