Gauss's Theorem: The net electric flux passing through any closed surface is

times, the total charge q present inside it.

Mathematically, Φ= ε o 1 ⋅q

Proof: Let a charge q be situated at a point O within a closed surface S as shown. Point P is situated on the closed surface at a distance r from O. The intensity of electric field at point P will be

Eˉ=4πεo1⋅r2q .......(1)

Electric flux passing through area ds enclosing point P,

dΦ=E⋅ds

or dΦ=E⋅dscosθ

[where θ is the angle between E and ds]

Flux passing through the whole surface S,

∫∫sdΦ=∫∫sE⋅dscosθ ........(2)