A solid nonconducting sphere of radius R carries a non-uniform charge distribution, the charge density being , where is a constant and r is the distance from the sphere. We have to show that (a) the total charge on the sphere is and (b) the electric field inside the sphere is given by

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(a)

The charge density in the solid nonconducting sphere of radius R is given by the function

,

where is a constant and r is the distance from the sphere.

The total charge in the sphere can be obtained by integrating the density over the volume of the sphere

(b)

The electric field at a point which is at a distance r from the centre of the sphere will be determined by the total charge contained inside the sphere of radius r. We therefore have