AC through inductors and capacitors

Inductors and capacitors also limit ac current, but differently from a resistor. Their opposition is called reactance X, measured in ohms, and it depends on the frequency.

Inductor: X_L = ωL = 2πνL.

• The back emf opposes the change of current, and faster changes are opposed more, so X_L rises with frequency. For dc (ν = 0), X_L = 0.
• The current lags the voltage by 90° (a quarter cycle).

Example: L = 0.1 H at 50 Hz: X_L = 2π × 50 × 0.1 ≈ 31.4 Ω. On 220 V, I = 220/31.4 ≈ 7 A. Pushing an iron rod into the coil raises L and X_L, so the current falls.

Capacitor: X_C = 1/(ωC) = 1/(2πνC).

• At higher frequency the capacitor charges and discharges more often, so more current flows: X_C falls with frequency. For dc, X_C is infinite: a capacitor blocks dc.
• The current leads the voltage by 90°.

Example: C = 10 µF at 50 Hz: X_C = 1/(2π × 50 × 10⁻⁵) ≈ 318 Ω.

A way to remember the phases: in an inductor, voltage comes first (ELI: E before I in L); in a capacitor, current comes first (ICE: I before E in C).

Power: in a pure inductor or a pure capacitor, the average power is zero. Energy is stored in the field for a quarter cycle and handed back in the next. Only resistance turns electrical energy into heat.