Question

A circular coil of 'N' turns and diameter 'd' carries current 'I'. It is unwound and rewound to make another coil of diameter '2d', current 'I' remaining the Same. The ratio of magnetic moments of the new coil to the original coil is

• A. \(2:1\)
• B. \(1:1\)
• C. \(1:2\)
• D. \(4:1\)

Hint:

The wire length is fixed, so turns halve when diameter doubles.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The magnetic moment of a coil with \(N\) turns, current \(I\) and area \(A\) is \(m=NIA\). Unwinding and rewinding keeps the total length of the wire the same.

Step 2: Wire length:
Original length \(=N\cdot\pi d\). New coil has \(N'\) turns of diameter \(2d\), so \(N'\cdot\pi(2d)=N\pi d\). This gives \(N'=\dfrac N2\).

Step 3: Magnetic moments:
Original: \(m_1=NI\cdot\dfrac{\pi d^2}4\).
New: \(m_2=\dfrac N2I\cdot\dfrac{\pi(2d)^2}{4}=\dfrac N2I\cdot\pi d^2=2\cdot NI\dfrac{\pi d^2}4\).

Step 4: Ratio:
\(\dfrac{m_2}{m_1}=2\), so the ratio is \(2:1\), option (A).

Final Answer:
The new coil has twice the magnetic moment. \[ \boxed{2:1} \]