what is gauss law with proof?

Question

In easy words

Answer 1

Do you mean Gauss' law in electrostatics?  If so, you want to prove that the electric flux Φ_E through a simply connected closed surface S is Q/ε₀, where Q is the total electric charge enclosed by the surface.  Let ρ be the volume density of the charge enclosed by S.

 

Φ_E = ∫∫E·ndS (integral over the entire closed surface)

 

where E is the electric field vector crossing the surface S, and n is an outward pointing unit vector normal to the surface S.  By the divergence theorem (which I assume you saw in your vector calculus),

 

∫∫E·ndS = ∫∫∫∇·EdV

 

where V is the volume enclosed by S.

 

Maxwell's first law states that

 

∇·E = ρ/ε₀

 

Then

 

Φ_E = ∫∫∫(ρ/ε₀)dV = (1/ε₀) ∫∫∫ρdV

 

But 

 

∫∫∫ρdV = Q

 

Therefore,

 

Φ_E = Q/ε₀