Self-Induction of a Coil

Multiply the inductance by the rate of change of the current: U_ind = −L·(dI/dt). For a uniform change you may replace dI/dt by ΔI/Δt. Example: in a coil with L = 0.5 H the current collapses from 2 A to 0 within 10 ms at switch-off, so ΔI/Δt = −200 A/s and U_ind = −0.5 × (−200) = +100 V. The sign says: the voltage opposes the change in current (Lenz rule). For magnitudes in exams |U| = L·|ΔI|/Δt often suffices. The faster the change, the higher the voltage, which is why abrupt switch-offs in particular are critical.

The magnetic field of a current-carrying coil stores the energy E = ½·L·I², and it cannot vanish instantly. When the switch opens the circuit, the current tries to fall to zero within microseconds; the rate dI/dt becomes enormous, and with it the induced voltage U = −L·dI/dt. It can reach hundreds to thousands of volts although the operating voltage was only a few volts, arcing across the opening contact or destroying transistors. The car ignition coil uses exactly this effect constructively to create spark voltages above 20 kV from 12 V. To protect circuits, an antiparallel freewheeling diode lets the coil current decay gently after switch-off.

L is the electrical inertia of a coil: it states how strongly the coil resists changes of current. A coil of 1 henry produces 1 V of back voltage when its current changes by 1 A per second. Large inductance means a sluggish, slowly rising current, just as large mass means sluggish acceleration; in the mechanical analogy L corresponds to the mass m and the current to the velocity. Geometrically, for a long coil L = μ₀·N²·A/l: the number of turns enters squared, and an iron core multiplies by μ_r. Typical values range from microhenries (wire loops, RF coils) through millihenries (chokes) to several henries (mains transformer windings).

Because the number of turns acts twice. First, more turns at the same current create a stronger magnetic field, since B = μ₀·(N/l)·I grows linearly with N. Second, this flux also threads more turns: the total flux linkage is N·Φ, again proportional to N. Both effects together give L = N·Φ/I ∝ N². Concretely: doubling the turns of a coil at the same length and area quadruples L. Sample calculation: N = 1000, A = 20 cm², l = 0.4 m gives L = 1.257×10⁻⁶ × 10⁶ × 2×10⁻³/0.4 ≈ 6.3 mH; with N = 2000 it would be about 25 mH.

In ordinary induction an external cause produces the flux change: a moving magnet, another coil or a loop rotating in a field. The voltage follows from U = −N·dΦ/dt with the external flux Φ. In self-induction the coil is its own cause: its own current creates the flux, and any change of this current induces the back voltage U = −L·dI/dt in the same coil. Physically it is the same law of induction, only applied to the coil own flux; L bundles the geometry (N²·A/l). In a transformer both occur at once: self-induction in the primary coil and mutual induction across to the secondary.