We can also show that the charges present outside the surface do not contribute to the electric flux. For this, suppose a charge + q is situated at a point O outside the closed surface.
Let a cone of small solid angle dΩ from O cut areas ds1 and ds2 of the closed surface at A and B respectively (Fig.)
As \(\vec E\) is inwards at A,
Flux at A = \(-\frac{q}{4πɛ_o}\) dΩ
(Negative sign because outdrawn normal at A makes obtuse angle with \(\vec E\) for which cos θ is negative).
As \(\vec E\) is outwards at B,
Flux at B = \(-\frac{q}{4πɛ_o}\) dΩ
Total electric flux over these elements of closed surface =\(-\frac{qdΩ}{4πɛ_o} + \frac{qdΩ}{4πɛ_o} = 0\)
Hence total electric flux over the closed surface due to charge outside the surface is zero.
