Gauss's Law: Definition, Formula, Application & Notes

Gauss's law offers a completely different way to calculate the electric field passing through a region of space. 

Imagine a water pipe of a certain cross-sectional area. Let's assume it fills a water tank in a specific time. If you want to measure the quantity of water and the measurement of the tank is unknown, there is a simple way. The product of the quantity of water passing through the pipe per minute and the time it was open in minutes.

In the same way, Gauss proposed a law;

It states that the total electric flux passing through a closed surface is directly proportional to the enclosed electric charge within that closed surface.

In other way, the total electric flux ${\mathrm{\Phi }}_E$ passing through an enclosed surface will be 1/∈ of the enclosed charge in the closed surface.

Mathematically, it is expressed as:

${\mathrm{\Phi }}_E=\text{∮}\mathrm{}\mathrm{}\vec{E}\cdot d\vec{A}=\frac{q_{\mathrm{e}\mathrm{n}\mathrm{c}}}{{ϵ}_0}$

where:

${\mathrm{\Phi }}_E$ = electric flux through the closed surface

$\vec{E}$ = electric field

$d\vec{A}$ = area vector on the surface

$q_{\text{enc}}$ = charge enclosed

${ϵ}_0\approx 8.854\times {10}^{-12}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^2$ permittivity of free space.

The gauss law applies to any real or imaginary closed Gaussian surface. This law is specially very effective for symmetry, spherical, cylindrical, or planar charge distributions.

Usually, electric flux is understood as the total number of electric field lines passing through a given area. Since we know that the electric field lines is an imaginary concept so it is not possible to count the specific number of lines passing through the area.

In reality, electric flux can be calculated as the scalar product of the electric field passing through a given area and area vector of the given area.

Mathematically, electric flux through a small area dA:

$$d{\mathrm{\Phi }}_E=\mathbf{E}\cdot d\mathbf{A}=\mathrm{\mid }\mathbf{E}\mathrm{\mid }\mathrm{\mid }d\mathbf{A}\mathrm{\mid }\cos \theta$$where $\theta$ = angle between electric field and area vector $\hat{\mathbf{n}}$.

The electric flux for a uniform electric field $\mathbf{E}$passing through flat surface with area $A$make an angle $\theta$ in between, then

$$\Phi_E = E\,A\cos\theta.$$

Gauss's Law is essential in calculating the electric fields around objects with regular shapes such as flat surfaces, cylinders, and spheres. Below are the applications of Gauss law.

• Electric Field due to a Point Charge
  • Using Gauss law, the calculation of the electric field at a distance r from a point charge q.

• • The electric field is inward for negative charges and outward for positive charges.
• Electric Field due to a Spherical Shell of Charge
  • Outside the shell: The behaviour of the electric field is as if the charge were toward the centre.
  • Inside the shell:  The electric field is zero.
• Electric Field due to a  uniformly charged solid Sphere
  • Outside the sphere: The electric field is the same as point charge

• • Inside the sphere: Inside the sphere

• Electric Field due to an Infinite Line of Charge
  • A long wire with linear charge density.

• • The electric field is outward from the wire.
• Electric Field between parallel sheets of charge
  • Two parallel sheets carrying equal and opposite charges.

• • The electric field is zero outside.
  • The electric field is constant between the plates.