Question

Two concentric circular coils of radii \( r_1 \) and \( r_2 \) (\( r_2 \gg r_1 \)) are placed coaxially with their centres coinciding. If a current \( I \) is passed through the outer coil, obtain the expression for mutual inductance of the arrangement.

Hint:

To derive mutual inductance, first compute magnetic field from the source coil, then calculate flux through the secondary coil using uniform field approximation.

Answer 1

Let: - Outer coil (large radius) has radius \( r_2 \), - Inner coil (small radius) has radius \( r_1 \), - Current \( I \) flows in the outer coil, - Number of turns in the inner coil = \( n \) (assume 1 turn if not given), - Magnetic field at center of a circular coil of radius \( r \) due to current \( I \) is: \[ B = \frac{\mu_0 I}{2r} \] Since \( r_1 \ll r_2 \), the magnetic field due to the outer coil is almost uniform across the small inner loop. Step 1: Magnetic field at center of large coil: \[ B = \frac{\mu_0 I}{2r_2} \] Step 2: Magnetic flux through the inner coil: \[ \Phi = B \cdot A = \frac{\mu_0 I}{2r_2} \cdot \pi r_1^2 \] Step 3: Mutual Inductance \( M \): \[ \Phi = MI \Rightarrow M = \frac{\Phi}{I} = \frac{\mu_0 \pi r_1^2}{2r_2} \] \[ \boxed{M = \frac{\mu_0 \pi r_1^2}{2r_2}} \]