Self and Mutual Inductance: Coefficient, Formula, Uses

Self inductance is the property of a coil by which it opposes any change in the current flowing through itself, because the changing current creates a changing magnetic flux that induces a back EMF in the same coil. Mutual inductance is the corresponding effect between two coils: a changing current in one coil induces an EMF in a neighbouring coil. Together they explain how inductors smooth currents and how transformers transfer energy.

These ideas matter because every motor, transformer, charger and radio circuit depends on them. The coefficient of inductance tells an engineer exactly how many volts of opposition a coil will produce for a given rate of change of current, which decides whether a circuit will spark, hum, filter or transfer power cleanly.

Key Concepts

• Self inductance (L): the ratio of the magnetic flux linkage of a coil to the current producing it; unit is the henry (H).
• Coefficient of self inductance: numerically, the back EMF induced when the current changes at 1 ampere per second.
• Mutual inductance (M): the flux linkage set up in one coil per unit current in the other coil; also measured in henrys.
• Coefficient of coupling (k): a number between 0 and 1 telling what fraction of one coil flux links the other; k = 1 is perfect coupling.
• Inductor energy: energy stored in the magnetic field of an inductor carrying current.
• Back EMF: the EMF a coil induces in itself opposing the change in current, the reason inductors resist sudden switching.

Formulas

Self inductance is defined by L = NΦ / I, and the induced back EMF is ε = −L (dI/dt), where L is in henrys, I in amperes and dI/dt in amperes per second. For a long solenoid, L = μ₀ n² A l, where n is turns per metre, A the cross-sectional area in m² and l the length in metres.

For two coils, the EMF induced in coil 2 by a changing current in coil 1 is ε₂ = −M (dI₁/dt). The mutual inductance is related to the two self inductances by M = k √(L₁ L₂), where k is the coefficient of coupling, 0 ≤ k ≤ 1. The energy stored in an inductor is U = ½ L I², in joules.

Worked Example

A coil has a self inductance of 0.5 H. The current through it changes uniformly from 0 to 4 A in 0.2 s. Find the back EMF induced and the final stored energy.

1. Rate of change of current: dI/dt = 4 / 0.2 = 20 A/s.
2. Back EMF magnitude: ε = L (dI/dt) = 0.5 × 20 = 10 V, directed to oppose the increase.
3. Stored energy at 4 A: U = ½ L I² = 0.5 × 0.5 × 16 = 4 J.

So while the current is rising, the coil fights it with 10 V, and once 4 A flows steadily it stores 4 joules in its magnetic field, which will be returned if the current is allowed to fall.

Common Misconceptions

• Inductance opposes current. Wrong. Inductance opposes the change in current. A steady DC current flows through an ideal inductor with no opposition at all.
• Mutual inductance depends on the current in the coils. Wrong. M depends only on the geometry, number of turns, separation and core material of the coil pair, not on how much current flows.
• A larger core always means proportionally larger inductance. Wrong. Once a ferromagnetic core saturates, extra current produces little extra flux and the effective inductance falls sharply.

Real-World Applications

Self inductance is used in choke coils that block AC while passing DC, in filters that smooth the output of power supplies, in fluorescent lamp ballasts, and in the ignition coil of petrol cars where collapsing current produces a high-voltage spark. Mutual inductance is the working heart of every transformer, from phone chargers to grid substations, and it also powers wireless phone charging, RFID cards, and metal-free energy transfer in some medical implants. Radio tuning circuits use adjustable inductors whose values set the resonant frequency of the receiver.

Self inductance quantifies how a coil resists changes in its own current, and mutual inductance quantifies how two coils talk to each other through their shared flux. With L, M and the coupling coefficient, engineers can predict back EMFs, stored energy and transformer behaviour exactly.