Question

A square loop and a circular loop are each made from the same length of wire. Each loop carries the same amount of current. The loops are placed in the same uniform magnetic field. What is the ratio of the maximum torque on the square loop to the maximum torque on the circular loop? Question 2 options: π/4 2/π None of the other answers are correct. 4/π π/2

Answer 1

Explanation:

Let l is the length of the wire. such that, side of square is l/4. Area of square loop is, $A_s=\dfrac{l^2}{16}$

Radius of the circular loop,

$l=2\pi r\\\\r=\dfrac{l}{2\pi}$

Area of the circular loop,

$A_a=\pi r^2\\\\A_a=\pi \times (\dfrac{l}{2\pi})^2\\\\A=\dfrac{l^2}{4\pi}$

Torque in magnetic field is given by :

$\tau=NIAB\sin\theta$

It is clear that, $\tau\propto A$

So,

$\dfrac{\tau_s}{\tau_a}=\dfrac{A_s}{A_a}\\\\\dfrac{\tau_s}{\tau_a}=\dfrac{\dfrac{l^2}{16}}{\dfrac{l^2}{4\pi}}\\\\\dfrac{\tau_s}{\tau_a}=\dfrac{\pi}{4}$

So, the ratio of maximum torque on the square loop to the maximum torque on the circular loop is $\pi :4$.