Question

A thin circular wire carrying current \(I\) has magnetic moment \(M\). The shape of the wire is changed to a square and it carries the same current. It will have magnetic moment \(M_1\). The ratio \(M\) to \(M_1\) is

• A. \(\frac{2}{π}\)
• B. \(\frac{4}{π}\)
• C. \(\frac{π}{4}\)
• D. \(\frac{π}{2}\)

Hint:

Same wire length, so compare the areas of a circle and a square.

Answer 1

The Correct Option is B

Step 1: Approach
Put in a number for the length to see the ratio.

Step 2: Take $L=4$
Circle: radius $\dfrac{4}{2\pi}=\dfrac2\pi$, area $\pi\cdot\dfrac4{\pi^2}=\dfrac4\pi$. Square: side 1, area 1.

Step 3: Ratio
$\dfrac{M}{M_1}=\dfrac{4/\pi}{1}=\dfrac4\pi$. Option (B).

Final Answer:
For the same wire length the circle has area L squared over 4 pi and the square L squared over 16, giving 4/pi, option (B). \[ \boxed{\frac{4}{\pi}} \]