First slide

Magnetism

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

Moderate

A

$\frac{1}{2 \sqrt{2}}$
B

$\frac{1}{\sqrt{2}}$
C

$\frac{1}{2}$
D

$2 \sqrt{2}$

Solution

We know time period
$T = 2 π \sqrt{\frac{I}{M B_{H}}}$
Where $I = \frac{m L^{2}}{12}$ and $M = M a g n e t i c M o m e n t$
$G i v e n : Γ_{1} = Γ_{2} \Rightarrow M_{1} B \sin 90^{o} = M_{2} B \sin 90^{o} [M a g n e t i c f i e l d i s s a m e] \Rightarrow M_{1} = M_{2} H e n c e, M a g n e i c M o m e n t i s a l s o s a m e .$
$\begin{array}{l} ∴ T \propto \sqrt{m} L \\ \Rightarrow \frac{T_{1}}{T_{2}} = \sqrt{\frac{m_{1}}{m_{2}}} \times \frac{L_{1}}{L_{2}} \end{array}$
Given $\frac{m_{1}}{m_{2}} = \frac{1}{2}, \frac{L_{1}}{L_{2}} = \frac{1}{2}$
$\Rightarrow \frac{T_{1}}{T_{2}} = \sqrt{\frac{1}{2}} \times \frac{1}{2} = \frac{1}{2 \sqrt{2}}$

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