First slide

Magnetic dipole in a uniform magnetic field

Question

Two small magnets have their masses and lengths in the ratio 1:2. The maximum torques experienced by them in a uniform magnetic field are the same. For small oscillations, the ratio of their time periods is (Ignore width of magnet)

Difficult

A

$1 / 2 \sqrt{2}$
B

$1 / \sqrt{2}$
C

$(1 / 2)$
D

$2 \sqrt{2}$

Solution

$T = 2 π \sqrt{\frac{I}{MB}} = 2 π \sqrt{\frac{\frac{m}{12} (l^{2} + b^{2})}{M B}} \Rightarrow T α l \sqrt{m}$
$∴ \frac{T_{1}}{T_{2}} = \frac{l_{1}}{l_{2}} \cdot \sqrt{\frac{m_{1}}{m_{2}}} = \frac{1}{2} \sqrt{\frac{1}{2}} = 1 / 2 \sqrt{2}$

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