Coulombs law states that the electrostatic force between two stationary point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Vector for of Couloms law: F₁ = k q₁q₂ ř₂₁ / [r₂₁]²

From Gauss Theorem:

Draw a Gaussian sphere of radius r at the centre of which charge +q is located

All the points on this surface are equivalent. Due to symmetry, the electric field E has the same magnitude at every point on the surface of the sphere and it is radially outward in direction. Therefore, for a area element dS around any point P on the Gaussian surface both E and dS are directed radially outward, Hence, the angle between E and dS is zero.

The flux passing through the area element dS ,that is,

d φ =E.dS= EdS cos 00=EdS

Hence, the total flux through the entire Gaussian sphere is obtained as,

Φ=∫EdS =E∫dS

But ∫dS is the total surface area of the sphere and is equal to 4πr²,

Φ=E(4πr²)                                                      ------Eqn (1)

But according to Gauss law,

Φ=q/ε0                                                         ------Eqn (2) where q is the charge enclosed within the closed surface

By comparing equation (1) and (2) ,we get E(4πr²)=q/ε0

Hence, E=q/4πε0r² -----------------Eqn(3)

The equation (3) is the expression for the magnitude of the intensity of electric field E at a point which is at distant r from the point charge +q.

In a second point charge q₀ be placed at the point at which the magnitude of E is computed, then the magnitude of the force acting on the second charge q₀ would be

F=q₀E

By substituting value of E from equation (3), we get F=q_oq/4πε0r² ------------Eqn(4)

The equation (4) represents the Coulomb’s Law and it is derived from gauss law.