Question

A circular loop of radius R carries a current I so that its magnetic moment is M.
Now, the loop is changed to a square, keeping other conditions the same. Its magnetic moment will becomes/remain
A. \[\dfrac{4}{\pi }M\]
B. M
C. \[\dfrac{\pi }{4}M\]
D. \[2M\]

Answer 1

Hint:Express the magnetic moment of the circular loop. After converting the circular loop into the square loop, the total length of the circular loop remains the same. Use this to express the magnetic moment of the square loop in terms of the magnetic moment of the circular loop.

Formula used:
Magnetic moment, \[M = IA\]

Here, I is the current and A is the area.

Complete step by step answer:

We have given the circular loop of radius R whose magnetic moment is M. Let the current
through the loop is I.

We have the expression for the magnetic moment,
\[M = IA\]

Here, I is the current and A is the area.

We have the area of the circular loop is \[\pi {R^2}\]. Therefore, the magnetic moment of
the circular loop will become,
\[M = I\left( {\pi {R^2}} \right)\]
\[ \Rightarrow M = \pi {R^2}I\]

Now, we have given that the same circular loop is converted into the square loop. Therefore,
the length of the square loop will be the same as circumference of the circular loop.
\[4a = 2\pi R\]
\[ \Rightarrow a = \dfrac{{\pi R}}{2}\] ……. (1)

Here, a is the length of each side of the square loop.

Let’s express the magnetic moment of the square loop as follows,
\[M' = I{a^2}\] …… (Since the area of the square is the square of one of its side)

Using equation (1) we can rewrite the above equation as,
\[M' = I{\left( {\dfrac{{\pi R}}{2}} \right)^2}\]
\[ \Rightarrow M' = \left( {\pi {R^2}I} \right)\left( {\dfrac{\pi }{4}} \right)\]
\[ \Rightarrow M' = \dfrac{\pi }{4}M\]

So, the correct answer is option (C).

Note: After converting a circular loop into a square loop, only the length remains the same and not the area of the circular loop. The magnetic moment is a vector quantity which always directs from the South Pole to the North Pole. The magnetic moment is used to determine the torque on the magnet if placed in the external magnetic field.