RC Time Constant

Q: Does the capacitor discharge with the same time constant?

Yes. Its voltage falls as $V_C(t) = V_0 e^{-t/\tau}$, and after one time constant it has dropped to about 36.8% of its starting voltage.

In one sentence: The RC time constant tells you how long a capacitor takes to fill up through a resistor: multiply the resistance by the capacitance.

$\tau = R C$

Read it out loud: “Tau equals R times C. The time constant equals resistance times capacitance.”

What each symbol means

Symbol	Meaning
$\tau$	The time constant (the Greek letter tau), measured in seconds (s). It sets how fast the capacitor charges or discharges.
$R$	Resistance of the resistor in the circuit, measured in ohms ( $\Omega$ ).
$C$	Capacitance, how much charge the capacitor holds for each volt, measured in farads (F). A microfarad is $10^{-6}$ F.

What the formula is doing

A capacitor stores electric charge, a bit like a small reservoir. When a capacitor is connected through a resistor to a battery, it does not fill up at once. The resistor limits how fast charge can flow in, so the voltage across the capacitor rises gradually.

The product $\tau = R C$ is the time constant, in seconds. Ohms times farads give seconds, because $\Omega \cdot \text{F} = \text{s}$ . After one time constant the capacitor voltage has reached about 63.2% of the supply voltage. The charging curve is $V_C(t) = V_s \left(1 - e^{-t/\tau}\right)$ , where $V_C(t)$ is the capacitor voltage at time $t$ , $V_s$ is the supply voltage and $e$ is Euler’s number, about 2.718.

The curve is steep at the start and flattens out as it nears the supply voltage. After about five time constants the capacitor is more than 99% charged, which is why five time constants is a common rule of thumb for full charge.

Bigger numbers mean slower charging. A larger resistor limits the current more, and a larger capacitor holds more charge and takes longer to fill. Doubling either $R$ or $C$ doubles $\tau$ .

An everyday picture

Think of filling a bucket through a hose. The resistor is the hose: a narrower hose fills the bucket more slowly. The capacitor is the bucket: a bigger bucket takes longer to fill. The time constant is the time it takes the water level to rise most of the way up.

Worked example: Charging a 100 microfarad capacitor through 10 kiloohm

A 5 V supply charges a capacitor with $C = 100\ \mu\text{F}$ through a resistor with $R = 10\ \text{k}\Omega$ , which is 10 000 ohm. Find the time constant, then the capacitor voltage after 1 s and after 5 s.

Time constant: $\tau = R C = 10 \times 10^{3}\ \Omega \times 100 \times 10^{-6}\ \text{F} = 1\ \text{s}$ .
Fraction charged after 1 s: $1 - e^{-1} \approx 1 - 0.3679 = 0.632$ , so 63.2%.
Voltage after 1 s: $V_C(1\ \text{s}) = 5\ \text{V} \times 0.632 \approx 3.16\ \text{V}$ .
Fraction charged after 5 s: $1 - e^{-5} \approx 1 - 0.0067 = 0.993$ , so 99.3%.
Voltage after 5 s: $V_C(5\ \text{s}) = 5\ \text{V} \times 0.993 \approx 4.97\ \text{V}$ .

Answer: The time constant is 1 s. After 1 s the capacitor is at 3.16 V, which is 63.2% of 5 V. After 5 s it is at 4.97 V, which is 99.3% of the supply, so it is almost fully charged.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Forgetting the prefixes. 100 microfarad is $100 \times 10^{-6}$ F, and 10 kiloohm is 10 000 ohm. Skipping either one changes the answer by a factor of a thousand or a million.
Thinking the capacitor is full after one time constant. After one $\tau$ it is only at about 63.2%, and it takes about five $\tau$ to get above 99%.
Using the supply voltage for the capacitor voltage at every moment. The capacitor starts at 0 V and only reaches the supply voltage after a long time.
Using $\tau = R C$ with the capacitance in farads and the resistance in kiloohms without converting. Convert to ohms and farads first, or the time comes out in the wrong scale.

Where you will meet it

Simple timing circuits, such as the delay in a blinking LED circuit or a short timed switch.
Power supply smoothing and audio filters, where a resistor and capacitor set the frequency at which the signal starts to be cut.
Button debouncing, where the slow charging of a capacitor smooths out the rapid switching when mechanical contacts bounce.
Sensors that measure a resistance by timing how long a capacitor takes to charge.

Questions people ask

What is a farad?

A farad (F) is one coulomb of charge stored per volt. Real capacitors are often rated in microfarads ( $\mu\text{F}$ ) or picofarads, because one whole farad is a very large capacitor.

When is the capacitor fully charged?

Not after one time constant. After about five time constants it is more than 99% charged, which is the usual rule of thumb for fully charged.

Does the capacitor discharge with the same time constant?

Yes. Its voltage falls as $V_C(t) = V_0 e^{-t/\tau}$ , and after one time constant it has dropped to about 36.8% of its starting voltage.

Why does the formula have an e in it?

The charging speed is proportional to how much voltage is still missing. A rule like that gives an exponential curve, and $e$ is the natural base for exponential curves.

Ohm's Law: Ohm’s law says the push on a wire, the voltage, equals the flow through it, the current, times how hard that flow is to get through, the resistance.
Resistors in Series and Parallel: Resistors in a line add up, and resistors side by side always give a total smaller than the smallest one.
Kirchhoff's Current Law: Kirchhoff’s current law says that whatever electricity flows into a junction has to flow out again, because nothing piles up there.