Question

For a given length of the wire, carrying a current $I$, how many circular turns will produce maximum magnetic moment ?

• A. $1$
• B. $2$
• C. $4$
• D. $16$

Answer 1

The correct option is A $1$

Let $l$ be the total length of the wire.

$\therefore l=N\left(2\pi a\right)$

$a=\frac{l}{2\pi N}$

Where, $N:$ number of turns,
$a:$ radius of coil

So, magnetic dipole moment will be

$M=NIA=NI\left(\pi a^2\right)$

$M=NI(\pi \frac{l^2}{4{\pi }^2{N}^2})$

$\therefore M=\frac{I{l}^2}{4\pi N}$

For $M$ to be maximum, $N$ should be minimum. Here, the smallest possible value of $N$ is $N=1$ as $N$ must be an integer.

$\therefore M_{max}=\frac{I{l}^2}{4\pi }$

Therefore, option $\left(A\right)$ is the correct answer.