The magnetic moment of an iron bar is . It is now bent in such a way that it forms an arc section of a circle subtending an angle of at the centre. The magnetic moment of this arc section is [2024]

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Solution

Q.
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 $60 °$ at the centre. The magnetic moment of this arc section is [2024]

1 $\frac{3 M}{π}$

2 $\frac{4 M}{π}$

3 $\frac{M}{π}$

4 $\frac{2 M}{π}$

Ans.
(1)

Let the length of the iron bar be $L$ .
The magnetic moment of bar is given by

$M = m L$ ...(i)

Where, $m$ is pole strength.

When the bar is bent in the form of an arc,

$L = \frac{θ}{360} \times 2 π r = \frac{60 °}{360 °} \times 2 π r$

$L = \frac{π r}{3} \Rightarrow r = \frac{3 L}{π}$

From $∆ P Q S$ in the figure,

$\sin 30 ° = \frac{x}{r} \Rightarrow x = \frac{r}{2}$

Distance between $P R = 2 x = r$

New magnetic moment

$M' = m \times r$

$\Rightarrow m \times (\frac{3 L}{π}); M' = \frac{3 m L}{π}$

From equation (i), $M' = \frac{3 M}{π}$

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