Question

A 2 amp current is flowing through two different small circular copper coils having radii ratio 1:2. The ratio of their respective magnetic moments will be:

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

Hint:

Remember that magnetic moment is proportional to the square of the radius when the current is constant (\(M \propto r^2\)). If the radii ratio is 1:2, the magnetic moment ratio will be \(1^2 : 2^2 = 1:4\).

Answer 1

The Correct Option is D

Step 1: Magnetic moment of a current loop
The magnetic moment (M) of a current loop is given by:
M = I × A, where I is the current and A is the area of the loop.
For a circular coil of radius r, the area is A = πr².

Step 2: Magnetic moment of the first coil
Let the radii of the two coils be r₁ and r₂, with r₁:r₂ = 1:2.
Let r₁ = k and r₂ = 2k, where k is a constant.
The current in both coils is I = 2 amp.
The magnetic moment of the first coil is:
M₁ = I × πr₁² = 2 × π(k)² = 2πk².

Step 3: Magnetic moment of the second coil
The magnetic moment of the second coil is:
M₂ = I × πr₂² = 2 × π(2k)² = 2 × π(4k²) = 8πk².

Step 4: Ratio of the magnetic moments
The ratio of their magnetic moments is:
M₁ / M₂ = (2πk²) / (8πk²) = 1 / 4.

So, the ratio M₁:M₂ = 1:4.