Derivation of coulomb's law by guess' theorem

*Derivation of coulomb's law by guess' theorem*=

Consider a point charge q is placed at point 0. We are to find the electric field intensity due to this point charge at a distance  r  at point P, for this we draw an imaginary sphere of centre O with radius r. This spherical surface behaves as Gaussian surface. We taking a small element of area ds around the point P . the direction of which is normal to the surface. let the electric field intensity due to point charge at point P be E.

According to Gaussian surface
         φε = ∫∫ E ds cosθ
Here E and ds are in same direction thus θ = 0
        φε = ∫∫E ds cos0
        φε = ∫∫ E ds
        φε = E ∫∫ ds
        φε = E ×S
∵ surface area of sphere S = 4πr²

∴  φε = E×4πr²--------------(1)

By gauss theorem
    φε = q/εo
putting the value of φε  in enq (1)
    q/εo = E ·4πr²
      E = q/ 4πr²εo
     [ E = 1/4πεo × q/r²] N/C--------(2)

let any point Q be placed at point p thus electrostatic force experienced bu this charge due to electric field will be
    F =  Q·E
putting the value of E
    F = Q ( 1/4πεo × q/r² )
   
    [F =  1/ 4πεo × qQ/r²] N
  THIS IS COULOMBS LAW