LC and RLC Oscillations

Charge a capacitor, connect it across a coil, and step back. No battery, no generator, nothing driving the circuit — and yet the charge does not simply drain away. It surges off the plates, through the coil, piles up on the opposite plates, and comes racing back, over and over, millions of times per second for a well-built circuit. The electrical world has a pendulum, and this is it. That discovery genuinely startled the physicists who made it. Joseph Henry noticed in 1842 that a discharging capacitor threw sparks that seemed to reverse direction; William Thomson worked out the theory in 1853 and predicted the frequency; Berend Feddersen needed a rotating mirror just to photograph oscillations too fast for any eye. By 1887 Heinrich Hertz was deliberately building these circuits to ring at tens of megahertz — and using them to launch the first radio waves. You are ready for this in one step. You know that a capacitor stores energy in its electric field and resists sudden changes in voltage; you know from RC and RL circuits that an inductor stores energy in its magnetic field and resists sudden changes in current. Put the two stubbornnesses in one loop and neither can win: the result is oscillation, governed by exactly the same differential equation as a mass on a spring. Mastering this article buys you the resonant heart of every radio, filter, and wireless charger — and the on-ramp to AC circuits, where we drive this pendulum on purpose.

Key takeaways

A charged capacitor discharging through an inductor oscillates: charge and current swap back and forth at the natural angular frequency $\omega_0 = 1/\sqrt{LC}$ .
Energy sloshes between the capacitor's electric field ( $q^2/2C$ ) and the inductor's magnetic field ( $Li^2/2$ ); with no resistance the total is exactly constant.
The circuit obeys the same equation as a mass on a spring, with the dictionary $q \leftrightarrow x$ , $i \leftrightarrow v$ , $L \leftrightarrow m$ , $1/C \leftrightarrow k$ , $R \leftrightarrow b$ .
Adding resistance gives a decaying ring: amplitude falls as $e^{-Rt/2L}$ and the frequency shifts only slightly until $R$ approaches the critical value $R_c = 2\sqrt{L/C}$ , where oscillation dies entirely.
The quality factor $Q = \omega_0 L/R$ counts (within a factor of $\pi$ ) how many times the circuit rings before the amplitude decays away.
KEY IDEA: an inductor and a capacitor in a loop form an electrical harmonic oscillator — charge plays position, current plays velocity, and energy swings between electric and magnetic form at $\omega_0 = 1/\sqrt{LC}$ .

The core insight

An LC loop is a harmonic oscillator built from fields: the capacitor's charge oscillates as $q(t) = q_0\cos\omega_0 t$ with $\omega_0 = 1/\sqrt{LC}$ , energy trading places between electric and magnetic storage, and adding resistance damps the ring exactly as friction damps a mass on a spring.

Where this lives today

The LC oscillator has migrated to the quantum frontier. A superconducting qubit is, at heart, an LC circuit cooled until it has to be treated quantum mechanically: the charge and flux that slosh classically in this article become non-commuting operators, the energy ladder $\hbar\omega_0(n + \tfrac12)$ replaces the continuous slosh, and a Josephson junction supplies just enough nonlinearity to single out the bottom two rungs as $|0\rangle$ and $|1\rangle$ . The readout resonators wired to each qubit on a 2026-era processor are literal LC circuits — lithographed spirals and pads ringing at gigahertz with quality factors in the millions — and their design still starts from $\omega_0 = 1/\sqrt{LC}$ . At more everyday scales, the wireless charging pad under a phone is a driven LC pair straight out of Hertz's playbook: transmit and receive coils, each resonated with a capacitor near 100–360 kHz, exchange energy efficiently only because both sides ring at the same frequency. And every MRI scanner's receive coil is an LC circuit tuned to the Larmor precession of protons — tens to hundreds of megahertz — where the quarter-cycle lag between charge and current that you learned in the hero figure sets the phase reference for the entire image.

The physical picture

Start with the capacitor fully charged and the switch just closed. All the circuit's energy sits in the electric field between the plates; the current is zero. The capacitor begins to push charge around the loop — but the inductor, true to its character, refuses to let the current jump. It rises gradually, and as it rises the capacitor drains. A quarter period in, the capacitor is empty — and this is the moment students misread. The story is not over: the current is now at its maximum, and all the energy has moved into the inductor's magnetic field. Now the inductor's stubbornness runs the show. Its current cannot stop any more suddenly than it could start, so charge keeps flowing — piling onto the far plate, charging the capacitor backwards. The current falls as the reverse voltage builds, reaching zero exactly when the capacitor holds its full original charge with reversed sign. Two quarter periods later the circuit is back where it began, and the cycle repeats. Nothing drives it. It rings because two energy stores keep handing the same energy back and forth, each store's "fullness" occurring at the other's "emptiness."

