Question

A straight wire carrying current $I$ is made into a circular loop. If $M$ is the magnetic moment associated with the loop, then the length of the wire is:
A. $\sqrt {\dfrac{{4\pi M}}{I}}$
B. $\sqrt {\dfrac{{2\pi M}}{I}}$
C. $\sqrt {\dfrac{{\pi M}}{{2I}}}$
D. $\sqrt {\dfrac{{\pi M}}{{4I}}}$

Answer 1

In the definition for the current loop, the magnetic moment is defined as the product of the current flowing and the area. We will take the idea of magnetic moment and insert the parameters provided into the formula to discover our desired response to the given issue, which is to know the length of the wire with magnetic moment.

Formula used:
Magnetic moment, $M = IA$.
Here, $I$ is the current and $A$ is the area of the cross section.

Complete step by step answer:
We've given the magnetic moment M to a circular loop of radius $R$ . we are going to let the current $I$ run through the loop. We have the magnetic moment's expression.
$M = IA$
The current is $I$ , and the area is $\,A$. We know that the circular loop's area is $\pi {R^2}$. As a result, the circular loop's magnetic moment will be,
$M = I\left( {\pi {R^2}} \right) \\\ \Rightarrow M = \pi {R^2}I\,\,\,........\left( i \right) \\\$
Let us consider length of the wire will be $L$
And this $L$ will converted to circular loop of radius $R$
$L = 2\pi R$
Therefore from here we will equate for $R$
$R = \dfrac{L}{{2\pi }}$
Hence, from equation $\left( i \right)$ we will put values in it and find for $L$

\Rightarrow M = \left( {\dfrac{{I{L^2}}}{{4\pi }}} \right)$$ From here, we will find for $L$ ${L^2} = \dfrac{{4\pi M}}{I} \\\ \therefore L = \sqrt {\dfrac{{4\pi M}}{I}}$ Therefore the length of the wire is $\sqrt {\dfrac{{4\pi M}}{I}} $. **Therefore, the correct option is A.** **Note:** It's worth mentioning that magnetometers are commonly used to measure the magnetic moments of things, albeit not all magnetometers do so: Instead of measuring magnetic field, some of them are set up to measure it. The magnetic moment can be estimated from the magnetic field surrounding an item if the magnetic field is known well enough.