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  4. A magnetic needle of negligible breadth and thickness compared to its length, oscillates in a horizontal plane with a period T . The period of oscillation of each part obtained on breaking the magnet into n equal parts and perpendicular to the length is

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Q. A magnetic needle of negligible breadth and thickness compared to its length, oscillates in a horizontal plane with a period $T$ . The period of oscillation of each part obtained on breaking the magnet into $n$ equal parts and perpendicular to the length is

NTA AbhyasNTA Abhyas 2020Moving Charges and Magnetism

Solution:

$T=2\pi \sqrt{\frac{I}{M B}}$
on breaking n equal parts perpendicular to length
$I′=\frac{1}{12} \, \left(\frac{m}{n}\right) \, \left(\frac{l}{n}\right)^{2}$ of each part
$I′=\frac{I}{n^{3}}, \, M′=\frac{M}{n}$
$\frac{T^{′}}{T}=\sqrt{\frac{I^{′}}{M^{′}} \times \frac{M}{I}}=\sqrt{\frac{1}{n^{3}} \, \times n}=\frac{1}{n}\Rightarrow \, T^{′}=\frac{T}{n}$