LC Oscillations
An LC circuit consists of just two components wired in a loop: an inductor (L) and a capacitor (C). Despite this simplicity, the circuit supports a remarkable phenomenon — energy sloshes back and forth between the electric field stored in the capacitor and the magnetic field stored in the inductor, producing a continuous oscillation of current and voltage. This is the electrical analogue of a swinging pendulum or a mass bouncing on a spring. Imagine charging a capacitor fully so that all the energy sits in its electric field. When you connect it to an inductor, the capacitor begins to discharge, driving current through the inductor. The growing current builds a magnetic field in the inductor. Once the capacitor is fully discharged, the inductor's collapsing field keeps the current flowing, now recharging the capacitor with the opposite polarity. The cycle then reverses, and the oscillation continues indefinitely — at least in the ideal (lossless) case. LC oscillations are foundational to radio and television technology, signal processing, and wireless communication. Every radio receiver uses an LC circuit whose resonant frequency is tuned to match the broadcast frequency of a chosen station. Understanding how the resonant frequency depends on L and C is therefore not just elegant physics — it is the engineering principle behind modern electronics.
- Initial Conditions: All Energy in the Capacitor
- Discharge Phase: Electric Energy Converts to Magnetic
- Recharge Phase: Magnetic Energy Converts Back
- The Second Half-Cycle and Periodicity
- Energy Conservation
Initial Conditions: All Energy in the Capacitor
Charge the capacitor to voltage V₀. The energy stored is U_E = ½CV₀². The inductor carries no current, so its magnetic energy is zero. This is like pulling a pendulum bob to one side and holding it — all potential energy, no kinetic energy.
Discharge Phase: Electric Energy Converts to Magnetic
Kirchhoff's voltage law around the loop gives: the capacitor voltage equals the back-EMF of the inductor, so V_C = L(dI/dt). As the capacitor discharges, I increases. The magnetic energy U_B = ½LI² grows as U_E shrinks. At the moment the capacitor is fully discharged (Q = 0), all energy is magnetic and the current is at its maximum value I_max = V₀√(C/L).
Recharge Phase: Magnetic Energy Converts Back
The inductor cannot let the current stop instantaneously (Lenz's law). It keeps current flowing in the same direction, which deposits charge on the capacitor with opposite polarity. The magnetic field collapses, rebuilding the electric field. When U_B = 0 again, the capacitor is fully charged to −V₀.
The Second Half-Cycle and Periodicity
The process repeats in reverse, returning the circuit to its original state after one full period T. The charge on the capacitor obeys Q(t) = Q₀ cos(ωt), and the current obeys I(t) = −Q₀ω sin(ωt). These are exactly the equations of simple harmonic motion, confirming the mathematical analogy: L plays the role of mass (inertia to change) and 1/C plays the role of spring constant (restoring force).
Energy Conservation
At every instant, U_E + U_B = ½CV₀² = constant. This can be verified: U_E = Q²/2C = (Q₀²/2C)cos²(ωt) and U_B = ½LI² = (Q₀²ω²L/2)sin²(ωt). Since ω² = 1/LC, the second term equals (Q₀²/2C)sin²(ωt). Adding them gives Q₀²/2C for all t — total energy is constant, exactly like a frictionless pendulum.
The Key Equation
- \omega_0 — Angular resonant frequency (rad s⁻¹): The number of radians of oscillation per second; determines how rapidly the circuit oscillates.
- f_0 — Resonant frequency (Hz (cycles per second)): The number of complete oscillations per second. Radio stations are identified by this frequency.
- L — Inductance (H (henries)): Measures how strongly the inductor resists changes in current; larger L slows the oscillation.
- C — Capacitance (F (farads)): Measures how much charge the capacitor can store per volt; larger C also slows the oscillation.
- T — Period (s (seconds)): The time for one complete cycle of oscillation.
Worked Example
Problem. A radio tuner circuit contains an inductor of L = 2.00 μH and a variable capacitor. You want the circuit to resonate at f₀ = 100 MHz (the FM band). (a) What capacitance C is required? (b) If the capacitor is initially charged to V₀ = 5.00 V, what is the maximum current in the circuit?
Answer. Required capacitance is approximately 1.27 pF; maximum current is approximately 3.97 mA.
Origins · 1 minJoseph Henry and how this was discovered
Henry independently discovered electromagnetic induction at the same time as Faraday, and crucially discovered self-inductance — the property of an inductor that underpins the oscillatory behavior of LC circuits. The SI unit of inductance, the henry (H), is named in his honor. His work on induced EMF and coil design laid the practical groundwork for understanding how inductors store and release energy. Henry built what may have been the world's first electric motor and used it to ring a bell via a mile-long wire — an early telegraph demonstration — years before Samuel Morse's famous system.
Related Concepts
- Faraday's Law of Electromagnetic Induction — The inductor's EMF that drives LC oscillations is directly produced by Faraday's law — the changing magnetic flux induces a voltage that keeps the current oscillating.
- Capacitors and Electric Field Energy — Half of the energy in an LC circuit resides in the capacitor's electric field; understanding how U_E = Q²/2C varies is essential to tracking the oscillation.
- Simple Harmonic Motion — LC oscillations are mathematically identical to SHM: L maps to mass and 1/C maps to spring constant, so all SHM results (period, energy, phase) transfer directly.
- Resonance in AC Circuits (RLC) — Adding a resistor to an LC circuit creates a damped RLC oscillator and introduces the concept of resonance frequency and quality factor, the natural extension of ideal LC theory.