Coulomb's theorem Explanation | proof |2021

 Coulomb's theorem

this theorem states that the electric field intensity at a point very near to charged conductor is

    Proof

>  Considering a small path of charge conductor and small part of charge conductor , we considering a point p very close to the surface of conductor,

Through the point p we concider a cylindrical surface whose plane surface dS¹ lie outside the conductor and dS² lie inside the conductor. and curved surface is perpendicular to the charged surface.

Let dS¹ = dS² ≃ dS 

dS is the area of the charged surface inside the imaginary cylinder.

    

     Now electric flux through the surface 

     dS¹ = ∈r E dS¹,

And electric flux through surface dS² = 0

Since there is is no electric flux inside a conductor.

 Electric flux through the the curved surface is is also zero, since the lines of force originated normally from the surface of charged conductor and the curved surface is also normal to the charged surface

  

   Total Electric flux through the surface of imaginary cylinder 

= ∈r Е dS¹ + 0+0,

   = ∈r E dS¹  ≃  ∈r E dS,    ; as dS¹ ≃ dS