SATHEE: Electromagnetic Induction and Alternating Current 7 Question 28

Electromagnetic Induction and Alternating Current 7 Question 28

31. An infinitesimally small bar magnet of dipole moment $M$ is pointing and moving with the speed $v$ in the positive $x$ -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. Find the force opposing the motion of the magnet, if 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 $a$ .

(1997C, 5M)

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Answer:

Correct Answer: 31. $F = \frac{21}{4} \frac{μ_{0}^{2} M^{2} a^{4} v}{R x^{8}}$ (repulsion)

Solution:

Given that $x » a$ .

Magnetic field at the centre of the coil due to the bar magnet is

$B = \frac{μ_{0}}{4 π} \frac{2 M}{x^{3}} = \frac{μ_{0}}{2 π} \frac{M}{x^{3}}$

Due to this, magnetic flux linked with the coil will be,

$φ = B S = \frac{μ_{0}}{2 π} \frac{M}{x^{3}} (π a^{2}) = \frac{μ_{0} M a^{2}}{2 x^{3}}$

$∴$ Induced emf in the coil, due to motion of the magnet is

$\begin{aligned} e & = \frac{- d φ}{d t} = - \frac{μ_{0} M a^{2}}{2} \frac{d}{d t} \frac{1}{x^{3}} \\ = \frac{μ_{0} M a^{2}}{2} \frac{3}{x^{4}} \frac{d x}{d t} = \frac{3}{2} \frac{μ_{0} M a^{2}}{x^{4}} v ∵ \frac{d x}{d t} = v \end{aligned}$

Therefore, induced current in the coil is

$i = \frac{e}{R} = \frac{3}{2} \frac{μ_{0} M a^{2}}{R x^{4}} v$

Magnetic moment of the coil due to this induced current will be,

$M^{'} = i S = \frac{3}{2} \frac{μ_{0} M a^{2}}{R x^{4}} v (π a^{2}) \Rightarrow M^{'} = \frac{3}{2} \frac{μ_{0} π M a^{4} v}{R x^{4}}$

Potential energy of $M^{'}$ in $B$ will be

$\begin{aligned} U & = - M^{'} B \cos 180^{\circ} \\ U & = M^{'} B = \frac{3}{2} \frac{μ_{0} π M a^{4} v}{R x^{4}} \frac{μ_{0}}{2 π} \cdot \frac{M}{x^{3}} \end{aligned}$

$U = \frac{3}{4} \frac{μ_{0}^{2} M^{2} a^{4} v}{R} \frac{1}{x^{7}} \Rightarrow F = - \frac{d U}{d x} = \frac{21}{4} \frac{μ_{0}^{2} M^{2} a^{4} v}{R x^{8}}$