RC Capacitor Charge/Discharge Calculator

Calculate voltage, charge, and time at any point during RC capacitor charging or discharging.

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RC Circuits Change Exponentially, Not Linearly

A resistor-capacitor circuit has a natural time scale τ=RC that controls how quickly capacitor voltage and charge approach a new value. During charging from a DC source V, capacitor voltage is VC=V(1−e−t/RC) and current starts at V/R then decays. During discharge from initial voltage V0, VC=V0e−t/RC.

After one time constant, a charging capacitor has reached about 63.2% of its final voltage, while a discharging capacitor retains about 36.8% of its initial voltage. Five time constants is a common practical approximation for being very close to the final state.

τ=RC,   charge: VC=V(1−e−t/τ),   discharge: VC=V0e−t/τ
SymbolMeaningWhy it appears / units
RResistanceΩ; limits current.
CCapacitanceF; stores charge and electric-field energy.
τTime constants when R is in ohms and C in farads.
tElapsed times or any unit consistent with τ.

The capacitor voltage cannot jump instantaneously in an ideal finite-current RC circuit because that would require infinite current. Current can change abruptly when switching occurs, while capacitor voltage remains continuous.

RC transients should approach their final values smoothly. At one time constant a charging capacitor reaches about 63.2% of its final voltage, while a discharging capacitor retains about 36.8% of its initial voltage. Those two landmark values are fast checks for a reversed exponential or an incorrect sign.

Worked Examples

Example 1: Charge: V₀=12V, R=10kΩ, C=100μF, t=0.5s
τ=1s, V=12(1-e^(-0.5))
Result: V=4.71V at t=0.5s (39.3% of V₀)
At t=τ=1s: V=7.58V (63.2%)
Example 2: Discharge from 12V, target 1V, same RC
t=-1s×ln(1/12)=-ln(0.083)
Result: t=2.485s = 2.49 time constants
Time to discharge to 8.3% of initial
Example 3: One time constant charging
t=RC
Result: VC=0.632Vsource
The 63.2% value is a useful benchmark for checking RC calculations.
Example 4: Five time constants discharging
t=5RC
Result: VC≈0.00674V0
Only about 0.67% of the initial voltage remains after five time constants.

Common Mistakes

⚠️
Using milliseconds with seconds without conversion

R and C determine τ in seconds when SI units are used. Match the unit of t to τ.

⚠️
Using the charging formula for discharge

Charging approaches a nonzero source value; discharge decays from an initial value toward zero in the basic circuit.

⚠️
Assuming the capacitor is fully charged at one time constant

One τ reaches only 63.2% of the final voltage, not 100%.

Frequently Asked Questions

Why 5τ for 'fully charged'?
At 5τ: V=V₀(1-e⁻⁵)=V₀×0.993 — 99.3% charged. Engineers use 5τ as practical 'full' threshold. At 3τ: 95%. At 7τ: 99.9%.
Applications?
Camera flash: charges slowly (RC large), discharges instantly. Timer circuits: voltage threshold triggers action after one RC period. Filters: RC sets cutoff frequency f_c=1/(2πRC). Signal smoothing: large C averages voltage fluctuations.
Why is the time constant RC?
Resistance controls how quickly charge can flow, while capacitance controls how much charge is needed to change voltage. Their product has units of seconds and emerges from the circuit differential equation.
What happens to current during charging?
It begins at its maximum V/R for an initially uncharged ideal capacitor and decays exponentially toward zero as capacitor voltage approaches the source voltage.
Why is five time constants often called fully charged?
The mathematical exponential never reaches the final value exactly, but after 5τ a charging capacitor is about 99.3% of the way there, sufficient for many practical estimates.
Can capacitor voltage change instantly?
An ideal capacitor voltage cannot jump without infinite current. Real circuits also have parasitic resistance and inductance, but voltage continuity remains a key circuit-analysis principle.
Why is five time constants used as a practical charging benchmark?
During charging, the remaining gap to the final voltage decays as e−t/RC. At t=5τ, only e−5≈0.0067 of the original gap remains, so the capacitor is about 99.3% charged. It is a practical approximation, not a moment when charging mathematically stops.

Formula Explorer connections

Interpretation: This formula links charge, voltage, current, resistance, capacitance, power or circuit time response. Assumption: Confirm DC versus AC conditions, RMS versus peak values, component topology and steady-state versus transient behavior. Ideal components may be assumed.

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