A sphere carries a uniformly distributed electric charge. Prove that the field inside the sphere is zero

Explanation

Suppose that there is a field inside the sphere. It is obvious from considerations of symmetry that in this case equipotential surfaces must be spherical sursaces concentric with the charged sphere. Accordingly, the lines of force coincide with the radii, i.e. they must either begin or end at the centre of the sphere. This would have been possible, if there were a positive or a negative charge at the centre of the sphere. But since there is no charge inside the sphere, the lines of force cannot begin or end there. Consequently there is no field inside the sphore.