Profile
Settings
Refer your friends
Sign out
Gauss’s law says that in a closed surface, the total flux of an electric field would be directly proportional to the electric charge enclosed inside the surface. Maxwell gave four equations on electromagnetism, and it is one of them. Primarily, it was formulated by the famous German mathematician, Carl Friedrich Gauss, in 1835. We can define electric flux as the product of the electric field that passes through a given area, and the surface area of a plane that is perpendicular to the electric field. Generally, a positive electric field would be generated by a positive charge.
Gauss law derivation
We can now perform the Gauss law derivation. We would use an integral equation to study Gauss’s Law:
Where the electric field vector is denoted by E, Q denotes the enclosed electric charge, ε0 denotes the electric permittivity of free space, and the outward pointing area vector is denoted by A.
We can define flux as the measure of the magnitude of a field that passes through a surface. The equation for electric flux is:
The electric field could be understood as the density of flux. Gauss’s Law states that the total electric flux via any closed surface would be zero if the volume of that surface has a net charge of zero.
Gauss Law Application
Gauss’s law has numerous applications. In many cases, the calculation of electric flux requires through integration and the calculations get quite complicated as well. Gauss’s law is used to make the calculations easy. Before trying to calculate the electric flux, we need to choose a Gaussian surface. A Gaussian surface is a closed surface by which we calculate the flux of a vector field, generally the electric field, magnetic field, or gravitational field.
We must choose a Gaussian surface carefully, so that it is easy to calculate the surface integral. It must be remembered that it is not a necessity for a Gaussian surface to coincide with the surface we want to calculate the flux of.
Chief Gauss law applications are:
- Electric field because of a uniformly charged infinite straight wire
- Electric field because of a uniformly charged infinite plate sheet
- Electric field because of a uniformly charged thin spherical shell
- Electric field because of uniformly charged infinite plate wire: If we consider an infinite plane sheet, that has a cross-sectional area A and σ as surface charge density.
Infinite Charge Sheet
The direction of the electric field would be perpendicular to the plane of the sheet. We would also consider a cylindrical Gaussian surface that has a normal axis to the plane of the sheet. Then, the law states:
We would consider the electric flux from only the two ends of the pre-assumed Gaussian surface. The curved surface area and the electric field are normal to each other, hence electric flux produced would be zero.
Φ = EA + EA
Φ = 2EA
2EA = σA/ ϵ0
In the equation, A cancels out, which means is that the electric field because of the plane sheet is not dependent on cross-sectional area A:
E = σ/2ϵ0
Electric field because of infinite wire
If we imagine a wire of infinite length that has linear charge density λ and L as its length. We need to assume a cylindrical Gaussian surface for the calculation of the electric field. The flux through the end of the assumed cylindrical surface would be zero because the direction of the electric field is radial. The reason for this is the angle between the electric field and area vector is 90 degrees. The magnitude of the electric field is constant because It is perpendicular to all the points on the curved surface.
Conclusion
Gauss’s law is used to find out the electric field and electric charge of a closed surface. The law states that the total flux of an electric field is directly proportional to the electric charge that is enclosed inside the closed surface. Gauss’s law helps in the simplification of calculations relating to the electric field. There are many applications of Gauss’s law. When we try to calculate the electric field of an object, we need to use a Gaussian surface to make the calculations easy. The famous mathematician, Carl Friedrich Gauss, formulated the law in 1835.
