Deduce Coulomb's law from Gauss theorum.

Gauss theorem states that total electric flux over a closed surface equals enclosed charge

divided by permittivity of enclosed medium.

 

Let us consider a sphere of radius R has charge at centre of sphere.

 

Then by Gauss theorem ,    .....................(1)

Where E is uniform electric field intensity over the surface of sphere , q is enclosed charge

and ε is permittivity of medium

 

Since Electric field intensity E is uniform over the surface of sphere ,

we can write the RHS of eqn.(1) as

 

= E × 4π R² ........................(2)

where 4π R² is surface area of sphere.

 

Using eqn.(1) and (2) , we write

 

E × 4π R² = ( q / ε )

 

Hence Electric field E is written as

 

 

Above expression equals electric field at a distance R from a charge q.

 

Since electric field is force / unit charge , force experienced by a test charge q' at a distance R

from the given charge q  is written as

 

Above expression is Columbs law that gives electrical force between two charges q and q'

that are separated by a distance R