Question

An iron rod of length L and magnetic moment M is bent in the form of a semicircular. Now its magnetic moment will be
( a ) M
( b) $\dfrac{2M}{\pi }$
( c ) $\dfrac{M}{\pi }$
( d ) $M\pi $

Answer 1

Hint:
In this question, we are given an iron rod and we have to find its magnetic moment when it is bent in the shape of semicircular. A magnetic dipole consists of two not like poles of equivalent strength and separated by a small distance . A magnetic moment is the product of pole strength and the distance between the poles of the magnet. Hence to find the magnetic moment, we use the formula of magnetic moment and then we find the magnetic moment when wire is in the form of semicircle and by putting the values, we get the desirable answer.



Complete step by step solution:
Given M = magnetic moment of the iron rod.
         L is the Length of the iron rod
The magnetic moment of the iron rod is given as
$M=m\times l$---------------------------(1)
Where m is the pole strength of the rod.
On bending a pole, its pole strength remains unchanged whereas its magnetic moment changes.
When each part is bent in the form of a semicircle, the distance between the poles is given as ( d = 2 radius of the semicircle )
${{l}_{s}}=2r$------------------------------(2)
The magnetic moment of the semicircular part is given as ( ${{m}_{s}}=m$)
Therefore, ${{M}_{s}}={{m}_{s}}\times {{l}_{s}}$
That is ${{M}_{s}}=m\times 2r$-------------------------- (3)
The length of the circumference of each semicircular part is given as
$l=\pi r$
r = $\dfrac{l}{\pi }$ --------------------------------- (4)
From equation (3) and (4), we get
${{M}_{s}}=m\times 2\dfrac{l}{\pi }$
${{M}_{s}}=\dfrac{2ml}{\pi }$ ----------------------------- (5)
Now from the equation (3) and equation (5), we get
${{M}_{s}}=\dfrac{2M}{\pi }$
Thus, Option (B) is correct.


Therefore, the correct option is B.




Note:
In this question, we know the circumference of a circle is $2\pi r$.
Then circumference of semicircle is $\dfrac{2\pi r}{2}=\pi r$
Since the length of rod is L. then L = $\pi r$
Then we get $r=\dfrac{l}{\pi }$
Keep in mind that magnetic moment depends on the distance between two poles hence if shape changes magnetic moment may change.
Keep in mind these things while solving these types of questions.