Question

A straight wire carrying a current (I) is turned into a circular loop. If the magnitude of magnetic moment associated with it is 'M', then the length of the wire will be

• A. \(4\pi \text{ IM}\)
• B. \(\sqrt{\frac{4\pi M}{I}}\)
• C. \(\sqrt{\frac{4\pi I}{M}}\)
• D. \(\frac{M\pi }{4\text{I}}\)

Hint:

Express the loop radius through the wire length, then write the magnetic moment as current times area.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
A loop with current \(I\) and area \(A\) has magnetic moment \(M=IA\).

Step 2: Geometry of the loop
A wire of length \(L\) bent into one circle has \(2\pi r=L\), so \(r=\dfrac{L}{2\pi}\) and \(A=\pi r^2=\dfrac{L^2}{4\pi}\).

Step 3: Substitute
\(M=\dfrac{IL^2}{4\pi}\Rightarrow L=\sqrt{\dfrac{4\pi M}{I}}\)

Final Answer:
The length of the wire is \(\sqrt{\dfrac{4\pi M}{I}}\). \[ \boxed{\text{(B) }\sqrt{\dfrac{4\pi M}{I}}} \]