Deriving Coulomb's law from Gauss's law

Coulomb's law:
                                    Coulomb's law states that: The magnitude of the electrostatic force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distance between them.
                                   
 Coulomb's Law Formula is given as

                                       


Gauss's Law:
                               The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity. The electric flux through an area is defined as the electric field multiplied by the area of the surface projected in a plane perpendicular to the field.
Gauss's law may be expressed as:

                                                   

Deriving Coulomb's law from Gauss's law:

                       Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of E.However, Coulomb's law can be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically-symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).
                     
                       
                   Taking S in the integral form of Gauss's law to be a spherical surface of radius r, centered at the point charge Q, we have

                By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is

                   where is a unit vector pointing radially away from the charge. Again by spherical symmetry, E points in the radial direction, and so we get

               which is essentially equivalent to Coulomb's law.
Thus the inverse-square law dependence of the electric field in Coulomb's law follows from Gauss's law.