Figure

One full cycle of an LC oscillation — energy sloshes between electric and magnetic storage

If this choreography feels familiar, it should. A mass on a spring does precisely the same dance: all the energy in the spring at maximum stretch (velocity zero), all of it kinetic at the center (stretch zero). The correspondence is not a loose analogy — it is the same differential equation wearing different symbols, and the full dictionary is tabulated below. The mechanics track has already solved that equation in every regime in aq-mech-damped-harmonic-oscillator and its siblings; our job here is to earn the electrical version of the equation honestly, then let the dictionary deliver the solutions for free.

The loop equation and the natural frequency

Everything follows from one statement: go once around the loop and the potential changes must sum to zero — Kirchhoff's loop rule, whose fine print (it holds when the circuit is small and slow compared to radiation) is spelled out in aq-em-kirchhoffs-rules. The capacitor contributes $q/C$ (that is what capacitance means — see aq-em-capacitance), and the inductor contributes its back-EMF $L\,di/dt$ , the "electrical inertia" derived in aq-em-self-inductance. Take a loop containing only $L$ and $C$ , define $i = dq/dt$ with $q$ the charge on the capacitor, and the loop rule reads

L\frac{d^2q}{dt^2} + \frac{q}{C} = 0

where: $L$ is the inductance in henries, $C$ the capacitance in farads, and $q(t)$ the instantaneous plate charge. Read it physically: the capacitor term is a restoring "force" — charge on the plates pushes current in the direction that empties them — while the inductor term is inertia, resisting any change in the flow. Restoring force plus inertia, and nothing else, is the recipe for simple harmonic motion. Dividing by $L$ puts the equation in the standard oscillator form $\ddot q = -\omega_0^2\, q$ , and the constant reads off as

\omega_0 = \frac{1}{\sqrt{LC}}, \qquad f_0 = \frac{1}{2\pi\sqrt{LC}}

where: $\omega_0$ is the natural angular frequency and $f_0$ the frequency in hertz. The product $LC$ has units of seconds squared — a useful sanity check — and both dials work the intuitive way: a bigger inductor (more inertia) or a bigger capacitor (a softer spring, since stiffness is $1/C$ ) slows the oscillation. A 10 mH coil with a 1 μF capacitor rings at $f_0 \approx 1.6\ \mathrm{kHz}$ , in the audible range; shrink both to radio-circuit values (microhenries and picofarads) and $f_0$ climbs to megahertz. Here is the dictionary that makes the mechanics track's entire oscillation chapter available to you at zero cost:

circuit quantity	mechanical twin	role
charge $q$	position $x$	the oscillating coordinate
current $i = \dot q$	velocity $v = \dot x$	rate of change
inductance $L$	mass $m$	inertia
inverse capacitance $1/C$	spring constant $k$	stiffness
resistance $R$	drag coefficient $b$	dissipation
capacitor energy $q^2/2C$	spring energy $kx^2/2$	potential store
inductor energy $Li^2/2$	kinetic energy $mv^2/2$	kinetic store
$\omega_0 = 1/\sqrt{LC}$	$\omega_0 = \sqrt{k/m}$	natural frequency

Derivation AThe LC loop rings: solving the electrical pendulum

We want the full time history of an LC discharge — charge, current, and energy — starting from a capacitor charged to $q_0$ with the switch closed at $t=0$ . The strategy is to solve the loop equation with these initial conditions, then audit the energy books to confirm that the oscillation conserves energy exactly, quarter period by quarter period. The route is deliberately the elementary one (guess-and-check against the SHM form) rather than an import from mechanics, because the initial conditions — charge full, current zero — are set by circuit physics and deserve to be handled in circuit language. Watch one subtlety: maximum current does not occur when the capacitor voltage is large, but precisely when it vanishes.

Step 1. The loop equation is $\ddot q = -\omega_0^2 q$ with $\omega_0 = 1/\sqrt{LC}$ . Any function whose second derivative is minus $\omega_0^2$ times itself is a combination of $\cos\omega_0 t$ and $\sin\omega_0 t$ , so write $q(t) = A\cos\omega_0 t + B\sin\omega_0 t$ .

