A straight wire carrying a current (I) is turned into a circular loop. If the magnitude of the magnetic…

A straight wire carrying a current (I) is turned into a circular loop. If the magnitude of the magnetic moment associated with it is ' $\mathrm{M}$ ', then the length of the wire will be
  1. $\frac{\mathrm{M} \pi}{4 \mathrm{I}}$
  2. $\left[\frac{4 \pi \mathrm{I}}{\mathrm{M}}\right]^{\frac{1}{2}}$
  3. $\left[\frac{4 \mathrm{M} \pi}{\mathrm{I}}\right]^{\frac{1}{2}}$
  4. $4 \pi \mathrm{MI}$

Solution

Magnetic moment is given as $\mathrm{M}=\mathrm{IA}$ $\therefore \quad \mathrm{M}=\mathrm{I}\left(\pi \mathrm{R}^2\right)$ Now, length of the wire is, $L=2 \pi R$ $\begin{aligned} & \therefore \quad \mathrm{R}=\frac{\mathrm{L}}{2 \pi} \\ & \Rightarrow \quad \mathrm{M}=\mathrm{I} \pi\left(\frac{\mathrm{L}}{2 \pi}\right)^2 \\ & \therefore \quad \mathrm{L}=\sqrt{\frac{4 \mathrm{M} \pi}{\mathrm{I}}} \end{aligned}$

Asked in: MHT CET 2023 (14 May Shift 1)

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