What this calculator does
Find out how quickly a capacitor fills up or empties through a resistor. A capacitor is a part that stores electric charge. As it charges or discharges, its voltage doesn't change at a steady rate. It moves quickly at first, then more and more slowly, along a curve that flattens out (an exponential curve).
One number sets the pace of that curve: the time constant τ (tau). It's the resistance times the capacitance. After one time constant, a charging capacitor reaches about 63.2% of the supply voltage. After five, it's within 1% of full.
The calculator finds τ for your resistor and capacitor. Then it finds either the capacitor's voltage at a time you choose, or the time it takes to reach a target voltage. It works for charging or discharging. It also reports the current flowing at that moment and the charge and energy stored. And it gives the cutoff frequency the same parts would have as a simple RC filter, a circuit that blocks some frequencies.
Where it's used
Why this math matters outside the classroom:
- Timing circuits. Hobbyists pick 47 kΩ and 22 µF for a delay of about one time constant, 1.03 seconds.
- Button debouncing. Engineers choose 10 kΩ and 100 nF (τ = 1 ms) so a switch's contact bounce is smoothed before a microcontroller reads it.
- Camera flash. Physics classes estimate how long a flash capacitor takes to recharge from its charging resistor and capacitance.
- Lab exponentials. Students log a discharging capacitor's voltage every second and fit V₀e^(−t/τ) to measure τ.
- Audio filters. A designer uses 1.6 kΩ and 100 nF for a tone filter with a cutoff near 1 kHz.
Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSF.LE.A.4, HSF.LE.B.5. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose whether the capacitor is charging from a supply or discharging from a starting voltage.
- Choose whether to find the voltage at a given time or the time to reach a given voltage.
- Enter the resistance and capacitance with their units, and the supply (or starting) voltage.
- Enter the time, or the target voltage. Then read the result, the steps and the table for one to five time constants (1τ–5τ).
What the inputs mean
- Resistance (R)
- The total resistance the capacitor charges or discharges through, in ohms, kilohms or megohms.
- Capacitance (C)
- The capacitor's value. Electrolytic capacitors (a common, larger type) often vary from their labeled value by up to ±20%. So real timings vary by about that much.
- Supply voltage (charging) or starting voltage (discharging)
- When charging, this is the voltage of the source the capacitor is connected to through the resistor. When discharging, it's the voltage across the capacitor at the start.
- Time since the switch closed
- Time since the switch was closed. Charging starts from an empty capacitor; discharging starts from the starting voltage.
- Target capacitor voltage
- The capacitor voltage to reach. A charging capacitor never quite reaches the supply voltage, so the target must be below it.
Formula and method
τ = R × C charging: Vc(t) = Vs × (1 − e^(−t/τ)) discharging: Vc(t) = V₀ × e^(−t/τ) time to reach Vc: t = −τ × ln(1 − Vc/Vs) when charging, t = −τ × ln(Vc/V₀) when discharging current: I(t) = (V/R) × e^(−t/τ); cutoff frequency f = 1 ÷ (2πRC)
In the formulas, R is the resistance and C is the capacitance. t is the time since the switch closed. Vc is the capacitor's voltage, Vs is the supply voltage, and V₀ is the starting voltage when discharging. e is a special math constant that describes natural growth and decay. I is the current and f is the cutoff frequency.
With R in ohms and C in farads, τ comes out in seconds. The curve depends only on t ÷ τ, the time counted in time constants. So every RC circuit follows the same curve: 63.2% after 1τ, 86.5% after 2τ, 95.0% after 3τ, 98.2% after 4τ and 99.3% after 5τ.
Solving for time uses ln, the natural logarithm, which undoes e. The charge stored is Q = C × Vc, and the energy is ½ × C × Vc². The current falls along the same kind of curve, from V ÷ R at the start toward zero.
Worked example
A 100 µF capacitor charges through a 10 kΩ resistor from a 5 V supply. What is its voltage after 2 seconds?
- Time constant: τ = 10,000 Ω × 0.0001 F = 1 s.
- Time in time constants: 2 s ÷ 1 s = 2.
- Voltage: 5 × (1 − e^−2) = 5 × 0.864665 = 4.3233 V.
- To reach 4 V instead: t = −1 × ln(1 − 4 ÷ 5) = ln 5 = 1.6094 s.
Result: After 2 seconds the capacitor is at about 4.3233 V, 86.47% of the supply.
Understanding the result
Five time constants (5τ) is the usual rule of thumb for fully charged or fully empty. Strictly speaking, the capacitor only gets closer and closer to the end voltage.
The cutoff frequency is the point where a filter built from these parts lets signals through at about 70.7% of their full strength (−3 dB). That holds for both RC low-pass and high-pass filters.
Assumptions and limitations
- The resistor and capacitor are perfect, with no leakage. The source is perfect too, with no resistance of its own (or with that included in R).
- Charging starts from 0 V. Discharging starts from the starting voltage with no supply connected.
- Part values are exact; real parts can differ a little from their labels.
- Circuits with several resistors or capacitors must first be boiled down to one matching R and one C.
- Real-part effects aren't modeled. These include leakage current, the capacitor's own small internal resistance (equivalent series resistance) and dielectric absorption (charge that creeps back after discharge).
- For safety: large capacitors can hold a dangerous charge. Never rely on a calculated discharge time instead of measuring.
Frequently asked questions
Why 63.2%?
After one time constant, the term e^(−t/τ) equals e^(−1), about 0.368. So a charging capacitor has 1 − 0.368 = 0.632 of the supply voltage. A discharging one keeps 36.8% of its starting voltage.
How long does a capacitor take to charge fully?
About five time constants, when it reaches 99.3% of the supply. With 10 kΩ and 100 µF, τ is 1 second. So it's practically full after about 5 seconds.
What units should I use?
Any of the listed ones. The calculator converts ohms, kilohms and megohms, and picofarads up to farads, before multiplying. In the basic units, ohms times farads gives seconds.
How is this related to an RC filter?
The same R and C set a filter's cutoff frequency, f = 1 ÷ (2πRC). A 1-second time constant gives a cutoff of about 0.159 Hz. A 1-millisecond one gives about 159 Hz.
How this calculator is checked
Method: First-order RC circuit solution: τ = RC, Vc = Vs(1 − e^(−t/τ)) charging and V₀e^(−t/τ) discharging; filter cutoff 1/(2πRC). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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