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# Gauss' Theorem Tejas Academy Centre
03/04/2017 0 0

# Gauss' Theorem:

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

## Proof

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.

## Case II :- When Gaussian Surface is not Spherical in Shape

##### 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

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