Resonance Frequency (LC Circuit)⚠ unverified

Understanding the Parameters

• Inductance $L$ and Capacitance $C$ are the two energy‑storage elements; resonance depends only on their product $LC$, not on resistance. Larger $L$ or $C$ lowers the resonant frequency (the circuit "sloshes" more slowly).

• $\omega_0$ is an angular frequency (radians per second); divide by $2\pi$ for hertz.

Derivation (Approaching a Proof)

Consider a capacitor of charge $q$ discharging through an inductor (a series LC loop with no resistance). Kirchhoff's voltage law sets the inductor voltage equal and opposite to the capacitor voltage:

$$L\frac{di}{dt} + \frac{q}{C} = 0, \qquad i = \frac{dq}{dt},$$

so $\;L\dfrac{d^2q}{dt^2} + \dfrac{q}{C} = 0$, i.e.

$$\frac{d^2q}{dt^2} + \frac{1}{LC}\,q = 0.$$

This is the equation of simple harmonic motion; its solutions oscillate at $\omega_0 = 1/\sqrt{LC}$. Energy shuttles between the capacitor's electric field ($\tfrac12 q^2/C$) and the inductor's magnetic field ($\tfrac12 L i^2$), their sum conserved.

Equivalent view (reactance balance). Resonance is where the inductive and capacitive reactances cancel: $X_L = X_C \Rightarrow \omega_0 L = \dfrac{1}{\omega_0 C} \Rightarrow \omega_0^2 = \dfrac{1}{LC}$, giving the same result. At $\omega_0$ a series LC looks like a short and a parallel LC looks like an open.

History

William Thomson (Lord Kelvin) derived the oscillation period of a charged capacitor discharging through an inductor in 1853 — "Thomson's formula," $T = 2\pi\sqrt{LC}$. The LC resonant circuit became the heart of wireless telegraphy (Hertz, Lodge, Marconi, Tesla): tuning a receiver means matching its $\omega_0$ to the transmitter's, the founding principle of radio.

Related Concepts: Impedance Capacitor, Impedance Inductor, Quality Factor, Capacitive Reactance, Inductive Reactance

Notes: Registry calculator resonance-frequency (unverified). Ideal loss‑free result; real circuits have resistance that damps the oscillation and slightly shifts the damped frequency (see Quality Factor).