PhysicsCBSE Class 12 · Board exam 2026, set 55/5/1 · 3 marks · Short answer

CBSE Class 12 Physics 2026: question 24

Set in the CBSE Class 12 board exam of 2026, paper 55/5/1, section C, question 24, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q24 · 3 marks · Short answerElectromagnetic Induction

A long solenoid of length L and radius $r_{1}$ having $N_{1}$ turns is surrounded symmetrically by a coil of radius $r_{2}$ ( $> r_{1}$ ) having $N_{2}$ turns ( $N_{2} << N_{1}$ ) around its mid-point. Derive an expression for the mutual inductance of solenoid and coil. Is $M_{12} = M_{21}$ valid in this case ?

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 55/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Pass current through the solenoid, not the coil — its field is easy to find.

Hint 2

The field exists only inside the solenoid, so the coil's effective area is $π r_{1}^{2}$ , not $π r_{2}^{2}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let current $I_{1}$ flow in the solenoid. Field inside the long solenoid (uniform, along axis): $B_{1} = \frac{μ _{0} N _{1} I _{1}}{L}$ ; outside it is negligible.
1 mark
The coil (radius $r_{2} > r_{1}$ ) links flux only over the solenoid's cross-section $π r_{1}^{2}$ :
Total flux linkage $N_{2} ϕ_{2} = N_{2} B_{1} π r_{1}^{2} = \frac{μ _{0} N _{1} N _{2} π r _{1}^{2}}{L} I_{1}$
1 mark
Since $N_{2} ϕ_{2} = M_{21} I_{1}$ : $M = M_{21} = \frac{μ _{0} N _{1} N _{2} π r _{1}^{2}}{L}$
½ mark
Yes, $M_{12} = M_{21}$ is valid here (mutual inductance is reciprocal for any pair of coils). However, $M_{12}$ cannot be easily calculated directly, since the field of the short coil is non-uniform over the solenoid; so we compute $M_{21}$ and use the equality.
½ mark

Answer: M = μ₀N₁N₂πr₁²/L; yes, M₁₂ = M₂₁ holds

Where the marks are

Field of solenoid $μ_{0} N_{1} I_{1} / L$ (1 mark)
Flux linkage with coil using area $π r_{1}^{2}$ (1 mark)
Expression $M = μ_{0} N_{1} N_{2} π r_{1}^{2} / L$ (½ mark)
States $M_{12} = M_{21}$ is valid, with reason (½ mark)

Revise the chapter

Chapter: Electromagnetic Induction (notes, formulas, where marks are lost and every past question)
Topic: Self and Mutual Inductance

Practise this chapterBoard-style questions from Electromagnetic Induction, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 55/5/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Physics 2026: question 24

Hints

Step-by-step solution

Revise the chapter

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