Let a charge of q coulombs be placed inside a hollow enclosure as shown in figure:
The flux density D' (considered in the direction of the radius vector d) will be:
=q4πd2 Coulombs/ sq.meter or C/m2
and its component D, normal to the surface will be
=q4πd2cos 𝜃 C/m2
where 𝜃 is the angle between D and D'.
Now, the flux d𝜓 crossing normally the surface of area dA = D ⤬ area
or dψ=q4πd2cos θ⋅dA coulombs
By definition,
solid angle=surface arearadius
∴ dω=dA cos θd2
∴ dψ=q4π⋅dω
or the total flux 𝜓 , crossing the total surface of the imagined enclosure is
∫dψ=q4π ∫dω
ψ=q4π×4π
or ψ=q coulombs
Thus the total electric flux due to a charge at any point in an enclosure is equal to the charge enclosed in that enclosure.
If there are a number of charges +𝑞₁ , +𝑞₂ , -𝑞₃ and +𝑞₄ placed in an enclosure, the total electric flux due to all the charges will be
𝜓 = 𝑞₁ + 𝑞₂ - 𝑞₃ + 𝑞₄
or 𝜓 = 𝛴𝑞