Question

Two concentric circular coils having radii \(r_1\) and \(r_2\) (\(r_2<<r_1\)) are placed co-axially with centres coinciding. The mutual inductance of the arrangement is (Both coils have single turn, \(μ_0\) = permeability of free space)

• A. \(\frac{μ_0πr_2}{2r_1}\)
• B. \(\frac{μ_0πr_1^2}{2r_2}\)
• C. \(\frac{μ_0πr_1}{2r_2}\)
• D. \(\frac{μ_0r_2^2π}{2r_1}\)

Hint:

Large coil gives a nearly uniform field mu0 I/(2 r1) through the small coil.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Mutual inductance is \(M = \dfrac{\phi_2}{I_1}\), the flux linked with the small coil per unit current in the large coil.

Step 2: Compute
Current \(I\) in the large coil gives at its centre \(B = \dfrac{\mu_0I}{2r_1}\). Since \(r_2 \ll r_1\), the field is almost uniform across the small coil.
\[ \phi_2 = B\cdot\pi r_2^2 = \frac{\mu_0 I\pi r_2^2}{2r_1} \]
\[ M = \frac{\phi_2}{I} = \frac{\mu_0\pi r_2^2}{2r_1} \]

Final Answer:
The mutual inductance is \(\dfrac{\mu_0\pi r_2^2}{2r_1}\), option (D). \[ \boxed{\frac{\mu_0\pi r_2^2}{2r_1}} \]