RC Circuits — Time Constant, Charging and Discharging

Intermediate⏱ 10 min readElectricity & Magnetism

RC circuits explained: the time constant τ = RC, exponential charging and discharging curves, why a capacitor blocks DC in steady state, and a worked timing…

Quick answer

An RC circuit is a resistor and capacitor in series. The capacitor charges exponentially: after one time constant τ = RC it reaches 63% of the supply voltage, and it is effectively full after about 5τ. In steady state with DC, the capacitor acts as an open circuit.

$\tau = RC, \qquad V_C(t) = V_0\left(1 - e^{-t/\tau}\right)$

Unit: time constant τ in seconds (Ω·F = s)

Example: R = 10 kΩ, C = 100 μF → τ = 1 s. After 1 s the capacitor is at 63% of 9 V = 5.7 V; after 5 s it is essentially at 9 V.

What you will learn

State the RC time constant τ = RC and its meaning
Write the exponential charging and discharging equations
Explain why a capacitor blocks DC in steady state
Calculate charge and voltage at any time t
Apply RC analysis to a timing example

A resistor, a capacitor, and a clock

Wire a battery, a resistor, and an uncharged capacitor in series and close the switch. Current surges — the empty capacitor acts almost like a plain wire — then fades away as the capacitor fills. When the capacitor's voltage equals the battery's, current stops entirely.

The resistor is the throttle: it limits how fast charge can arrive, so a bigger R means a slower fill. The capacitor is the tank: a bigger C holds more charge per volt, so it takes longer to reach full voltage. Together they set a single number that governs everything about the circuit's timing.

The RC loop: the resistor throttles the flow while the capacitor fills up.

The time constant τ = RC

The product of resistance and capacitance has the units of time:

$\tau = RC$

$\tau$ (tau), the time constant, is the characteristic time scale of the circuit. It does not depend on the battery voltage — a bigger battery fills the capacitor to a higher final voltage, but the fraction of the journey completed in time $\tau$ is always the same.

Unit check: ohms times farads = $(\mathrm{V/A}) \times (\mathrm{C/V}) = \mathrm{C/A} = \mathrm{s}$ . Resistance times capacitance is always seconds — a dimensional guarantee that the formula is well-formed.

Charging: the exponential curve

While the capacitor charges, its voltage $V_C$ climbs toward the battery voltage $V_0$ exponentially:

$V_C(t) = V_0\left(1 - e^{-t/\tau}\right)$

The shape is steep at first (empty capacitor, full current) and flattens as it fills (rising back-voltage throttles the current). Benchmark points worth memorizing:

$t = \tau$ : 63% charged
$t = 2\tau$ : 86% charged
$t = 3\tau$ : 95% charged
$t = 5\tau$ : 99.3% charged — effectively full

Engineers treat $5\tau$ as 'done'. The current follows the mirror image, $I(t) = I_0 e^{-t/\tau}$ , starting high and decaying to zero.

The 63% rule

One time constant gets you 63% of the way (charging) or down to 37% (discharging). It is the single most useful number for estimating RC timing without a calculator.

Discharging: the mirror image

Short the charged capacitor through a resistor (no battery). Now charge drains out, and the voltage falls exponentially:

$V_C(t) = V_0\,e^{-t/\tau}$

Same time constant, mirrored curve. After $5\tau$ the capacitor is down to 0.7% of its starting voltage — effectively empty. This is the curve behind camera flashes: a slow charge (large effective R) followed by a near-instant discharge through the flash tube (tiny R), dumping all the stored energy in milliseconds.

One caution for real life: large capacitors can hold a dangerous charge long after the power is off. Technicians short them with a resistor before handling them — and they wait a few time constants first.

Why capacitors block DC in steady state

Look at the charging curve's end state: current has decayed to zero. A fully charged capacitor in a DC circuit is an open circuit — it passes no steady current at all.

