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Two concentric circular coils, one of small radius r₁ and the other of large radius r₂, such that r₁ r₂ are placed co-axially with centres coinciding. The mutual inductance M of the arrangement is proportional to _______.

Asked in GUJCET 2026 · Standard coil geometries

• (1) (r₁)/(r₂²)
• (2) (r₁²)/(r₂)
• (3) (r₂)/(r₁²)
• (4) (r₂²)/(r₁)

Step-by-step solution

Why the other options are wrong

• (1) Both powers are wrong. The flux through the small coil is the field multiplied by that coil's area π r₁², not by r₁; and the large coil's field at its centre is (μ₀I)/(2r₂), which falls as 1/(r₂), not as 1/(r₂²) — doubling r₂ halves M, it does not quarter it.
• (3) Turns the correct ratio upside down. It would make M grow as the outer coil widened, but a wider outer coil makes a weaker field at the centre, so M must fall as r₂ grows — and shrink, not blow up, as the inner coil is made smaller.
• (4) Swaps the roles of the two coils, pairing the large coil's area π r₂² with a field set by the small coil. M is the same either way round, but the small coil's field is nowhere near uniform over the large coil, so it cannot be worked out like this; the honest route is the large coil's field through the small coil.

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