Question

A circular loop A of radius \( R \) carries a current \( I \). Another circular loop B of radius \( r = \frac{R}{20} \) is placed concentrically in the plane of A. The magnetic flux linked with loop B is proportional to:

• A. R
• B. \(√R\)
• C. \(R^{\frac{3}{2}}\)
• D. \(R^{2}\)

Hint:

In the case of two concentric loops, the magnetic flux linked with the smaller loop is proportional to the radius of the larger loop.

Answer 1

The Correct Option is B

Magnetic Flux for Loop B
Step 1: Magnetic Field from Loop A
- The magnetic field \( B \) at the center of a circular loop with current \( I \) is \( B = \frac{\mu_0 I}{2R} \), where \( R \) is the loop's radius and \( \mu_0 \) is the permeability of free space.

Step 2: Magnetic Flux in Loop B
- The magnetic flux \( \Phi_B \) for loop B is \( \Phi_B = B \times A_B \). The area of loop B is \( A_B = \pi r^2 \), with radius \( r = \frac{R}{20} \). Substituting these into the flux equation:
\[ \Phi_B = \left( \frac{\mu_0 I}{2R} \right) \times \pi \left( \frac{R}{20} \right)^2 \]
Simplifying the expression:
\[ \Phi_B = \frac{\mu_0 I \pi R^2}{2R \times 400} = \frac{\mu_0 I \pi R}{800} \]
Therefore, the magnetic flux linked with loop B is directly proportional to \( R \).

Step 3: Conclusion
Since the magnetic flux linked with loop B is proportional to \( R \), the correct answer is: \[ \boxed{(A) \, R} \]