Question

A coil in the shape of an equilateral triangle of side $l$ 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. $\frac{2}{\sqrt{3}}\left(\frac{\tau }{Bi}\right)$
• B. $2{\left(\frac{\tau }{\sqrt{3}Bi}\right)}^{1/2}$
• C. $\frac{2}{\sqrt{3}}{\left(\frac{\tau }{Bi}\right)}^{1/2}$
• D. $\frac{1}{\sqrt{3}}\frac{\tau }{Bi}$

Answer 1

The correct option is B $2{\left(\frac{\tau }{\sqrt{3}Bi}\right)}^{1/2}$

The current flowing clockwise in an equilateral triangle has a magnetic field in the direction of $\hat{k}$.
$\tau =BiNA\sin \theta$
$\tau =BiNA\sin {90}^o$
$\tau =Bi\times \frac{\sqrt{3}}{4}{I}^2\times 1$
[$\because$ Area of equilateral triangle $=\frac{\sqrt{3}}{4}{I}^2$ and $N=1$]
(assuming)
$\Rightarrow I^2=\frac{4\tau }{\sqrt{3}Bi}\Rightarrow I=2{\left[\frac{\tau }{Bi\sqrt{3}}\right]}^{1/2}$