MHT CET · Physics · Electromagnetic Induction

The planar concentric rings of metal wire having radii \(r_1\) and \(r_2\) (with \(r_1 \gt r_2\)) are placed in air. The current I is flowing through the coil of larger radius. The mutual inductance between the coils is given by (\(\mu_0=\) permeability of free space)

A \(\frac{\mu_0 \pi\left(r_1+r_2\right)^2}{2 r_2}\)
B \(\frac{\mu_0 \pi\left(r_1-r_2\right)^2}{2 r_1}\)
C \(\frac{\mu_0 \pi r_1^2}{2 r_2}\)
D \(\frac{\mu_0 \pi r_2^2}{2 r_1}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{\mu_0 \pi r_2^2}{2 r_1}\)

Step-by-step Solution

Detailed explanation

The magnetic field at the centre of a loop is given by
\(B=\frac{\mu_0 N I}{2 R}...(i)\)
\(\therefore \quad\) Magnetic field produced by ring \(A, B_A=\frac{\mu_0 I}{2 r_1}\)
\(\therefore \quad\) Magnetic flux produced in ring B due to \(\mathrm{B}_{\mathrm{A}}\),
\(\phi_B=B_A A_B\)
\(\ldots\left(\mathrm{A}_{\mathrm{B}}\right.\) is area)
\(\therefore \quad \phi_B=\frac{\pi_0 I}{2 r_1} \times \pi r_2^2=\frac{\mu_0 \pi r_2^2}{2 r_1} I\)
...[From(i)]
Mutual Inductance \(M=\frac{\phi}{I}\)
\(\therefore \quad M=\frac{\phi_B}{I}=\frac{\mu_0 \pi r_2^2 I}{2 r_1 I}=\frac{\mu_0 \pi r_2^2}{2 r_1}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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