AC in Inductors

When alternating current flows through an inductor — a coil of wire — something fascinating happens: the current and voltage fall out of step with each other. Unlike a simple resistor, where voltage and current peak at exactly the same moment, an inductor causes the current to lag behind the voltage by a quarter cycle, or 90°. This phase shift is the defining feature of inductive behaviour in AC circuits and has enormous practical consequences for power delivery and circuit design. The reason for this lag lies in Faraday's law of electromagnetic induction. An inductor opposes any change in current through it by generating a back-EMF proportional to the rate of change of current. When AC voltage is applied, the inductor is always fighting the change, so the current builds up slowly after the voltage has already peaked — always playing catch-up. A useful analogy: think of turning a heavy flywheel. You push hard (voltage) but the wheel takes time to spin up (current) because of its rotational inertia. Inductance is the electrical equivalent of that inertia. The opposition an inductor offers to AC is called inductive reactance, symbol X_L. Unlike resistance, reactance does not dissipate energy as heat; instead it stores energy in the magnetic field and returns it to the circuit. Inductive reactance depends on frequency: the higher the frequency, the harder the inductor fights the current, so X_L increases with frequency. This makes inductors natural high-frequency blockers and low-frequency passers — the opposite of capacitors — a property exploited in filters, transformers, and radio tuners everywhere.

Key takeaways

The Back-EMF and Faraday's Law
Inductive Reactance X_L
Impedance and Phasor Representation
Energy Storage — Not Dissipation

The Back-EMF and Faraday's Law

An inductor of inductance L (in henries) generates a back-EMF equal to ε = −L(dI/dt). When the AC current is changing most rapidly (near the zero-crossing), this back-EMF is largest. When current is at its peak, dI/dt = 0 and the back-EMF is zero. The applied voltage must equal this back-EMF at every instant, so voltage peaks when current is crossing zero — exactly 90° (a quarter cycle) earlier. Mathematically: if I(t) = I₀ sin(ωt), then V(t) = L·dI/dt = LI₀ω cos(ωt) = V₀ cos(ωt). Since cos(ωt) = sin(ωt + 90°), the voltage leads the current by 90°.

Inductive Reactance X_L

From V₀ = LωI₀, we can define inductive reactance X_L = ωL = 2πfL, measured in ohms (Ω). This plays the same role as resistance in Ohm's law for the amplitudes: V₀ = X_L · I₀, or equivalently V_rms = X_L · I_rms. The crucial difference from resistance: no energy is permanently lost. The inductor stores energy in its magnetic field during one quarter-cycle and releases it during the next. Because X_L = 2πfL, doubling the frequency doubles the reactance — high-frequency signals are blocked while low-frequency signals pass relatively freely.

Impedance and Phasor Representation

In a real circuit an inductor also has some resistance r in its wire. The total opposition to AC is called impedance, Z, and combines resistance and reactance as vectors (phasors). For a pure inductor, Z = X_L and the phase angle φ = 90°. For a real inductor with resistance r, Z = √(r² + X_L²) and tan φ = X_L/r. Phasors are rotating vectors in the complex plane: voltage phasor leads current phasor by φ. The impedance in complex notation is Z = r + jX_L, where j = √(−1). This phasor picture makes AC circuit analysis systematic and is the foundation of all AC engineering.

Energy Storage — Not Dissipation

The instantaneous power delivered to a pure inductor is P(t) = V(t)·I(t) = V₀cos(ωt)·I₀sin(ωt) = (V₀I₀/2)sin(2ωt). This oscillates symmetrically about zero, so the average power over a full cycle is zero. The energy stored in the magnetic field at any instant is U = ½LI². The inductor is continually charging and discharging its magnetic field, acting as a temporary energy reservoir. This is why inductive (and capacitive) loads in power grids are problematic — they draw reactive current that the generators must supply even though no net work is done, wasting transmission capacity.

The Key Equation

X_L = \omega L = 2\pi f L

X_L — Inductive reactance (Ω (ohms)): The frequency-dependent opposition an inductor presents to AC current; analogous to resistance but stores rather than dissipates energy.
\omega — Angular frequency (rad s⁻¹): Equals 2πf; describes how rapidly the AC oscillates in radians per second.
L — Inductance (H (henries)): The inductor's ability to store magnetic energy; determined by the number of turns and geometry of the coil.
f — Frequency (Hz (hertz)): The number of complete AC cycles per second; mains supply is 50 Hz (UK/EU) or 60 Hz (USA).

Worked Example

Problem. A 200 mH inductor is connected to a 240 V (rms), 50 Hz AC supply. (a) Calculate the inductive reactance. (b) Find the rms current through the inductor. (c) State the phase relationship between voltage and current.

X_L \approx 62.8\,\Omega,\quad I_{rms} \approx 3.82\,\text{A},\quad \phi = 90°

Answer. Inductive reactance X_L ≈ 62.8 Ω; rms current ≈ 3.82 A; current lags voltage by 90°; average power = 0 W.

Figure

Diagram

Origins · 1 minJoseph Henry and how this was discovered

Joseph Henry

1797–1878 · American

Joseph Henry independently discovered electromagnetic induction at almost the same time as Faraday, and he discovered the phenomenon of self-inductance — the very effect responsible for inductive reactance in AC circuits. The SI unit of inductance, the henry (H), is named in his honour. Henry also built some of the most powerful electromagnets of his era and invented an early form of the electric relay, which became the foundation of the telegraph. Henry discovered self-inductance before Faraday's published result, but because he delayed publication he received less historical credit. He was famously generous about this, saying scientific discovery belongs to whoever publishes it first.

AC in Capacitors — Capacitors in AC circuits show a complementary phase shift: current leads voltage by 90°, and capacitive reactance decreases with frequency — the exact mirror behaviour of inductors.
AC in Resistors — Resistors in AC circuits have zero phase shift and their opposition (resistance) is frequency-independent — forming the baseline comparison for understanding reactive elements like inductors.
Series LCR Circuits and Resonance — When inductors, capacitors, and resistors are combined in series, inductive and capacitive reactances can cancel at the resonant frequency, producing the sharp resonance used in radio tuners.
Faraday's Law of Electromagnetic Induction — The 90° phase lag in inductors is a direct consequence of Faraday's law; understanding the law gives the physical foundation for all inductive behaviour in AC circuits.