An alternating voltage v = vₘ sin ωt is applied across a pure inductor. Show that the current lags the voltage by π/2, find the inductive reactance and draw the phasor diagram.

1. 1 · Write the loop rule. v − L di/dt = 0, so di/dt = (vₘ/L) sin ωt.
2. 2 · Integrate. i = −(vₘ/ωL) cos ωt = iₘ sin(ωt − π/2), with iₘ = vₘ/ωL.
3. 3 · Define reactance. X_L = ωL = 2πνL, measured in ohm.
4. 4 · Draw the phasor diagram. [V phasor; I phasor π/2 clockwise from V; angle π/2 marked; ω shown anticlockwise]
5. 5 · State the result. The current lags the voltage by π/2, and the average power is zero.
6. 6 · Close in one line. So, an inductor opposes AC more at higher frequency.

Scoring tip: Name the phase difference in words and symbols: 'lags by π/2, a quarter cycle'.