What is the time constant of an RC circuit?

The time constant τ = R·C (resistance times capacitance) is the natural timescale of the circuit — the time for the capacitor voltage to cover 1 − 1/e ≈ 63.2% of the remaining gap to its final value. With R in ohms and C in farads, τ comes out in seconds: 200 Ω with 50 µF gives 10 ms, while 10 kΩ with 1000 µF gives 10 s. Doubling either component doubles τ and slows everything proportionally; τ is the only quantity that sets the speed.

How long until a capacitor is “fully” charged?

Never exactly — the charge approaches the supply voltage asymptotically as V = ε(1 − e^(−t/τ)) — but it gets close fast. After one time constant it is 63.2% charged, after two 86.5%, three 95%, four 98.2%, and after five time constants 99.3%. Engineers treat 5τ as “fully charged” (and 5τ as “fully discharged”) because the last 0.7% rarely matters. The current does the reverse: it starts at its maximum ε/R the instant you connect and decays as e^(−t/τ).

Why does an RC circuit charge fast then slow down?

Because the charging current is driven by the voltage still missing across the capacitor. At the start the capacitor is empty, the full supply voltage pushes current, and it charges quickly. As it fills, less voltage is left to drive current, so the flow tapers and the approach slows — an exponential, not a straight line. That same exponential, in reverse, governs discharge, and it is the basis of timers, filters and debouncing circuits everywhere in electronics.

Electronics

The RC Circuit, by the Clock

By Lucian — builder & engineer, LK Forge

Connect a battery to a resistor and a capacitor and the capacitor doesn’t just fill up — it rushes, then dawdles, on a curve set by a single number. That number, τ = R·C, is the heartbeat of half of electronics: timers, filters, flashes and debouncers all run on it. We took the RC-circuit lab’s equations and clocked the charge.

September 10, 2026 · 5 min read · exact from V = ε(1 − e^(−t/τ))

τ = R·C

the one number that sets the whole timescale

63%

charged after a single time constant

5τ → 99.3%

the “fully charged” rule of thumb

ε / R

the inrush current the instant you connect

Rush, then dawdle

Plotted against time measured in time constants, the curves are universal — the same for a 10 ms circuit and a 10 second one. Charging climbs steeply then flattens toward the supply voltage; discharging falls the same way toward zero. The crossing point, where a charging and discharging capacitor briefly share the same voltage, sits at one time constant and 63% / 37%.

Charging and discharging in time-constant units — one universal pair of curves for every RC circuit ever built.

Why five is “full”

Each time constant closes 63.2% of whatever gap to full remains, so the charge climbs in ever-smaller steps and never quite arrives. But the shortfall shrinks by a factor of e each step, and after five it is under 1%. That is the origin of the ubiquitous engineering rule of thumb: wait 5τ and treat the job as done.

Percent charged at each time constant. The last bar (5τ, 99.3%) is the practical finish line; discharge follows the same steps down to 0.7%.

Three circuits, three speeds

Only the product R·C sets the pace, and it spans milliseconds to seconds with ordinary parts. The initial current is the other half of the picture: it is ε/R the instant you connect, before the capacitor has any voltage to push back.

Resistance	Capacitance	τ = R·C	Inrush current (9 V)
200 Ω	50 µF	10 ms	45 mA
1000 Ω	470 µF	0.47 s	9 mA
10000 Ω	1000 µF	10 s	0.9 mA

τ = R·C with R in ohms and C in farads (µF × 10⁻⁶) gives seconds directly. A bigger capacitor stores more charge so takes longer; a bigger resistor throttles the current so also takes longer.

Reproduce it yourself

Two lines give the whole curve and the ladder.

const tau = (R, C) => R * C;                    // ohms × farads = seconds
const Vc  = (eps, t, R, C) => eps * (1 - Math.exp(-t / tau(R, C)));

tau(1000, 470e-6);          // 0.47 s
[1,2,3,4,5].map(n => (100 * (1 - Math.exp(-n))).toFixed(1));
// ["63.2","86.5","95.0","98.2","99.3"]  → 5τ ≈ fully charged

The charge, current and time-constant formulas are lifted straight from the RC-circuit solver; the ladder and component table are the small script behind this post.

Charge a capacitor

Open the lab, change R and C to feel the time constant stretch and shrink, then flip between charge and discharge to watch the mirror-image curves.

Open the RC-circuit lab →

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