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:
=q4ฯd2 Coulombs/ sq.meter or C/m2
and its component D, normal to the surface will be
=q4ฯd2cos ๐ C/m2
where ๐ is the angle between D and D'.
Now, the flux d๐ crossing normally the surface of area dA = D โคฌ area
or dฯ=q4ฯd2cos ฮธโ dA coulombs
By definition,
solid angle=surface arearadius
โด dฯ=dA cos ฮธd2
โด dฯ=q4ฯโ dฯ
or the total flux ๐ , crossing the total surface of the imagined enclosure is
โซdฯ=q4ฯ โซdฯ
ฯ=q4ฯร4ฯ
or ฯ=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 +๐โ , +๐โ , -๐โ and +๐โ placed in an enclosure, the total electric flux due to all the charges will be
๐ = ๐โ + ๐โ - ๐โ + ๐โ
or ๐ = ๐ด๐