Gauss TheoremPublished on 24/2/2012 and last updated on 8/10/2018
This theorem states that the total electric flux through any closed surface surrounding a charge, is equal to the net positive charge enclosed by that surface. Suppose the charges Q1, Q2_ _ _ _Qi, _ _ _ Qn are enclosed by a surface, then the theorem may be expressed mathematically by surface integral as Where, D is the flux density in coulombs/m2 and dS is the outwardly directed vector.
Explanation of Gauss's TheoremFor explaining the Gauss's theorem, it is better to go through an example for proper understanding. Let Q be the charge at the center of a sphere and the flux emanated from the charge is normal to the surface. Now, this theorem states that the total flux emanated from the charge will be equal to Q coulombs and this can be proved mathematically also. But what about when the charge is not placed at the center but at any point other than the center (as shown in the figure).
At that time, the flux lines are not normal to the surface surrounding the charge, then this flux is resolved into two components which are perpendicular to each other, the horizontal one is the sinθ component and the vertical one is the cosθ component. Now when the sum of these components is taken for all the charges, then the net result is equal to the total charge of the system which proves Gauss's theorem.