Question

A bar magnet of length l and magnetic dipole moment M is bent in a form of an arc as shown in the following fig. The new magnetic dipole moment will be:

A. M
B. ${\displaystyle \frac{3}{\pi }}$M
C. ${\displaystyle \frac{2}{\pi }}$M
D. ${\displaystyle \frac{M}{2}}$

Answer 1

Hint: A bar magnet is a rectangular fragment of an object that has permanent magnetic properties, composed of iron, steel or some other ferromagnetic material or ferromagnetic composite. As the size of the source is reduced to zero while maintaining the magnetic moment constant, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles. In this question, a bar magnet has been bent to form an arc, the angle of the arc is given which can be used to find out the new length. After finding the new length, it can directly be applied to the formula for the magnetic moment.

Formula used:
For solving the question, that has been given to us can be solved by using the formula that is used for finding the magnetic moment of a magnet. The formula is:
M=ml

Complete step-by-step answer:
Now before we start solving the above given question, let us take a look at all the parameters that are given to us in the question.
We have been given a bar magnet in which,
Length of the bar magnet = l
Magnetic dipole moment of the bar magnet = M
Now, the bar magnet is bent in a way such that it forms an arc
So, to find out its magnetic dipole moment, we will use the formula for the magnetic moment of a magnet that we have mentioned above
So,
The magnetic moment, M = ml
From the figure,
Let the new length be l’ so it will be
$l^{\prime }={\displaystyle \frac{\pi }{3}}r$
so,
$\Rightarrow r={\displaystyle \frac{3l}{\pi }}$
Now,
Let the new magnetic dipole moment be M’
So, we have
$M^{\prime }=m{l}^{\prime }$
Substituting the values from above
$\Rightarrow M^{\prime }=m\times {\displaystyle \frac{3l}{\pi }}$
$\Rightarrow M^{\prime }={\displaystyle \frac{3}{\pi }}\times M$
As, M = ml
So, the new magnetic dipole moment will be ${\displaystyle \frac{3}{\pi }}M$, i.e., Option - B

So, the correct answer is “Option B”.

Note: A determination of the propensity to be organised in a magnetic field is the magnetic moment. As we all remember, there are two poles of a magnet, i.e., north and south. It is possible to describe the magnetic moment as: the magnetic power and orientation of a magnet or other material creating a magnetic field.