Question

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single turn. The long side of the rectangle is 1.8 times the short side. Find the ratio (τsquare/τrectangle) of the maximum torques that these coils experience in the same magnetic field when they contain the same current.

Answer 1

As we know that the flux lines cut by the conductor is proportional to the area of the loop and so Torque is proportional to the area of the loop.

Now, consider the side of the square loop is A.
Area of square is A^2
perimeter of square = 4A
Again, if l = length and b is the breadth of the rectangle loop
l = 1.8b
then perimeter = 2*(1.8b + b ) = 5.6 b

so -  
5.6 b = 4A
=> b = (4 / 5.6)A = 0.71 A

and l = 1.8*(4 / 5.6)A = 1.28 A
Area of rectangle = 0.71* 1.28*A^2 = 0.91 * A^2

Therefore, the ratio of Square's area/ rectangle's area = 1 / 0.91 = 1.10

So, the ration of the torque generated by the square loop qnd by the rectangular loop = 1.10