The magnetic moment of an iron bar is $M$. It is now bent in such a way that it forms an arc section

Question

The magnetic moment of an iron bar is $M$. It is now bent in such a way that it forms an arc section of a circle subtending an angle of $60^{\circ}$ at the centre. The magnetic moment of this arc section is

Professor PC Thomas Classes · Answer

(a) : Let the length of the iron bar be $L$. The magnetic moment of bar is given by $ M=m L $ \  Where, $m$ is pole strength. When bar is bent in form of an arc, \ $ \begin{aligned} & L=\frac{	heta}{360} 	imes 2 \pi r=\frac{60^{\circ}}{360^{\circ}} 	imes 2 \pi r \ & L=\frac{\pi r}{3} \Rightarrow r=\frac{3 L}{\pi} \end{aligned} $ \ From $	riangle P Q S$ in figure, $ \sin 30^{\circ}=\frac{x}{r} \Rightarrow x=\frac{r}{2} $ \ Distance between $P R=2 x=r$ New magnetic moment $$ \begin{aligned} & M^{\prime}=m 	imes r \ & \Rightarrow \quad m 	imes\left(\frac{3 L}{\pi}ight) ; M^{\prime}=\frac{3 m L}{\pi} \end{aligned} $$ From equation (i), $M^{\prime}=\frac{3 M}{\pi}$