A small bar magnet of dipole moment M is moving with speed v along x direction towards a small closed circular conducting loop of radius ' a ' with its centre O at x = 0 (see figure). Assume x a and the coil has a resistance R . Then which of the following statemenmt(s) is/are true?

Question

A small bar magnet of dipole moment M is moving with speed v along x direction towards a small closed circular conducting loop of radius ' a ' with its centre O at x = 0 (see figure). Assume x >> a and the coil has a resistance R . Then which of the following statemenmt(s) is/are true?, Magnetic field at the centre O of the circular coil due to the bar magnet is x 3 M ​, Induced EMF is proportional to x 4 1 ​, The magnetic moment μ due to induced current in the coil is proportional to a 4, The heat produced is proportional to x 6 1 ​

MARKS App · Accepted Answer

Magnetic field at an axial point is 2 ( 4 π μ 0 ​ ​ ) x 3 M ​ $ ϕ = 2 ( 4 π μ 0 ​ ​ ) x 3 M ​ ⋅ π a 2 \mathrm{EMF}=\frac{-\mathrm{d} \phi}{\mathrm{dt}}=-2\left(\frac{\mu_{0}}{4 \pi}\right) \mathrm{M} \pi \mathrm{a}^{2}\left(\frac{-3}{\mathrm{x}^{4}}\right) \frac{\mathrm{dx}}{\mathrm{dt}}=6\left(\frac{\mu_{0}}{4 \pi}\right) \frac{\mathrm{M} \pi \mathrm{a}^{2}}{\mathrm{x}^{4}} \mathrm{~V} \quad \mathrm{EMF} \propto \frac{1}{\mathrm{x}^{4}} C u rre n t in co i l =\frac{E M F}{R} \quad i=6\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M \pi a^{2}}{x^{4}} \frac{V}{R} M a g n e t i c m o m e n t \mu=\mathrm{i} \pi \mathrm{a}^{2}, \mu \propto \mathrm{a}^{4} R a t e a tw hi c hh e a t i s p ro d u ce d =\mathrm{i}^{2} \mathrm{R} . \propto \frac{1}{\mathrm{x}^{8}}$

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