Question

Current $ i $ is carried in a wire of length $ L $ . If the wire is turned into circular coil, the maximum magnitude of torque in a given magnetic field $ B $ will be
(A) $ \dfrac{{Li{B^2}}}{2} $
(B) $ \dfrac{{L{i^2}B}}{2} $
(C) $ \dfrac{{{L^2}iB}}{2} $
(D) $ \dfrac{{{L^2}iB}}{{4\pi }} $

Answer 1

Hint: The measure of the force that can cause an object to rotate about an axis is torque. A simple way to calculate the torque magnitude is to determine the lever arm first and then multiply it by multiplying the applied force. The torque unit is the Newton meter ( $ N - m $ ). It is possible to represent the above equation as the vector product of the vector of force and position.

Formula used:
We will use the following formula to solve this question
 $ \mathop \tau \limits^ \to = \mathop \mu \limits^ \to \times \mathop B\limits^ \to $
Where
 $ \tau $ is the torque
 $ \mu $ is the magnetic moment
 $ B $ is the magnetic field

Complete step by step solution:
According to the question, The length of the wire is $ L $ and the same wire is now converted into a circular coil of radius $ r $ .
Since both the wires are the same , the length of the wire will be equal to the circumference of the circular coil.
The circumference of the circular coil will be $ 2\pi r $
Then
 $ L = 2\pi r $
 $ \therefore r = \dfrac{L}{{2\pi }} $
In this circular coil, current $ i $ passes through it and the direction of magnetic field is known to us as per the question.
Now according to the right-hand screw rule, the direction of the magnetic moment $ \mu $ will be vertically outwards out of the plane.
So, the angle between magnetic moment and magnetic field will be $ {90^\circ } $ .
Now magnetic moment, $ \mu = iA $
Where
 $ A $ is the area of the circular coil
 $ \Rightarrow \mu = i\dfrac{{\pi {L^2}}}{{4{\pi ^2}}} $
Which can also be written as
 $ \Rightarrow \mu = i\dfrac{{{L^2}}}{{4\pi }} $
Now the question is asking for the torque. It can be calculated by using the formula
 $ \mathop \tau \limits^ \to = \mathop \mu \limits^ \to \times \mathop B\limits^ \to $
Now the magnitude of the torque can be calculated by
 $ |\mathop \tau \limits^{} | = |\mathop \mu \limits^{} | \times |\mathop B\limits^{} |\sin \theta $
We already know that value of angle $ \theta = {90^\circ } $ and $ \sin {90^\circ } = 1 $
So, we can write the above expression as
 $ \Rightarrow |\mathop \tau \limits^{} | = |\mathop \mu \limits^{} | \times |\mathop B\limits^{} | $
 $ \therefore |\mathop \tau \limits^{} | = i\dfrac{{{L^2}}}{{4\pi }} \times B $
Hence the correct option is (D).

Note:
In this question, the same wire has been bent and converted into a circular coil. So, the length of the wire will be equal to the circumference of the circular coil. Understanding this simple and easy concept can enable us to solve the problem easily and faster.