# Gaussian Surface, Gauss՚s Law, Gaussian Surface of Cylinder (For CBSE, ICSE, IAS, NET, NRA 2022)

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# Gaussian Surface

## What is Gaussian Surface?

• Part of the power of Gauss՚ law in evaluating electric fields is that it applies to any surface.
• It is often convenient to construct an imaginary surface called a Gaussian surface to take advantage of the symmetry of the physical situation.
• It is known as a closed surface in three-dimensional space such that the flux of a vector field is calculated.
• These vector fields can either be the gravitational field or the electric field or the magnetic field.

## Gauss՚s Law

• The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.
• Using Gauss law, Gaussian surface can be calculated:

### Gaussian Surface of a Sphere

• When a flux or electric field is produced on the surface of the spherical Gaussian surface due to any of the following:
• A point charges
• A spherical shell with the uniform charge distribution
• Charge distribution with spherical symmetry
• Consider a spherical shell S with uniform distribution of charge Q, radius R and with negligible thickness.
• Using Gauss law to find electric field E at a distance r from the centre of the charged shell.
• As , net flux, and the magnitude of the electric field on the Gaussian surface are zero.
• Following is the flux out of the spherical surface S with surface area of radius r is given as:

The above equation shows the spherical distribution of charge which acts as a point charge verifying Coulomb՚s law.

## Gaussian Surface of Cylinder

• When a flux or electric field is produced on the surface of a cylindrical Gaussian surface due to any of the following:
• Uniform distribution of charge in an infinitely long line
• Uniform distribution of charge in an infinite plane
• Uniform distribution of charge on an infinitely long cylinder
• Consider a point charge P at a distance r having charge density λ of an infinite line charge.
• The axis of rotation for the cylinder of length h is the line charge, following is the charge q enclosed in the cylinder:

## Gaussian Pillbox

• The pillbox is the surface with an infinite charge of uniform charge density is used to determine the electric field.
• It is of a cylindrical shape consisting of three components; the disk at one end with area , the disk at the other end with the equal area and the side of the cylinder.
• As stated by Gauss law, the sum of electric flux through each component is proportional to the enclosed charge of the pillbox.

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