question_answer

A coil in the shape of an equilateral triangle of side (is suspended between the pole pieces of a permanent magnet such that \[\vec{B}\] is in plane of the coil. If due to a current i in the triangle a torque t acts on it, the side 2 of the triangle is:                                                                                [AIPMT (S) 2005]

A)  \[\frac{2}{\sqrt{3}}{{\left( \frac{\tau }{Bi} \right)}^{1/2}}\]

B)                   \[\frac{2}{\sqrt{3}}\left( \frac{\tau }{Bi} \right)\]

C)  \[2{{\left( \frac{\tau }{\sqrt{3}\,Bi} \right)}^{1/2}}\]

D)                   \[\frac{1}{\sqrt{3}}\,\frac{\tau }{Bi}\]

Correct Answer: C

Solution :

[c] Torque acting on equilateral triangle in a magnetic field \[\vec{B}\] is

\[\tau =i\,AB\sin \theta \]

Area of triangle LMN

\[A=\frac{\sqrt{3}}{4}{{l}^{2}}\] and \[\theta ={{90}^{o}}\]

Substituting the given values in the expression for torque, we have

\[\tau =i\times \frac{\sqrt{3}}{4}{{l}^{2}}B\,\sin {{90}^{o}}\]

\[=\frac{\sqrt{3}}{4}i\,{{l}^{2}}\]                    \[(\because \sin {{90}^{o}}=1)\]

Hence, \[l=2{{\left( \frac{\tau }{\sqrt{3}Bi} \right)}^{1/2}}\]