Question

A wire of length $L$ is bent in form of a circular coil and current $i$ is passed through it. If this coil is placed in a magnetic field, then the torque on coil will be maximum when number of turns is:

• A. As large as possible
• B. $4$
• C. $2$
• D. $1$

Answer 1

The correct option is D $1$

The torque on the coil is given by,

$\overrightarrow{\tau }=\overrightarrow{M}\times \overrightarrow{B}$

The magnitude of torque depends on both magnitude of magnetic moment, magnetic field and the angle between $\overrightarrow{M}$ & $\overrightarrow{B}$

$\tau =MBsin\theta$

${\tau }_{max}=MB(when\theta ={90}^o)$

Let the number of turns in the coil be $n$.

$L=n\left(2\pi r\right)$

or, $r=\frac{L}{2n\pi }$

${\tau }_{max}=\left(niA\right)B$

${\tau }_{max}=ni\left(\pi r^2\right)B$

${\tau }_{max}=niB\pi \frac{L^2}{4n^2{\pi }^2}$

$L$, $i$ and $B$ are constants.

${\tau }_{max}\alpha \frac{1}{n}$

Thus, $n_{min}$ will lead to ${\tau }_{max}$

$\therefore n=1$

Hence, option $\left(D\right)$ is the correct answer.