Question

An infinitesimally small bar magnet of dipole moment $P_m$ is pointing and moving with the speed v in the x-direction. A small closed circular conducting loop of radius a and negligible self–inductance lies in the y-z plane with its centre at x = 0, and its axis coinciding with the x-axis. The resistance of the loop is R. Assume that the distance x of the magnet from the centre of the loop is much greater than a. The force opposing the motion is $\propto \frac{1}{x^y}$. Value of y is ___

Answer 1

Flux $\Phi$ linked with the loop
$=\frac{2p_m}{x^3}\pi a^2\times \frac{{\mu }_0}{4\pi }$
Induced emf
$\epsilon =-\frac{6\pi a_m^{p}v}{x^4}\times \frac{{\mu }_0}{4\pi }$
Current in the ring
$=\frac{3{\mu }_0{a}^{2}v{p}_m}{2x^{4}R}$
This current will set up a field at the site of the magnet gives by
$B^{{}^{\prime }}=\frac{{\mu }_0}{2\pi }\frac{i\pi a^2}{x^3}=\frac{3{\mu }_0^2{a}^{4}v{p}_m}{4x^{7}R}$
This will give rise to a force in the dipole
$\left|F\right|=\left|p_m\frac{\delta B^{{}^{\prime }}}{\delta x}\right|=\frac{27{\mu }_0^2{a}^{4}v{p}_m^2}{4x^{8}R}$
By Lenz'z law this forces opposes the motion of the dipole in all cases.