Question

A coil in the shape of an equilateral triangle of side \[0.02\,m\] is a suspended plane between the pole pieces of permanent magnet producing a uniform field of \[5 \times {10^{ - 2}}\,T\]. If a current of \[0.01\,A\] is passed through the coil, the couple acting is
A. \[5\sqrt 3 \times {10^{ - 7}}\,Nm\]
B. \[5\sqrt 3 \times {10^{ - 10}}\,Nm\]
C. \[\dfrac{{\sqrt 3 }}{5} \times {10^{ - 7}}\,Nm\]
D. \[3\sqrt 5 \times {10^{ - 10}}\,Nm\]

Answer 1

Hint: The torque on a loop placed in a magnetic field is equal to the product of the current, area, number of turns, strength of the magnetic field, angle between the field and the loop.

Formula used:
The torque acting on a current loop placed in a magnetic field is given by,
\[\tau = IBNA\sin \theta \]
where, \[I\] is the current flowing through the loop, \[B\] is the magnetic field, \[A\] is the area of the current loop, \[N\] is the number of turns and \[\theta \] is the angle between the loop and the magnetic field.

Complete step by step answer:
We have given here an equilateral triangle of side \[0.02\,m\] placed in between the poles of a permanent magnet of strength \[5 \times {10^{ - 2}}\,T\]. Now, we know that the torque due to a magnetic field on a current loop is given by,
\[\tau = IBNA\sin \theta \]
So, we have here, strength of the magnetic field \[B = 5 \times {10^{ - 2}}T\], area \[A = \dfrac{{\sqrt 3 }}{4}{0.02^2}{m^2}\], number of turns \[N = 1\], current \[I = 0.01A\] and \[\theta = {90^ \circ }\].
Now putting the values we have,
\[\tau = 0.01 \times 5 \times {10^{ - 2}} \times 1\dfrac{{\sqrt 3 }}{4}{0.02^2} \times \sin {90^ \circ }\]
So, simplifying we have,
\[\therefore \tau = 5\sqrt 3 \times {10^{ - 7}}\]
Hence, the value of the torque acting on the triangle is \[5\sqrt 3 \times {10^{ - 7}}\,Nm\].

Hence, option A is the correct answer.

Note: If the mass of the loop is negligible this much torque by the magnetic field will be able to rotate the loop. The same principle is used in electric motors to rotate the motor, there the strength of the magnetic field, current through the loop and the number of turns are kept high to increase the torque on the loop.