A magnetic needle has magnetic moment of \( 5\times10^{-2}\,\mathrm{A\,m^{2}} \) and Moment of inertia \( 12\times10^{-6}\,\mathrm{kg\,m^{2}} \). It performs 60 oscillations in 1 minute when suspended in a magnetic field. What is the magnitude of the magnetic field?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 94.7\times10^{-4}\,\mathrm{T} \)
A magnetic needle oscillating in a field B has the time period \( T = 2\pi\sqrt{\frac{I}{MB}} \), where I is its moment of inertia and M its magnetic moment.
\( \displaystyle T = \frac{60\,\mathrm{s}}{60} = 1\,\mathrm{s} \)
\( \displaystyle B = \frac{4\pi^{2}I}{MT^{2}} = \frac{4\times9.87\times12\times10^{-6}}{5\times10^{-2}\times1^{2}} \)
\( B \approx 9.47\times10^{-3}\,\mathrm{T} = 94.7\times10^{-4}\,\mathrm{T} \)