Question

Two concentric loops of radii 'r' and 'R' are in same plane such that \( R \gg r \). If a current \( I \) is maintained in the loop of radius \( r \), then the magnetic flux associated with the loop of radius 'R' due to this current is

• A. \( \frac{\pi \mu_0 I r^2}{R} \)
• B. \( \frac{\pi \mu_0 I r}{2} \)
• C. \( \frac{\pi \mu_0 I r^2}{2R} \)
• D. \( \frac{\pi \mu_0 I R}{2} \)

Hint:

When calculating magnetic flux due to one loop over another, use the approximate uniform magnetic field if the outer loop is much larger and coaxial.

Answer 1

The Correct Option is C

The magnetic field at the center of a loop of radius \( r \) carrying current \( I \) is: \[ B = \frac{\mu_0 I}{2r} \] When \( R \gg r \), the field at distance \( R \) due to the inner loop is approximately uniform across the outer loop, so the magnetic flux \( \Phi \) through the outer loop of radius \( R \) is: \[ \Phi = B \cdot A = \frac{\mu_0 I}{2r} \cdot \pi r^2 = \frac{\pi \mu_0 I r^2}{2R} \]