( iv ) APPLICATION OF GAUSS'S LAW

( iv ) Electric Field due to a Uniformly Charged Sphere: -
Let r  = uniform volume charge density on a sphere of radius R.
( a ) For points inside the sphere: Applying Gauss’s Theorem to a sphere of radius r £ R,
4 pr2 E( r ) = (4p r 3  r) / 3e₀ 

The direction of the field is radially outwards if r  > 0 and inwards if r  < 0.

( b ) For points outside the sphere: -

Applying Gauss’s Theorem to a sphere of radius r, concentric with charged sphere of radius R ( r > R ),

4 pr2 E( r ) = (4p R 3  r) / 3e_{0 = Q /} e₀ 

where Q is the charge on the sphere_. Thus, for points outside the sphere, the entire charge of the sphere can be treated as concentrated at its centre.