Question

A straight wire carrying current I is turned into a circular loop. If the magnitude of the magnetic moment associated in the MKS units in M, then the length of the wire is
A. $4\pi IM$
B. $\sqrt {\dfrac{{4\pi M}}{I}}$
C. $\sqrt {\dfrac{{4\pi I}}{M}}$
D. $\dfrac{{M\pi }}{{4I}}$

Answer 1

In this question, a wire of some finite length has been turned to form a circular loop. Thus, the circumference of the loop shall be equal to the length of the wire. On equating these two, we will obtain the radius of the circular loop in terms of the length of the wire. Then we shall apply the formula of the magnetic moment of a circular loop. Making proper substitutions would get us the correct answer.

Complete answer:
Magnetic moment is defined as the measure of an object to align with the applied magnetic field. The magnetic moment is a vector quantity. For a current carrying loop, it is the product of the current flowing through the loop and the area vector.
Mathematically it is expressed as $M = IA$ where M is the magnetic moment, I is the current flowing through the loop and A is the area vector.
The unit of magnetic moment is $A\,{m^2}$ .
Let the loop carry a current I and has the radius r.
Since the same length of the wire was circled into a loop, the circumference of the loop shall be equal to the length of the wire.
Assume the length of the wire to be L.
Hence, we can say that $L = 2\pi r$
This can be rewritten as $r = \dfrac{L}{{2\pi }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,.........(1)$
Now using the definition of the magnetic moment, we have,
$M = IA$
Substituting the formula for area of a loop,
$M = I(\pi {r^2})$
Substituting the value of r from equation 1,
$M = I\pi {(\dfrac{L}{{2\pi }})^2}$
Further solving this equation, we get,
${L^2} = \dfrac{{4\pi M}}{I}$
$\Rightarrow L = \sqrt {\dfrac{{4\pi M}}{I}}$
Hence, option B is the correct answer.

Note:
The magnetic moment is a vector quantity. So, along with its magnitude, its direction also holds importance. Generally, the direction of magnetic moment is along the direction of the magnetic flux. For a loop, the direction of the magnetic moment is along the area vector which is perpendicular to the plane of the loop.