Mutual Inductance
Two coupled coils. M = μ₀N₁N₂A/ℓ. ε₂ = −M·dI₁/dt with 90° phase shift.
Key Notes
Two coils nearby: a changing current in one induces an EMF in the other through shared flux. The constant is the mutual inductance M.
Defining relation: ε₂ = −M · dI₁/dt (and symmetrically ε₁ = −M · dI₂/dt).
Unit: henry (H), same as self-inductance.
Two coaxial solenoids (outer = primary, inner = secondary, same area A and length ℓ): M = μ₀·N₁·N₂·A/ℓ.
M depends only on geometry and core material — never on currents.
M is SYMMETRIC: M₁₂ = M₂₁. EMF induced in coil 2 per dI/dt in coil 1 equals EMF induced in coil 1 per dI/dt in coil 2.
Mutual inductance is the basic principle of transformers, induction cookers, and wireless charging.
Coupling coefficient k = M/√(L₁L₂), with 0 ≤ k ≤ 1. Perfect coupling (k=1) means ALL flux links both coils.
Formulas
Defining relation
Cross-induced EMF in each coil.
Two coaxial solenoids
Inner coil fully inside outer; A = cross-section area, ℓ = solenoid length.
Coupling coefficient
k = 1 for ideal transformer (no leakage).
Energy in two coupled coils
Sign depends on relative current directions (mutual stored / borrowed energy).
Transformer relation (k=1)
Ideal-transformer limit.
Important Points
M is GEOMETRIC and SYMMETRIC. Always M₁₂ = M₂₁.
Maximising M: tight coupling (no leakage), iron core, coaxial geometry. Used in transformers.
Minimising M: orthogonal coils, distance, magnetic shielding. Used in noise-sensitive electronics.
Coupling k = 1 is an idealisation. Real transformers reach k ≈ 0.98–0.999.
Mutual inductance is what makes 'induction' chargers work even with no wire between charger and device — flux from primary links the secondary in the phone.
If two inductors L₁ and L₂ are placed in series with k = 1: L_total = L₁ + L₂ + 2M (aiding) or L₁ + L₂ − 2M (opposing).
Mutual Inductance notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.