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:

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 :

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