This is not a property of the plates being 'full' in some absolute sense; it is because the capacitor's voltage now exactly opposes the battery, leaving zero net voltage to drive current through the resistor. Change the battery voltage, and a brief current flows again until the new balance is reached. This blocking behavior is what lets capacitors separate DC levels between circuit stages while passing changing signals — the foundation of AC coupling.

Worked example: timing the charge

A $10\ \mathrm{k\Omega}$ resistor and a $100\ \mu\mathrm{F}$ capacitor are wired in series with a $9\ \mathrm{V}$ battery. How long until the capacitor reaches $8\ \mathrm{V}$ , and what is the current at that moment?

Time constant: $\tau = RC = (10{,}000)(100 \times 10^{-6}) = 1.0\ \mathrm{s}$ .

Set $V_C = 8\ \mathrm{V}$ : $8 = 9(1 - e^{-t/1})$ , so $e^{-t} = 1/9$ and $t = \ln 9 \approx 2.2\ \mathrm{s}$ .

At that moment the resistor drops $9 - 8 = 1\ \mathrm{V}$ , so $I = V_R/R = 1/10{,}000 = 0.10\ \mathrm{mA}$ .

Notice the current started at $0.9\ \mathrm{mA}$ (empty capacitor, full $9\ \mathrm{V}$ across R) and has decayed nearly tenfold — the exponential in action.

Reality check

2.2 s is between τ (63% ≈ 5.7 V) and 3τ (95% ≈ 8.6 V), and 8 V is 89% — right where it should land. The current check closes the loop via Ohm's law.

Variables and units

Symbol	Name	Unit	Meaning
$τ$	Time constant	s	Characteristic time of the circuit
$R$	Resistance	Ω	Series resistance throttling the charge flow
$C$	Capacitance	F	Capacitance being charged
$V_C$	Capacitor voltage	V	Voltage across the capacitor at time t
$V₀$	Source voltage	V	Battery or supply voltage
$t$	Elapsed time	s	Time since the switch closed
$I$	Circuit current	A	Current through the resistor at time t

Common mistakes

✗ Current flows through a capacitor while it charges.

Charge piles up on one plate and leaves the other — the wires carry current, but no charge crosses the gap. Inside the capacitor it is displacement, not conduction.

✗ The time constant is the time to fully charge.

τ = RC is the time to reach 63% (charging) or drop to 37% (discharging). Full charge takes about 5τ.

✗ A bigger capacitor always charges faster.

Bigger C means a bigger time constant (slower), unless you also lower R. Speed is set by the product RC, never by one part alone.

Real-world applications

Camera flashes use an RC circuit tuned so the capacitor charges in a few seconds, then fires instantly.
Heartbeat pacemakers time their pulses with RC delays built from a resistor and capacitor.
Touch dimmers and blinking LED circuits use RC timing to set their on/off rhythm.
Engineers estimate how long a power supply's capacitors take to recharge after a blackout using the RC time constant.

Frequently asked questions

How long does a capacitor take to charge?

Seconds, roughly τ = RC. A 10 kΩ resistor with a 100 μF capacitor gives τ = 1 s: about 1 s to reach 63% and 5 s to finish. Swap in a 1 MΩ resistor and the same capacitor needs hours.

Does a capacitor ever finish charging?

Never perfectly — but after 5 time constants it is within 1% of full, which counts as 'done' for every practical purpose. Mathematically the exponential only reaches its limit at infinite time.

Does Ohm's law still apply in an RC circuit?

Yes. The circuit obeys Ohm's and Kirchhoff's laws at every instant — the quantities just change with time. The current follows I = (V₀ − V_c)/R, where V_c is the capacitor's growing voltage, and that is exactly what produces the exponential.

What if I charge a capacitor through a larger resistor?

It still fills in about five time constants, but the time constant itself changes: τ = R_eq·C where R_eq is the new total resistance. Bigger resistance means slower charging, which is why a resistor is the 'throttle' of an RC timer.