Gauss theorem and proof

*“Gauss'  theorms”: According to this theorem, The net electric flux passing through closed surface which is placed in air/vacuum is 1/Eo times the total charge enclosed by closed surface.

Mathematical form:  Φ = q. 1/Eo
Where,  q = total charge enclosed                          by closed surface
              Eo = permittivity of free                              space


* proof :

Let a charge +q be placed at centre 0  of a sphere of radius r . Any point P is placed on the surface of a sphere about this point ,we assuming an element of surface area ds . the solid angle subtended by area ds at the centre 0 be dw.

By the coulumb's Law :
Electric field intensity due to the point charge + q and the distance r is—
      E = 1/4πEo × q/r²   N/C

Thus the electric flux passing through small surface area ds—
       dΦ = E. ds cosθ

Putting the value of  E
     dΦ = (1/4πEo × q/r²).ds cosθ

Taking surface integration
    ∫∫dφ = ∫∫ (1/4πEo × q/r²) dscosθ
     
       φ =  q/4πEo ∫∫ ds cosθ/r²

we know that
[small solid angle dw = ds cosθ/r²]

∴ φ = q/4πEo ∫∫ dw

 φ =  q/4πEo ×W

∵ total solid angle W = 4π
∴  φ = q/4πEo × 4π

[ φ = q/Eo ]

HENCE PROVED