Step 2. Impose the initial conditions. At $t=0$ the capacitor holds $q_0$ , so $A = q_0$ . The current $i = \dot q$ must start at zero — the inductor forbids a jump — so $B = 0$ :

q(t) = q_0\cos\omega_0 t, \qquad i(t) = -\,\omega_0 q_0 \sin\omega_0 t

Why must the current start at zero even though the capacitor is pushing hard? Because a jump in $i$ would mean $di/dt \to \infty$ and hence an infinite back-EMF across the inductor. The inductor always wins arguments about sudden changes in current — the same fact that governs the switch-on transient in aq-em-rl-circuits.

Step 3. Identify the current amplitude. The peak current is $i_{\max} = \omega_0 q_0 = q_0/\sqrt{LC}$ , reached at $t = T/4$ , when $q = 0$ . Charge and current are a quarter cycle out of step: each peaks exactly when the other vanishes.

Step 4. Audit the energy. The two stores are $U_C = q^2/2C$ and $U_L = Li^2/2$ . Substituting the solutions:

U_C = \frac{q_0^2}{2C}\cos^2\omega_0 t, \qquad U_L = \frac{L\,\omega_0^2 q_0^2}{2}\sin^2\omega_0 t = \frac{q_0^2}{2C}\sin^2\omega_0 t

Why did the inductor's prefactor collapse to $q_0^2/2C$ ? Because $L\omega_0^2 = L/(LC) = 1/C$ — the same cancellation that makes $\omega_0$ the natural frequency makes the two energy amplitudes equal. The total is $U_C + U_L = (q_0^2/2C)(\cos^2 + \sin^2) = q_0^2/2C$ , constant for all time.

Step 5. Read the slosh timetable. At $t=0$ : all electric. At $T/8$ : $\cos^2(\pi/4) = 1/2$ — an even split. At $T/4$ : all magnetic. At $T/2$ : all electric again but with the capacitor's polarity reversed. Energy completes two full slosh cycles per oscillation period, because energy depends on the squares of $q$ and $i$ . Verified: $\tfrac12 L i_{\max}^2 = \tfrac12 L\,\omega_0^2 q_0^2 = q_0^2/2C$ — the magnetic store at peak current exactly equals the electric store at peak charge, and the numbers in Worked example 1 below check to three digits.

Adding resistance: the ring decays

Real coils have resistive wire, and real circuits lose energy. Put a resistance $R$ in the loop and the rule picks up the familiar $iR$ drop:

L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{q}{C} = 0

where: the new middle term is a drag — proportional to the "velocity" $\dot q$ and always opposing it, bleeding energy into heat at the rate $i^2R$ . Compare this, symbol by symbol against the dictionary, with $m\ddot x + b\dot x + kx = 0$ : it is the damped harmonic oscillator of aq-mech-damped-harmonic-oscillator, with nothing new to solve.

Go deeper · Derivation B · 2 minThe RLC ringdown, imported through the dictionary

Derivation BThe RLC ringdown, imported through the dictionary

We want $q(t)$ for the series RLC loop, the shifted ring frequency, and the boundary of oscillation. The strategy is the honest use of an already-solved problem: map our equation onto the damped-oscillator equation that the mechanics track has solved in full (exponential ansatz, all three regimes), transcribe the solution through the dictionary, and then verify by direct substitution that the transcription is right — so nothing rests on faith in the analogy. The payoff of doing it this way is that every regime fact from mechanics (fastest settling at critical damping, the taxonomy of roots) transfers instantly, and we can spend our effort on the circuit-specific readings.

Step 1. Map the coefficients: $m \to L$ , $b \to R$ , $k \to 1/C$ . The mechanics solution for weak damping (the underdamped branch of aq-mech-damping-regimes) is $x = X e^{-(b/2m)t}\cos(\omega' t + \varphi)$ with $\omega'^2 = k/m - (b/2m)^2$ .

Step 2. Transcribe. The decay rate is $b/2m \to R/2L$ and the stiffness-over-inertia is $k/m \to 1/LC = \omega_0^2$ :

q(t) = Q\,e^{-Rt/2L}\cos(\omega' t + \varphi), \qquad \omega'^2 = \omega_0^2 - \left(\frac{R}{2L}\right)^2

Why does the amplitude decay with $R/2L$ and not $R/L$ ? The factor 2 is the ansatz splitting its exponent between the two first-order roots; physically, the energy — which goes as amplitude squared — decays at the full rate $R/L$ , so the amplitude only carries half the exponent.

