Mixed Worksheet for Chapter 1 - Electric Charges and Fields

(i) Two infinitely long straight wires having linear charge densities − λ and 3λ are held vertically parallel to each other, distance r apart in free space. Find the nature and magnitude of the force/length exerted by one wire on the other.

(ii) A small hollow conducting sphere of radius \(r_{1}\) is given a charge Q. It is surrounded by a concentric conducting spherical shell of inner radius \(r_{2}\) and outer radius \(r_{3}\), having charge − 3q. If a point charge 2q were kept at the centre, find :

(I) the electric flux through a concentric spherical Gaussian surface of radius x for (1) x < \(r_{1}\), and (2) \(r_{1}\) < x < \(r_{2}\).

(II) electric field at a point distant x from the centre for (1) x > \(r_{3}\), and (2) \(r_{1}\) < x < \(r_{2}\).

(III) surface charge density on the inner surface of (1) sphere, and (2) shell.

An electric dipole \(\vec{P}=2qa\hat{i}\) is placed in x-y plane, centred at the origin. Let \(E_1\) and \(E_2\) be the magnitudes of electric field at a point distant \(r\;(\gg a)\) from its centre, along its axis and on its equatorial plane, respectively. Then the value of \(\frac{E_1}{E_2}\) is: