Question

(a) Three charges –q, Q and –q are placed at equal distances on a straight line. If the potential energy of the system of these charges is zero, then what is the ratio Q:q? 

(b) (i) Obtain the expression for the electric field intensity due to a uniformly charged spherical shell of radius R at a point distant r from the centre of the shell outside it.

(ii) Draw a graph showing the variation of electric field intensity E with r, for r > R and r < R.

Answer 1

(a) Diagram

(b) Electric field due to a uniformly charged thin spherical shell:

(i) When point P lies outside the spherical shell: Suppose that we have calculate field at the point P at a distance r (r > R) from its centre. Draw Gaussian surface through point P so as to enclose the charged spherical shell. Gaussian surface is a spherical surface of radius r and centre O.

Let \(\vec E\) be the electric field at point P, then the electric flux through area element of area \(\vec {ds}\) is given by

\(d\phi=\vec E.\vec {ds}\)

Since \(\vec {ds}\) is also along normal to the surface

\(d\phi= E. {ds}\)

∴ Total electric flux through the Gaussian surface is given by

Since the charge enclosed by the Gaussian surface is q, according to the Gauss’s theorem,

From equation (i) and (ii) we obtain

(ii) A graph showing the variation of electric field as a function of r is shown below.