A bar magnet placed in a uniform magnetic field making an angle $ heta$ with the field experiences a torque. If the angle made by the magnet with the field is doubled, the torque experienced by the magnet increases by $41.4 \%$. The initial angle made by the magnet with the magnetic field is

Question

A bar magnet placed in a uniform magnetic field making an angle $	heta$ with the field experiences a torque. If the angle made by the magnet with the field is doubled, the torque experienced by the magnet increases by $41.4 \%$. The initial angle made by the magnet with the magnetic field is

Aicharya · Accepted Answer

As we know, torque in a magnetic field, $ \begin{aligned} \tau & =M B \sin \theta \\ \theta & =\theta_l, \end{aligned} $

Similarly, if $\theta=\theta_2$ then $ \begin{aligned} \tau_2=M B \sin \theta_2 & =M B \sin 2 \theta_1 \\ \text { Given } \quad \therefore \quad \tau_2 & =\tau_1+\tau_1 \times \frac{41.4}{100} \\ & =1.414 \tau_1=\sqrt{2} \tau_1 \end{aligned} \quad\left(\because \text { Given, } \theta_2=2 \theta_1\right) $

From Eqs. (i) and (ii), we get $ \frac{1}{\sqrt{2}}=\frac{\sin \theta_1}{\sin 2 \theta_1} $

As we know that $\sin 2 \theta=2 \sin \theta \cos \theta$ Hence, $2 \sin \theta_1 \cos \theta_1=\sqrt{2} \sin \theta_1$ $ 2 \cos \theta=\sqrt{2} \Rightarrow \cos \theta=\frac{1}{\sqrt{2}} \Rightarrow \theta=45^{\circ} $ Hence, the correct option is (d)

A bar magnet placed in a uniform magnetic field making an angle $\theta$ with the field experiences a torque…

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