How does a radio pick one station out of the air? An inductor and a capacitor together ring at exactly one frequency, and turning the tuning knob (a variable capacitor) slides that frequency across the dial until it lands on the station you want. Turn the dial (or tap).
An LC tuned circuit. An inductor L and capacitor C in a loop resonate at f = 1 / (2Ļā(LC)) ā the one frequency at which their opposing reactances cancel and the circuit rings freely. A variable capacitor is the tuning knob: because f ā 1/āC, turning it sweeps the resonant frequency across the band, and the circuit responds strongly only near f (a bandpass peak). With L fixed at 100 ĀµH the readout tracks the dial across the AM band. A fail-loud self-check throws unless quadrupling C exactly halves the resonant frequency (the 1/āC law).
An ideal lossless LC (real circuits have resistance, so a finite-width peak / finite Q; antennas and amplifiers matter too); component values are illustrative. The f = 1/(2Ļā(LC)) resonance and the 1/āC scaling are exact.