Step 3. Verify by substitution (this closes the dictionary loop). Differentiate $q(t)$ twice, insert into $L\ddot q + R\dot q + q/C$ , and collect the $e^{-Rt/2L}\cos$ and $e^{-Rt/2L}\sin$ terms separately. The cosine coefficient is $L[(R/2L)^2 - \omega'^2] - R(R/2L) + 1/C$ , which vanishes exactly when $\omega'^2 = \omega_0^2 - (R/2L)^2$ ; the sine coefficient vanishes identically. The transcription is exact.

Step 4. Find the boundary of ringing. Oscillation requires $\omega'^2 > 0$ , i.e. $R/2L < \omega_0$ . Solving for the resistance:

R_c = 2\sqrt{\frac{L}{C}}

Why does the critical resistance grow with $L$ and shrink with $C$ ? $\sqrt{L/C}$ is the circuit's natural impedance scale (it has units of ohms — check: henries per farad is ohms squared). A circuit with large inertia and a stiff spring oscillates vigorously and takes a lot of drag to subdue. Above $R_c$ the discharge is a dead slump — the overdamped regime, with the critical case $R = R_c$ giving the fastest non-ringing return, exactly as aq-mech-damping-regimes proves for the mass on a spring.

Step 5. Count the rings. The dimensionless figure of merit is the quality factor $Q = \omega_0 L/R$ (not to be confused with charge — an unfortunate but universal notation clash). The amplitude falls to $1/e$ after $\omega' t \approx 2Q$ radians of phase, i.e. after about $Q/\pi$ full cycles. A circuit with $Q = 100$ rings roughly thirty times before fading; the "AC Power and Resonance" article (aq-em-ac-power-resonance) makes $Q$ the star of the show. Verified: for $L = 50$ mH, $R = 40\ \Omega$ , $C = 2\ \mu$ F — Worked example 2 — direct numerical integration of the loop equation reproduces $q(t) = Q e^{-400t}\cos(3137\,t)$ to plotting accuracy, and $R_c = 316\ \Omega$ confirms the circuit rings.

Figure

An RLC ringdown, computed: the decaying envelope is exponential, the ring frequency barely shifts

Two readings of the ringdown deserve emphasis. First, the frequency barely moves: with $Q \approx 4$ — quite lossy — the shift from $\omega_0$ is under one percent, because the shift enters at second order in $R/R_c$ . Tuned circuits keep their pitch even as they fade, which is why a struck bell (mechanical or electrical) plays a definite note. Second, the envelope is exponential, so equal time intervals lose equal fractions of amplitude: the decay looks the same at every scale until the ring disappears into the noise.

Go deeper · Derivation C · 2 minThe historic oscillators, computed from their geometry

Derivation CThe historic oscillators, computed from their geometry

We want to know what frequencies the nineteenth century actually built, using only $f_0 = 1/2\pi\sqrt{LC}$ and rough component values — partly as a plausibility audit of the story in the history band, and partly because estimating $L$ and $C$ from geometry is a skill this chapter should leave you with. The strategy: take a Leyden jar discharged through a wire loop (the Feddersen–Kelvin configuration), then Hertz's deliberately small resonator, estimate each element from its physical size, and see where the frequencies land. Expect order-of-magnitude fidelity, not three digits — the point is why sparks, mirrors, and radio were the right instruments.

Step 1. Estimate the Leyden jar. A pint-sized glass jar with foil coatings is a parallel-plate capacitor wrapped into a cylinder: plate area of order $0.02\ \mathrm{m^2}$ , glass a few millimetres thick with dielectric constant around 5, giving $C \sim 1\ \mathrm{nF}$ — the standard textbook value for a jar.

Step 2. Estimate the discharge loop. A wire loop tens of centimetres across has an inductance of order $1\ \mu\mathrm{H}$ (a useful rule of thumb: about a microhenry per metre of wire in an open loop).

