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The RC Time Constant

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Resistors, capacitors, and time

So far our resistor circuits settled immediately in our simplified examples. Add a capacitor and something new appears: time.

+ − 10 V DC source R = 10 kΩ resistor C = 100 µF capacitor + − VC charging current I voltage builds across the plates
The two separated parallel lines are the capacitor. The resistor limits how quickly the capacitor charges.

First, recognize the capacitor symbol

A capacitor is drawn as two separated plates. The gap is important: the plates do not make direct electrical contact.

Non-polarized capacitor C Polarized / electrolytic + C

Different drawing standards use slightly different symbols, especially for polarized electrolytic capacitors. The + mark identifies the positive terminal when polarity matters.

The symbols

R = resistance in ohms (Ω)
C = capacitance in farads (F)
τ = the Greek letter tau, used for the time constant

τ = RC

A useful first picture

A capacitor is somewhat like a small storage tank. The resistor limits how quickly the “tank” can fill or empty. Larger resistance or larger capacitance means a slower change.

What is really happening?

A capacitor stores energy in an electric field. Its voltage cannot jump instantly when charging through a finite resistance. Instead, the voltage approaches its final value exponentially.

One time constant does not mean “fully charged.” After one τ, the capacitor has reached about 63% of its final charging voltage. After about five τ, it is above 99% and is often treated as essentially settled.

What charging looks like over time

100% 0% VC time 1τ63%2τ86%3τ95%4τ98%5τ99% capacitor charging toward its final voltage

The first time constant makes the biggest jump. Each following time constant closes most of the remaining gap, so the curve approaches the final voltage rather than snapping to it.

Worked example

R = 10 kΩ = 10,000 Ω
C = 100 µF = 0.000100 F

τ = RC = 10,000 × 0.000100 = 1 s

So one time constant is 1 second, and five time constants are about 5 seconds.

A little history

The mathematical behavior of capacitor charging grew out of the broader 18th- and 19th-century study of stored electric charge, resistance, and transient circuits. The exponential form later became a standard engineering tool for timing, filtering, and control.

Try these yourself

For R = 10 kΩ and C = 100 µF, what is the RC time constant? + − 10 V DC source R = 10 kΩ resistor C = 100 µF capacitor + − VC charging current I voltage builds across the plates
The two separated parallel lines are the capacitor. The resistor limits how quickly the capacitor charges.
10,000 Ω × 0.000100 F = 1 second.
After about one time constant while charging, a capacitor has reached about: 100% 0% VC time 1τ63%2τ86%3τ95%4τ98%5τ99% capacitor charging toward its final voltage
After 1τ, a charging capacitor reaches about 63.2% of its final value.
After about five time constants, a charging capacitor is: 100% 0% VC time 1τ63%2τ86%3τ95%4τ98%5τ99% capacitor charging toward its final voltage
After 5τ it is above 99% of the final value and is commonly treated as essentially charged for practical work.

A little deeper

For a capacitor charging toward a DC source:

VC(t) = V(1 - e-t/RC)

The number e comes from the mathematics of exponential change. Beginners do not need to calculate this expression to use the extremely useful 1τ and 5τ rules.

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