A coil in the shape of an equilateral triangle of side 1 is suspended between the pole pieces of a permanent magnet such that $B$ is in the plane of the coil. If due to current $i$ in the triangle, a torque $\tau$ acts on it. The side $l$ of the triangle is

• A

  $\frac{2}{\sqrt{3}}\left(\frac{\tau}{B i}\right)$
• Correct answer
  B

  $2\left(\frac{\tau}{\sqrt{3} B i}\right)^{1 / 2}$
• C

  $\frac{2}{\sqrt{3}}\left(\frac{\tau}{B i}\right)^{1 / 2}$
• D

  $\frac{1}{\sqrt{3}} \frac{\tau}{B i}$

Solution

The current flowing clockw:se in an equilateral triangle has a magnetic field in the direction of $\hat{\mathbf{k}}$.
$\begin{aligned} & \tau=\text { BiNA } \sin \theta \\ & \tau=\text { BiNA } \sin 90^{\circ}\end{aligned}$ $\tau=B i \times \frac{\sqrt{3}}{4} I^2 \times 1$ $\left[\because\right.$ Area of equilateral triangle $=\frac{\sqrt{3}}{4} l^2$ and $N=1$, (assuming] $\Rightarrow \quad l^2=\frac{4 \tau}{\sqrt{3} B i} \Rightarrow l=2\left[\frac{\tau}{B i \sqrt{3}}\right]^{1 / 2}$

Hence, the correct answer is option: 'B'.