Lesson 2.2 · Module 2: Reactive Components in Time & Frequency

RC Time Constant: Capacitor Charging Explained

A capacitor charging and discharging through a resistor. This is how circuits make time delays, like a light that fades out slowly.

Think of it like this: A capacitor is like a bucket filled through a narrow pipe (the resistor). It fills fast at first, then slower and slower as it gets full.

The live, interactive simulation for this lesson opens here when JavaScript is on.

Key ideas

A capacitor stores charge. Its voltage cannot jump; it rises gradually.
The time constant τ = R × C sets the speed. After one τ it is 63 % full, after five τ it is practically full.
Current is biggest at the very start of charging and fades to zero once the capacitor is full.

Try this in the simulation

Click S1 to charge, then click again to discharge. Watch the curve on the oscilloscope.
Make R or C ten times bigger: everything happens ten times slower.
Hover any wire or part in the circuit to see its live voltage and current. Click to pin the readout.

What is the RC time constant?

When a capacitor charges through a resistor, its voltage does not jump: it rises quickly at first and then more and more slowly. The time constant τ = R × C tells you the speed. After one τ the capacitor is about 63 % charged, after five τ it is practically full, and discharging follows the same curve in reverse. Bigger resistance or capacitance means a slower change.

What you’ll learn

What a capacitor does, and why its voltage cannot change instantly.
How to calculate the time constant τ = R × C.
The 1 τ to 5 τ rule: 63 %, 86 %, 95 %, 98 %, 99 % charged.
How much energy a charged capacitor stores.

The formula, letter by letter

\begin{aligned}\htmlClass{mv mv-v_c}{\htmlStyle{font-size:0.80em}{\htmlData{sym=U_C}{U_{\mathrm{C}}}(\htmlData{sym=t}{t})}} &= \htmlClass{mv mv-v_in}{\htmlStyle{font-size:1.00em}{\htmlData{sym=U_s}{U_{\mathrm{s}}}}}\left(1 - \htmlData{sym=e}{\mathrm{e}}^{-\htmlData{sym=t}{t}/\htmlData{sym=tau}{\tau}}\right),\quad \htmlData{sym=tau}{\tau }= \htmlClass{mv mv-r}{\htmlStyle{font-size:1.19em}{\htmlData{sym=R}{R}}}\htmlClass{mv mv-c}{\htmlStyle{font-size:1.23em}{\htmlData{sym=C}{C}}} = \htmlData{funit=ms,v=0.1}{\text{100 ms}}\\ \htmlClass{mv mv-i}{\htmlStyle{font-size:1.00em}{\htmlData{sym=I}{I}}} &= \tfrac{\htmlData{sym=U_s}{U_{\mathrm{s}}} - \htmlData{sym=U_C}{U_{\mathrm{C}}}}{\htmlData{sym=R}{R}} = \htmlData{funit=µA,v=0.0005}{\text{500 }\mu\text{A}},\quad \htmlData{sym=E}{E} = \tfrac12 \htmlData{sym=C}{C} \htmlData{sym=U_C}{U_{\mathrm{C}}}^2 = \htmlData{funit=J,v=0}{\text{0.00 J}}\end{aligned}

Shown with this lesson’s starting values. In the simulation the numbers change as you move the sliders.

$U_{\mathrm{C}}$: Voltage across the capacitor (V · volt)
$t$: Time (s · second) : Time since something started.
$U_{\mathrm{s}}$: Supply voltage (V · volt) : The source or supply voltage.
$\mathrm{e}$: Euler’s number e ≈ 2.718 (no unit) : A fixed number used for anything that grows or decays smoothly, such as a capacitor charging. Not a variable.
$\tau$: Time constant τ (tau) (s · second) : How long an RC or RL circuit takes to get 63 % of the way to its final value. After 5τ it is basically done.
$R$: Resistance (Ω · ohm) : How strongly something opposes current.
$C$: Capacitance (F · farad) : How much charge a capacitor stores per volt.
$I$: Current (A · ampere) : How much charge flows each second.
$E$: Energy (J · joule) : An amount of energy.

Common questions

What does the RC time constant mean?

It is the time a capacitor needs to get about 63 % of the way to its final voltage when charging through a resistor. R in ohms times C in farads gives the answer in seconds.

How long does a capacitor take to fully charge?

In theory it never quite finishes, but after five time constants it is within 1 % of full, so that is the usual rule of thumb.

What happens if I make the resistor bigger?

The time constant grows in proportion, so the capacitor charges and discharges more slowly. Doubling R doubles the time.