Step 3. Compute the ring:

f_0 = \frac{1}{2\pi\sqrt{(10^{-6}\ \mathrm{H})(10^{-9}\ \mathrm{F})}} \approx 5.0\ \mathrm{MHz}

Why did this defeat every instrument of 1850? Five million reversals per second: no galvanometer needle, no eye, no photographic plate of the era could follow it. Kelvin could only predict the ring in 1853; Feddersen finally photographed it in 1858 by bouncing the spark's light off a mirror spinning fast enough to smear successive oscillations across the plate — the first oscilloscope, built from clockwork.

Step 4. Now Hertz's resonator: he shrank the capacitor to a pair of small spheres and plates ( $C \sim 5\ \mathrm{pF}$ ) on a metre-scale rod-and-loop ( $L \sim 1\ \mu\mathrm{H}$ ):

f_0 = \frac{1}{2\pi\sqrt{(10^{-6})(5\times10^{-12})}} \approx 71\ \mathrm{MHz}

Why did Hertz need such a high frequency? Radiated power grows steeply with frequency, and the wavelength $c/f \approx 4\ \mathrm{m}$ finally fit inside a lecture hall — standing waves he could walk along and map with his spark-gap receiver loop. The LC formula, pushed to small $L$ and $C$ , is what turned a laboratory curiosity into the radio spectrum. Verified: both frequencies recomputed from $f_0 = 1/2\pi\sqrt{LC}$ — $5.03$ MHz for $(1\ \mu\mathrm{H}, 1\ \mathrm{nF})$ and $71.2$ MHz for $(1\ \mu\mathrm{H}, 5\ \mathrm{pF})$ ; Hertz's published fundamental frequencies for his large resonators were indeed some tens of megahertz.

Limiting cases

case	result	meaning
$R = 0$	$q = q_0\cos\omega_0 t$ forever	the ideal LC pendulum; energy exactly conserved
$R \ll R_c$	ring at $\omega' \approx \omega_0$ , envelope $e^{-Rt/2L}$	high- $Q$ tuned circuit: keeps its pitch while fading
$R = R_c = 2\sqrt{L/C}$	fastest decay with no ring	critical damping — the design point for meter movements
$R > R_c$	monotonic slump, no oscillation	overdamped; energy crawls out through the resistor
$L \to 0$	first-order RC decay, $\tau = RC$	no inertia, no oscillation — you recover aq-em-rc-circuits
$1/C \to 0$ (capacitor huge)	first-order RL decay, $\tau = L/R$	no restoring force — you recover aq-em-rl-circuits
driven at frequency $\omega$	resonance near $\omega_0$	the subject of the next two articles

Common trap“The oscillation must be driven by the battery — something has to keep pushing”

Nothing external pushes an LC oscillation. The battery (if any) only sets the initial charge and is disconnected before the ring begins; from then on the circuit runs on its own stored energy, handing it back and forth between the capacitor's electric field and the inductor's magnetic field. The confusion usually comes from conflating oscillation with steady state: a resistor circuit indeed dies instantly without a source, but L and C are energy stores, not dissipators. The battery-free ring is precisely what makes the LC circuit a clock — and the related error, "the current stops the moment the capacitor is empty," fails for the same reason: at $q=0$ the current is at its maximum, and the inductor's inertia carries the charge through to reverse the capacitor. Only the resistor ever removes energy from the loop.

Worked examples

Example 1 — the numbers of a bench-top ring (confidence builder). A 1.0 μF capacitor is charged to hold $q_0 = 20\ \mu\mathrm{C}$ , then switched across a 10 mH inductor of negligible resistance. Find the frequency, the peak current, and the total oscillating energy.

Step 1. The angular frequency: $\omega_0 = 1/\sqrt{LC} = 1/\sqrt{(10^{-2})(10^{-6})} = 1.0\times10^{4}\ \mathrm{rad/s}$ , so $f_0 = \omega_0/2\pi = 1.59\ \mathrm{kHz}$ — an audible ring, just above a soprano's high C.

Step 2. Peak current: $i_{\max} = \omega_0 q_0 = (10^4)(2.0\times10^{-5}) = 0.20\ \mathrm{A}$ , reached a quarter period ( $T/4 = 0.157$ ms) after the switch closes.

Step 3. The energy, computed both ways: $U = q_0^2/2C = (2.0\times10^{-5})^2/(2\times10^{-6}) = \mathbf{2.0\times10^{-4}\ J}$ , and $\tfrac12 L i_{\max}^2 = \tfrac12(10^{-2})(0.20)^2 = 2.0\times10^{-4}\ \mathrm{J}$ . The two stores agree — the slosh audit in miniature.

