Question

A straight wire carrying current \( i \) is turned into a circular loop. If the magnitude of magnetic moment associated with it in MKS unit is \( M \), the length of wire will be

• A. \( \dfrac{4\pi}{M} \)
• B. \( \sqrt{\dfrac{4\pi M}{i}} \)
• C. \( \sqrt{\dfrac{4\pi i}{M}} \)
• D. \( \dfrac{M\pi}{i} \)

Hint:

Magnetic moment \(M\) of a current loop is always \(M = iA\). For circular loop, \(A=\pi r^2\) and wire length \(L=2\pi r\).

Answer 1

The Correct Option is B

Step 1: Write the formula of magnetic moment.
Magnetic moment of a current loop is given by:
\[ M = iA \] where \(A\) is the area of the loop.
Step 2: Express area for a circular loop.
For a circle of radius \(r\),
\[ A = \pi r^2 \] So,
\[ M = i\pi r^2 \] Step 3: Find radius \(r\) in terms of \(M\).
\[ r^2 = \dfrac{M}{i\pi} \Rightarrow r = \sqrt{\dfrac{M}{i\pi}} \] Step 4: Find the length of wire.
Length of wire = circumference of loop
\[ L = 2\pi r \] Substituting \(r\),
\[ L = 2\pi \sqrt{\dfrac{M}{i\pi}} = \sqrt{\dfrac{4\pi M}{i}} \] Final Answer: \[ \boxed{\sqrt{\dfrac{4\pi M}{i}}} \]