Electromagnetic Induction · Coupled Solenoids

Watch the shorted inner coil reduce the effective inductance

The outer solenoid forms an LC circuit, while the smaller solenoid is completely inside it and shorted on itself. A changing current in the outer coil induces an opposing current in the inner coil, so the capacitor does not see the full inductance L.

2026 Paper 1 Q2 · coupled solenoid resonance
Outer LC circuit drives flux · shorted inner coil opposes the flux change
Observation stage
Oscillation speed 1.0×
Nested solenoid geometry
The inner coil has length d/2, area S/2, and 2N turns per unit length. Its total turns are Nd, the same as the large coil.
ExpressionL₂ = μ₀(2N)²(S/2)(d/2)
MeaningL₂ = L
Outer solenoidself inductance L
Shorted inner coilinduced current opposes flux change
Capacitor branchresonates with Leff

Derivation path

01 / SELF INDUCTANCE

The smaller coil also has inductance L

For a long solenoid, L = μ₀n²Al. Substituting 2N, S/2, and d/2 gives the same inductance as the larger coil.

L₂ = μ₀(2N)²(S/2)(d/2) = L
02 / MUTUAL INDUCTANCE

Only half the large-coil flux links the inner coil

The large coil field is μ₀NI. The smaller coil has Nd turns and area S/2, so its flux linkage is (L/2)I.

M = L/2
03 / SHORTED COIL

The inner loop has zero voltage

Since the inner coil is shorted with no resistance, its net induced emf is zero, making its current oppose the outer current change.

0 = M dI₁/dt + L₂ dI₂/dt
Resonant angular frequency
Leff = L − M²/L₂ = L − (L/2)²/L = 3L/4
ω = 1/√(LeffC) = 2/√(3LC)
Correct option: (C)