Example 2 — will it ring, and for how long? (exam level). A series loop has $L = 50\ \mathrm{mH}$ , $C = 2.0\ \mu\mathrm{F}$ , and total resistance $R = 40\ \Omega$ . The capacitor starts charged. Does the circuit oscillate? If so, at what frequency, and how long until the amplitude falls to 10% of its start?

Step 1. Check against critical: $R_c = 2\sqrt{L/C} = 2\sqrt{(5\times10^{-2})/(2\times10^{-6})} = 2\sqrt{2.5\times10^{4}} = 316\ \Omega$ . Since $40 \ll 316$ , the circuit rings comfortably.

Step 2. Undamped frequency: $\omega_0 = 1/\sqrt{LC} = 1/\sqrt{10^{-7}} = 3162\ \mathrm{rad/s}$ . Decay rate: $R/2L = 40/0.1 = 400\ \mathrm{s^{-1}}$ .

Step 3. Ring frequency: $\omega' = \sqrt{\omega_0^2 - (R/2L)^2} = \sqrt{1.0\times10^{7} - 1.6\times10^{5}} = 3137\ \mathrm{rad/s}$ , i.e. $f' = 499\ \mathrm{Hz}$ — a shift of only 0.8% from $f_0 = 503\ \mathrm{Hz}$ despite the hefty resistor. The pitch is robust; only the loudness dies.

Step 4. Time to 10%: the envelope is $e^{-400t}$ , so $t = \ln(10)/400 = \mathbf{5.8\ ms}$ — about $499 \times 0.0058 \approx 2.9$ cycles of ring. Consistency check: $Q = \omega_0 L/R = (3162)(0.05)/40 = 3.95$ , and $Q/\pi \approx 1.3$ cycles per $e$ -fold times $\ln 10/1 \approx 2.3$ $e$ -folds gives the same $\approx 2.9$ cycles.

Origins · 1 minHeinrich Hertz and how this was discovered

Heinrich Hertz

1857–1894 · Karlsruhe Polytechnic

Hertz appears elsewhere in this track as the man who vindicated Maxwell; here he deserves credit as the first great circuit designer. His 1886–88 apparatus was, in modern language, a matched pair of LC oscillators: a driven primary — induction coil, capacitor plates, spark gap — and a passive receiving loop whose dimensions he adjusted, centimetre by centimetre, until its own $1/\sqrt{LC}$ matched the transmitter's. That tuning procedure was the point: sparks jumped in the receiver only near matching, and by plotting spark length against detuning Hertz traced what we would now call a resonance curve — the first one ever measured for an electrical oscillator. He understood the two knobs of this article completely, shrinking $C$ (small spheres instead of jars) and $L$ (short, fat conductors) to push $f_0$ from the kilohertz of Leyden-jar demonstrations to tens of megahertz, where his resonators became efficient radiators. Every tuned circuit since — crystal set to smartphone front end — is a descendant of that deliberately matched pair.

History

1745

— The Leyden jar gives physics its first capacitor, and its discharges their famous violence.

1842

— Joseph Henry, discharging jars through coils, notes sparks that magnetize needles first one way then the other: the discharge is "not instantaneous but oscillatory."

1853

— William Thomson (later Lord Kelvin) publishes the theory: discharge through an inductor obeys the pendulum equation, ringing at

1/2\pi\sqrt{LC}

1858

— Berend Feddersen photographs a spark in a rotating mirror, resolving individual oscillations — Kelvin's prediction made visible.

1887

— Hertz builds matched high-frequency LC resonators and launches the waves that vindicate Maxwell.

1901

— Marconi's transatlantic signal rides tuned LC circuits at both ends; "syntony" (tuning) becomes the radio engineer's craft.

1960

— Quartz crystals — electromechanical resonators with

Q

in the millions — displace LC tanks as frequency standards, but every radio still opens with an LC front end.

Going further

The driven version of this circuit — an AC source forcing the loop at an adjustable frequency — is where the story pays off practically: aq-em-ac-circuits-phasors builds the impedance algebra, and aq-em-ac-power-resonance develops the resonance curve, bandwidth, and the quality factor $Q$ in full. The element physics underneath came from aq-em-capacitance, aq-em-capacitor-energy-dielectrics, aq-em-self-inductance, and aq-em-energy-stored-in-magnetic-fields; the transient one-store circuits are aq-em-rc-circuits and aq-em-rl-circuits. On the mechanics side, the same equation is solved and dissected in Damped Harmonic Oscillator, Underdamped, Critically Damped, and Overdamped Regimes, and Resonance — everything proved there transfers here through the dictionary table. Provenance: Authored fresh for the SuperSTEM E&M track's circuits chapter (recon-§d gap heal); sources: v3 circuit pilots (rewritten to UG register) + legacy wiki circuit articles (narrative) — no AllQFT base exists for this topic.

