We have already learnt to find the electric field intensity due to a charged conductor using Coulomb's law. Gauss' theorem can also be used to calculate the electric field intensity provided there is a symmetry in the charge distribution.

Can be considered as an alternative to Coulomb's law for expressing the relationship between electric charge and electric field. This theorem was formulated by a German mathematician Karl Friedrich Gauss.

Gauss' theorem states that the total electric flux through any closed surface is proportional to the total electric charge inside the surface.

Total electric flux q

But we have already obtained the relation

This means the electric flux is independent of the radius of the surface but only depends on the charge enclosed by the surface

Let case I: - when Gaussian surface is spherical in shape, a positive charge q be placed at the centre of an imaginary spherical surface of radius R as shown in the figure.

 

 

By symmetry, the field due to the charge +q is radial and E is perpendicular to the sphere and is directed along the normal to the surface.

So the angle between the normal and the electric intensity is zero.

That is, df = E dA cos0

Which is nothing but the mathematical representation of Gauss' theorem.

The total electric flux is,

where dW is the solid angle subtended by the area dA at the point.

Which is nothing but Gauss' theorem