Question

A circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1T. The field lines make an angle of 60$^\circ$ with the normal of the coil. Calculate the magnitude of counter-torque that must be applied to prevent the coil from turning.

Answer 1

The torque produced in a coil placed in a magnetic is equal to the product of the number of turns in the coil, the current, the magnetic field, area of the coil and the sine of angle between the area vector and the magnetic field. We have all the required values from which we can obtain the required value of counter torque.

Complete answer:
We are given a circular coil. The number of turns in this coil are given as
$n = 30$
The radius of this circular coil is given as
$r = 8cm = 0.08m$
Therefore, the area of this coil can be obtained in the following way.
$A = \pi {r^2} = \pi {\left( {0.08} \right)^2}{m^2}$
The coil is carrying a current whose value is given as
$I = 6A$
This coil is suspended vertically in a uniform horizontal magnetic field whose magnitude is given as
$B = 1T$
The angle between the area vector of the coil and the external magnetic field is given as
$\theta = 60^\circ$
The torque acting on the coil due to the magnetic field is given by the following expression.
$\tau = nI\overrightarrow B \times \overrightarrow A = nBIA\sin \theta$
Now we can insert the known values in this expression to obtain the value of torque acting on the coil. Doing so, we get
$\tau = 30 \times 1 \times 6 \times \pi \times 0.08 \times 0.08 \times \sin 60^\circ \\\ = 3.13Nm \\\$
This is the required value of torque.

Note:
It should be noted that the counter torque is equal to the value of the torque that we have obtained above. The counter torque will be applied in direction opposite to the direction of the above torque but with the same magnitude as that of the torque produced in the coil when it is placed in the given magnetic field so that the coil does not turn in the magnetic field.