The magnetic moment and moment of inertia of a magnetic needle as shown are, respectively, and If it completes 10 oscillations in 10 s, the magnitude of the magnetic field is [2024]

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Solution

Q.
The magnetic moment and moment of inertia of a magnetic needle as shown are, respectively, $1.0 \times 10^{- 2}$ $A$ $m^{2}$ and $\frac{10^{- 6}}{π^{2}}$ $kg$ $m^{2} .$

If it completes 10 oscillations in 10 s, the magnitude of the magnetic field is [2024]

1 0.4 T

2 4 T

3 0.4 mT

4 4 mT

Ans.
(3)

Magnetic moment, $M = 1.0 \times 10^{- 2} {Am}^{2}$

Moment of inertia, $I = \frac{10^{- 6}}{π^{2}}$ $kg$ $m^{2}$

Time period for one oscillation $\frac{10}{10} = 1$ $s$

We know that, $T = 2 π \sqrt{\frac{I}{M B}}$

$∴$ $T^{2} = \frac{4 π^{2} I}{M B}$

$B = \frac{4 π^{2} I}{M T^{2}} = \frac{4 \times {(3.14)}^{2}}{1 \times 10^{- 2}} \times \frac{10^{- 6}}{{(3.14)}^{2}} = 4 \times 10^{- 4} = 0.4$ $mT$

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