Resonance

Systems that tend to oscillate at a natural frequency respond strongly when driven near it, like a swing pushed in time with its own motion. This is resonance.

In a series LCR circuit, XL = ωL grows and XC = 1/ωC shrinks as ω rises. At one frequency ω₀ they are equal, Z falls to its least value R, and the current amplitude reaches its greatest value vm/R.

Setting ω₀L = 1/ω₀C gives the resonant frequency ω₀ = 1/√(LC).

NCERT's graph uses L = 1.00 mH, C = 1.00 nF and vm = 100 V, so ω₀ = 1.00 × 10⁶ rad/s. With R = 100 Ω the peak current at resonance is twice what it is with R = 200 Ω, since im = vm/R there.

At resonance the voltages across L and C are equal and opposite and cancel; the whole source voltage appears across R. Resonance therefore needs both L and C: an RL or RC circuit cannot resonate.

Radio and TV tuning uses resonance. The aerial picks up many stations at once; turning the tuning capacitor moves the circuit's resonant frequency onto one station's frequency, so that station's current is the largest.

Example 7.10: an airport metal detector is a many-turn coil with a capacitor, tuned to resonance. Metal carried through the coil changes the impedance, the current changes noticeably, and the change sets off the alarm.