Question

A closely wound solenoid of $2000$turns and area of cross-section $1.5 \times {10^{ - 4}}{m^2}$ carries current of $2.0A$. It is suspended through its centre and perpendicular to its length, allowing it to turn in a horizontal plane in a uniform magnetic field $5 \times {10^{ - 2}}T$ making an angle of ${30^0}$ with the axis of the solenoid. The torque on the solenoid with the
(A) $3 \times {10^{ - 3}}N - m$
(B) $1.5 \times {10^{ - 3}}N - m$
(C) $1.5 \times {10^{ - 2}}N - m$
(D) $3 \times {10^{ - 2}}N - m$

Answer 1

Hint
There is a solenoid given, we can treat it as a closed current carrying loop which is placed in a magnetic field. Hence it will experience a torque. We can use the equation of torque with the given values to calculate its magnitude.
$\Rightarrow \tau = NiAB\sin \theta $

Complete step by step answer

We can see in the figure that a solenoid is kept in a magnetic field B which is ${30^0}$ with the axis of the solenoid
We know that torque $\tau = \overrightarrow M \times \overrightarrow B = MB\sin \theta $ where M is the magnetic moment and B is the magnetic field
The magnetic moment of a current loop is $\overrightarrow M = N\overrightarrow i A$ where i is the current, N is the number of turns and A is the current loop area.
From this the torque equation becomes, $\tau = NiAB\sin \theta $
In the question it is given that N=2000, $A = 1.5 \times {10^{ - 4}}{m^2}$, $i = 2.0A$, $B = 5 \times {10^{ - 2}}T$and $\theta = {30^0}$
Substituting these in torque equation, $\tau = 2000 \times 2 \times 1.5 \times {10^{ - 4}} \times 5 \times {10^{ - 2}}\sin {30^0}$
$ \Rightarrow \tau = 0.6 \times 5 \times {10^{ - 2}} \times \dfrac{1}{2} = 1.5 \times {10^{ - 2}}N - m$
Hence the correct option is (C).

Additional Information
Electric motors and galvanometer are based on the principle that a deflecting torque is experienced when a current loop is kept in a magnetic field, this torque depends upon the magnitude of current. Galvanometer is a device which is used to measure current.
We can find the direction of magnetic moment using the curled-straight right hand rule for any planar current carrying loop.

Note
A solenoid carrying current has behavior similar to a bar magnet. So, it is a magnetic dipole like a bar magnet. When a current loop experiences torque when it is suspended in a magnetic field, the torque rotates the loop and tends it to a position where the axis of the loop is parallel to the field.