The law relates the flux through any closed surface and the net charge enclosed within the surface.
The law states that the total flux of the electric field (E) over any closed surface is equal to 1/εo times the net charge (q) enclosed by the surface.
φ = ∮ E * ds = q /εo
This closed imaginary surface is called Gaussian surface.
Gauss’s law tells us that the flux of E through a closed surface S depends only on the value of net charge inside the surface and not on the location of the charges. Charges outside the surface will not contribute to flux.
Proof of Gauss's Law :
Let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre. Divide the sphere into small area elements, as shown in Fig.
The flux through an area element ΔS is
dφ = E * ΔS = q / 4π εo r^2 * ΔS ˆr (r-cap)
where unit vector ˆr is along the radius vector from the centre to the area element.
Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ΔS and ˆr have the same direction.
dφ = E ΔS cos 0 = q / 4π εo r^2 ΔS cos 0
or dφ = E ΔS = q / 4π εo r^2 ΔS
The total flux through the sphere is obtained by adding up flux through all the different area elements
so, φ = ∮ dφ = ∮ E ΔS = q / 4π εo r^2 ∮ΔS
or φ = q / 4π εo r^2 * 4π r^2 (∮ΔS= 4π r^2)
or φ = q / εo
It proves the Gauss's theorem.
