{"id":718,"date":"2026-08-06T15:16:06","date_gmt":"2026-08-06T15:16:06","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=718"},"modified":"2026-08-06T15:16:08","modified_gmt":"2026-08-06T15:16:08","slug":"rc-time-constant","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/electromagnetism\/rc-time-constant\/","title":{"rendered":"What Is the RC Time Constant?"},"content":{"rendered":"\n
Definition<\/div>

\nThe RC time constant is the time taken for a capacitor in a resistor-capacitor circuit to charge to about 63.2% of its final voltage, or to fall to 36.8% of its starting voltage while discharging. It equals resistance multiplied by capacitance: tau equals R times C, measured in seconds.\n<\/p><\/div>\n\n

Pull the plug on a desktop computer and watch the power LED. It does not snap off \u2014 it lingers, dims, then fades out over a second or so. Somewhere on that motherboard a capacitor is emptying itself through a resistor, and the pace of that fade is not random.<\/p>

It is set by two numbers you could read off the components with a magnifying glass. Multiply them together and you get a time. That product is the RC time constant, and once you can calculate it you can predict how long almost any charging or discharging circuit will take to settle.<\/p>

What Is the RC Time Constant?<\/h2>

The RC time constant is the characteristic timescale of a circuit containing resistance and capacitance, equal to the product of the two. It is written with the Greek letter tau (\u03c4), and despite being built from an ohm and a farad, it comes out in seconds.<\/p>

Think of filling a bucket through a narrow hose, where the tap pressure drops as the bucket fills. Early on, water pours in fast. As the level rises the flow eases off, so the last centimetre takes far longer than the first. A capacitor charges exactly like that \u2014 and \u03c4 is the natural yardstick for how sluggish the whole process is.<\/p>

Here is the part that catches people out. One time constant is not<\/em> the time to charge fully. It is the time to get 63.2% of the way there, and that figure is fixed by the mathematics, not by the components you picked.<\/p>

Why 63.2%?<\/h3>

Because the charging curve is exponential. After one time constant the fraction of the journey still remaining is e-1<\/sup>, which is 0.368 \u2014 so the fraction completed is 1 – 0.368 = 0.632. Every RC circuit ever built hits that same 63.2% mark at t = \u03c4, whether \u03c4 is a microsecond or an hour.<\/p>

The RC Time Constant Formula<\/h2>

The RC time constant formula is \u03c4 = R \u00d7 C: multiply the resistance in ohms by the capacitance in farads and the answer comes out directly in seconds.<\/p>\n\n

\u03c4 = R \u00d7 C<\/div>\n\n

Every symbol, with its SI unit:<\/p>

Symbol<\/th> Quantity<\/th> SI unit<\/th> Typical values<\/th> <\/tr> <\/thead>
\u03c4<\/strong><\/td> Time constant<\/td> second (s)<\/td> 1 \u00b5s to 10 s<\/td> <\/tr>
R<\/strong><\/td> Resistance in series with the capacitor<\/td> ohm (\u03a9)<\/td> 100 \u03a9 to 1 M\u03a9<\/td> <\/tr>
C<\/strong><\/td> Capacitance<\/td> farad (F)<\/td> 1 pF to 10 mF<\/td> <\/tr> <\/tbody> <\/table> <\/div>

Why Ohms Times Farads Gives Seconds<\/h3>

This one is worth doing once, because it makes the formula impossible to forget. An ohm is a volt per amp, and a farad is a coulomb per volt.<\/p>

Multiply them and the volts cancel: \u03a9 \u00d7 F = (V\/A) \u00d7 (C\/V) = C\/A. Since an amp is a coulomb per second, C\/A is a coulomb divided by a coulomb-per-second \u2014 which is a second. The result is exact, not approximate, and it follows straight from the SI definitions of the derived units<\/a> maintained by NIST.<\/p>

In practice, the arithmetic is easiest if you convert everything to base units first: kilohms to ohms, microfarads to farads. A 47 k\u03a9 resistor with a 22 \u00b5F capacitor gives \u03c4 = 47,000 \u00d7 0.000022 = 1.034 s. If you would rather skip the unit-juggling, our RC Time Constant Calculator<\/a> handles the prefixes and will also rearrange the formula to solve for R or C when you already know the timing you need.<\/p>

How the RC Time Constant Works<\/h2>

The exponential shape comes from a feedback loop hiding inside the circuit. Here is the circuit in question:<\/p> <\/g> <\/g> <\/g> <\/g> V0<\/text> R<\/text> C<\/text> switch<\/text> I<\/text> the same current flows through R and C<\/text> <\/g> <\/svg>

A series RC circuit. Closing the switch starts the charging process governed by \u03c4 = RC.<\/p>

Close the switch and, for an instant, the capacitor is empty. The full supply voltage appears across the resistor, so the current is at its maximum, V0<\/sub>\/R.<\/p>

That current pushes charge onto the capacitor, and charge on a capacitor is<\/em> voltage. As the capacitor voltage climbs, less of the supply is left over for the resistor \u2014 so the current falls. Less current means slower charging, which means the voltage climbs more gently still.<\/p>

The circuit throttles itself. That is the signature of exponential behaviour: the rate of change is proportional to how far you still have to go.<\/p>

The Derivation in Three Lines<\/h3>

Apply the loop rule around the circuit. The supply voltage is shared between the resistor and the capacitor:<\/p>\n\n

V0 = I\u00b7R + Q\/C<\/div>\n\n

Current is the rate at which charge arrives, I = dQ\/dt, so this becomes a differential equation in Q alone:<\/p>\n\n

R (dQ\/dt) + Q\/C = V0<\/div>\n\n

Separating variables and integrating gives the charging solution. The group RC sits in the exponent, which is precisely why it has the units of time:<\/p>\n\n

V(t) = V0 \u00d7 (1 – e^(-t\/\u03c4))<\/div>\n\n

For a capacitor already charged to V0<\/sub> and then discharged through R, the same algebra with the supply removed gives the mirror-image curve:<\/p>\n\n

V(t) = V0 \u00d7 e^(-t\/\u03c4)<\/div>\n\n

MIT’s introductory circuits course treats these as the standard first-order system response<\/a>, which is the general name for any system whose behaviour is governed by a single time constant like this one.<\/p>\n\n

<\/span>RC Time Constant Lab<\/span><\/div>