Lessons · circuits

The LC oscillator: the current is the slope of the charge

Regime 2 — calculus does what algebra cannot. The acceleration isn't constant, so the algebra formulas don't apply; calculus is the only road in, and SymPy proves the closed form solves the equation of motion.

Regime 2 · calculus does more Math machine-derived & checked by SymPy 2 modeling assumptions (author-asserted)

A capacitor $C = 100\ \mu$ F is charged to $Q_0 = 0.01$ C and then connected across an inductor $L = 10$ mH. With no resistance to drain it, the charge $Q(t)$ and the current $I(t)$ oscillate forever at angular frequency $\omega = 1/\sqrt{LC}$ . Plotted as a stacked $Q$ – $t$ over $I$ – $t$ , the slope of the charge is the current at every instant — the same slope↔value pivot as position and velocity, and the electrical twin of a mass on a spring. Drag $L$ and watch the period $T = 2\pi\sqrt{LC}$ stretch; the energy sloshes between the capacitor and the inductor while the total stays fixed.

Angular frequency

\omega = 1/\sqrt{LC}

1000 1/s

Period

T = 2\pi\sqrt{LC}

0.006283 s

Peak current

I_{\max} = Q_0\omega = Q_0/\sqrt{LC}

10 A

Total energy

E = Q_0^2/2C

(constant)

0.5 J

Kirchhoff's voltage law is a second-order differential equation

L\,\frac{dI}{dt} + \frac{Q}{C} = 0, \quad I = \frac{dQ}{dt} \ \Longrightarrow\ L\,\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0

Around the loop, the inductor's back-EMF $L\,dI/dt$ balances the capacitor voltage $Q/C$ . Because the current is the rate of change of charge, $I = dQ/dt$ , this is a second-order differential equation for $Q(t)$ — the exact shape of the mass–spring equation $x'' = -\omega^2 x$ , with $\omega^2 = 1/(LC)$ .

check $I = dQ/dt$ ; check $L\,Q'' + Q/C = 0$ ; energy $\tfrac{Q^2}{2C} + \tfrac12 LI^2$ constant; $T = 2\pi\sqrt{LC}$ ; quarter-period $90^\circ$ phase

✓ The current is the slope of the charge: $I = \dfrac{dQ}{dt}$ — exactly as velocity is the slope of position. [structural]
✓ The charge solves the LC equation $L\,\dfrac{d^2Q}{dt^2} + \dfrac{Q}{C} = 0$ — the electrical twin of the spring's $x'' = -\omega^2 x$ . [structural]
✓ Energy trades between the capacitor $\tfrac{Q^2}{2C}$ and the inductor $\tfrac12 LI^2$ ; the total stays at the initial $\tfrac{Q_0^2}{2C}$ . [simplify]
✓ The period is $T = 2\pi\sqrt{LC}$ : a full cycle later the charge repeats, $Q(t+T) = Q(t)$ . [structural]
✓ A quarter period in, the capacitor is empty ( $Q = 0$ ) while the current peaks — charge and current are $90^\circ$ out of phase. [structural]

Dimensional homogeneity: checked by SymPy (holds).

✗ Common misconception: “The charge and current both start large and fade away, the way they do when a capacitor charges through a resistor (RC).”

Not here. With no resistor there is nothing to dissipate the energy, so nothing decays — the charge and current oscillate forever. They are $90^\circ$ out of phase: when the charge $Q$ is at its peak the current $I$ is zero (the capacitor is full and momentarily not flowing), and a quarter cycle later $Q = 0$ while $I$ is at its peak (the capacitor is empty and the current is greatest). The energy trades back and forth between the capacitor's $Q^2/2C$ and the inductor's $\tfrac12 LI^2$ , but the total is constant. RC relaxes; LC rings.

Modeling assumptions — author-asserted, disclosed not discharged

An ideal LC circuit: a perfect inductor and capacitor with no resistance, so no energy is dissipated and the oscillation never decays.
The capacitor starts fully charged to $Q_0$ and the initial current is zero, $I(0) = 0$ — all the energy begins in the capacitor's electric field.

The stacked graph, fully annotated

A static rendering (Matplotlib) at the default parameters — the interactive version is in the Graph tab above.

Formulas used

Hover a formula to preview its reference entry; click to open it in the reference (or the concept graph):