Solveeit Logo
Login

Question

Question: A closely wound solenoid of \(2000\)turns and area of cross-section \(1.5 \times {10^{ - 4}}{m^2}\) ...

A closely wound solenoid of 20002000turns and area of cross-section 1.5×10−4m21.5 \times {10^{ - 4}}{m^2} carries current of 2.0A2.0A. It is suspended through its centre and perpendicular to its length, allowing it to turn in a horizontal plane in a uniform magnetic field 5×10−2T5 \times {10^{ - 2}}T making an angle of 300{30^0} with the axis of the solenoid. The torque on the solenoid with the
(A) 3×10−3N−m3 \times {10^{ - 3}}N - m
(B) 1.5×10−3N−m1.5 \times {10^{ - 3}}N - m
(C) 1.5×10−2N−m1.5 \times {10^{ - 2}}N - m
(D) 3×10−2N−m3 \times {10^{ - 2}}N - m

Explanation

Solution

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.
⇒τ=NiABsin⁡θ\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 300{30^0} with the axis of the solenoid
We know that torque τ=M→×B→=MBsin⁡θ\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 M→=Ni→A\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, τ=NiABsin⁡θ\tau = NiAB\sin \theta
In the question it is given that N=2000, A=1.5×10−4m2A = 1.5 \times {10^{ - 4}}{m^2}, i=2.0Ai = 2.0A, B=5×10−2TB = 5 \times {10^{ - 2}}Tand θ=300\theta = {30^0}
Substituting these in torque equation, τ=2000×2×1.5×10−4×5×10−2sin⁡300\tau = 2000 \times 2 \times 1.5 \times {10^{ - 4}} \times 5 \times {10^{ - 2}}\sin {30^0}
⇒τ=0.6×5×10−2×12=1.5×10−2N−m\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.