Question

(a) State the theorem which relates total charge enclosed within a closed surface S and the electric flux passing through it. Prove it for a single point charge. 

(b) An 'atom' was earlier assumed to be a sphere of radius having a positively charged point nucleus of charge +Ze at its centre. This nucleus was believed to be surrounded by a uniform density of negative charge that made the atom neutral as a whole. 

Use this theorem to find the electric field of this 'atom' at a distance r (r < a) from the centre of the atom.

Answer 1

(a) Electric flux through a closed surface S = q/ε₀ 

Here, q = total charge enclosed by S.

Proof:

Let us consider the total flux through a sphere of radius r, which encloses a point charge g at its centre. Divide the sphere into small area elements,as shown in the figure.

since the magnitude of a unit vector is 1.

The total flux through the sphere is obtained by adding up flux through all the different area elements.

Since each area element of the sphere is at the same distance r from the charge,