question_answer

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

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

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

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

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

Correct Answer: C

Solution :

 Torque acting an equilateral triangle in magnetic field B is \[\tau =iAB\sin \theta \]                ...(i) Area of triangle LMN \[A=\frac{\sqrt{3}}{4}{{l}^{2}}\] and       \[\theta =90{}^\circ \] \[\tau =i\times \frac{\sqrt{3}}{4}{{l}^{2}}B\sin 90{}^\circ \] \[=\frac{\sqrt{3}}{4}i{{l}^{2}}B\]    \[(\therefore \sin 90{}^\circ =1)\] \[l=2{{\left( \frac{\tau }{\sqrt{3}Bi} \right)}^{1/2}}\]