State and Prove Gauss Theorem in Electrostatics

Answer: In electrostatics, Gauss’ Law connects the electric flux going through a closed path with the charge contained within it. This formula is extremely useful for calculating the electric field produced by various charged substances of varied forms. By tracing a closed Gaussian surface across a point outside an equally thin charged spherical shell, we can determine the electric field.

The electric flux over any sealed surface is proportionate to the total electric charge encompassed by the surface, according to Gauss’s law. According to the law, isolated electric charges occur, and like charges resist each other but unlike charges attract. The magnetic flux over any closed surface is 0, according to Gauss’s law, which is compatible with the finding that independent magnetic poles do not appear.

Proof of Gauss’s Theorem

Let’s say the charge is equal to q.

Let’s make a Gaussian sphere with radius = r.

Now imagine surface A or area ds has a ds vector

At ds, the flux is:

dΦ  = E (vector) d s (vector) cos θ

But , θ = 0

Hence , Total flux:

Φ = E 4 π r2

Hence,  σ = 1 / 4πɛo q / r2 × 4π r2

Φ= q / ɛo

As per the Gauss law, the total flux associated with a sealed surface equals 1/ε0 times the charge encompassed by the closed surface.