Gauss Law

Statement: The gauss’s theorem states that the total electric flux through any closed surface is equal to  times the net charge enclosed by the surface.

The circle on the integral means that the surface S is closed. The net charge means the algebraic sum of the charge within the surface S.

Proof: Consider a charge +q placed at a point O. Let P be a point at a distance r from O. The electric intensity at P is given by

 along OP

Gauss law holds good for closed surface of any shape. For the sake of simplicity, let us consider a spherical surface with O as centre and r as its radius.

This is the Gaussian surface. By symmetry, the field of the charge +q is radial. E is perpendicular to the sphere and is directed along the normal to the surface. So the angle between the direction of E and the normal to the surface of the sphere is zero (cosθ = 1). Also symmetry requires that E has the same magnitude everywhere on this sphere.

Hence the theorem.