Where does the energy actually live while it "sloshes" — in the wires?

No — in the fields. When the capacitor is charged, the energy density $\varepsilon_0 E^2/2$ fills the gap between its plates; a quarter period later the energy density $B^2/2\mu_0$ fills the volume threaded by the inductor's field. The wires are just plumbing that lets the two field regions exchange energy. This is not pedantry: at high frequency you can remove the wires entirely and let the exchange happen through space — that is what Hertz's transmitter–receiver pair did, and it is why the LC circuit is the natural parent of radio.

Why is the frequency 1/√(LC) and not something like R/L — where did the resistor go?

The natural frequency is set by the two energy stores: inertia ( $L$ ) and stiffness ( $1/C$ ). The resistor is not a store — it cannot return energy — so it cannot set the pitch; it only drains the amplitude, at rate $R/2L$ , and drags the frequency down slightly through $\omega'^2 = \omega_0^2 - (R/2L)^2$ . The shift is second order: even at $Q \approx 4$ (a poor circuit) the pitch drops less than 1%. Only as $R$ approaches $R_c = 2\sqrt{L/C}$ does the drag win outright and the oscillation cease.

The solution says the current can't jump — but at t = 0 the capacitor voltage across the inductor is large. How can di/dt be big while i is zero?

That is exactly the resolution: $i$ and $di/dt$ are independent pieces of initial data. At the instant the switch closes, $i = 0$ (the inductor forbids a current jump) while $di/dt = V_0/L$ is large (the capacitor's full voltage sits across the inductor and accelerates the charge). It is the same distinction as a ball released from rest on a steep hill: velocity zero, acceleration large. Confusing the two is the most common error in every transient problem in this chapter.

Is the LC oscillation really the "same equation" as the mass on a spring, or just similar?

Identical, term for term: $L\ddot q + R\dot q + q/C = 0$ versus $m\ddot x + b\dot x + kx = 0$ under the dictionary $q\to x$ , $L\to m$ , $R\to b$ , $1/C\to k$ . "Identical equation" means every theorem transfers: the three damping regimes, the fastest-settling property of critical damping, the driven-oscillator response you will meet next. This is why physicists prize canonical equations — solve one, and you have solved every system that maps onto it, from a swaying bridge to a qubit readout resonator.

Why does the energy slosh twice per period? Shouldn't it go around once per cycle?

Energy depends on the squares $q^2$ and $i^2$ , and squaring doubles the frequency: $\cos^2\omega_0 t = \tfrac12(1 + \cos 2\omega_0 t)$ . Concretely, the capacitor is "full" both at $t = 0$ (plates $+,-$ ) and at $t = T/2$ (plates $-,+$ ) — full twice per period, empty twice per period, so the energy makes two round trips while the charge makes one. The same doubling shows up in the mass–spring system and, later, in the fact that AC power pulses at twice the line frequency.

What sets the amplitude of the oscillation? The formula has q0 in it but nothing fixes it.

Initial conditions — and nothing else. The homogeneous oscillator equation is linear: if $q(t)$ is a solution, so is any multiple of it. Whatever charge you place on the capacitor before closing the switch becomes $q_0$ , and the circuit rings at that amplitude (decaying if $R > 0$ ). This is different from the driven circuits of the next articles, where the source amplitude and the impedance fix a definite steady-state response independent of how you start.

My lab LC circuit rings at a slightly lower frequency than 1/2π√LC even though the coil resistance is tiny. What gives?

Three usual suspects. First, real coils have stray capacitance between windings and real capacitors have lead inductance, so the effective $L$ and $C$ are a little larger than the nameplate values. Second, anything conductive nearby (a steel bench, a shield can) shifts $L$ by induced currents. Third, if you measure with a probe, its ~10 pF input capacitance adds directly to $C$ — a big effect in picofarad-scale circuits. The damping correction $(R/2L)^2$ is usually the smallest of the four effects; blame the parasitics first.