Capacitor RC Circuit Formula and Time Constant

The capacitor RC circuit formula describes how voltage and current change over time when a resistor charges or discharges a capacitor. Enter resistance, capacitance, supply voltage, and elapsed time to calculate the capacitor voltage during charging; the result is in volts.

Formula

For a capacitor initially uncharged and connected to a constant voltage source through a resistor, its charging voltage is V_C(t) = V_S(1 − e^−t/(RC)). In the calculator, V_C is the capacitor voltage in volts, V_S is the supply voltage in volts, t is elapsed time in seconds, R is resistance in ohms, and C is capacitance in farads. The constant e is the base of natural logarithms.

The product τ = RC is the time constant, measured in seconds. The calculator also gives the charging current, I(t) = (V_S/R)e^−t/(RC), in amperes. These expressions assume an ideal resistor and capacitor, a fixed supply, and a capacitor starting at zero volts.

Worked Example

Using the default values, the circuit has R = 10,000 Ω, C = 0.0001 F, V_S = 5 V, and t = 1 s.

1. Calculate the time constant: τ = RC = 10,000 × 0.0001 = 1 s.
2. Find the charging voltage: V_C = 5 × (1 − e^−1/1) = 5 × (1 − e⁻¹) ≈ 3.1606 V, or 3.161 V rounded to three decimal places.
3. Find the current: I = (5/10,000) × e⁻¹ ≈ 0.00018394 A, or 0.000184 A rounded to six decimal places.

At one time constant, the capacitor has reached about 63.2% of its final voltage. The calculator’s three outputs match these values.

How an RC Circuit Charges

A resistor limits the flow of charge into a capacitor. At the instant an ideal uncharged capacitor is connected to a DC source, its voltage is zero and the current is largest: V_S/R. As charge accumulates, capacitor voltage rises, leaving less voltage across the resistor. Current therefore decreases, and the voltage approaches the supply voltage progressively rather than jumping to it.

The exponential term e^−t/τ captures this changing rate. Charging is mathematically asymptotic: the ideal capacitor approaches the supply voltage but does not reach it at a finite time. In practical circuits, it is often treated as nearly charged after several time constants.

Reading the Time Constant

The time constant gives a useful scale for the circuit’s response, not an exact “fully charged” time. For an initially uncharged capacitor charging toward a fixed supply:

• At 1τ, voltage is about 63.2% of its final value.
• At 2τ, it is about 86.5%.
• At 3τ, it is about 95.0%.
• At 5τ, it is about 99.3%.

For example, a 10 kΩ resistor and a 100 μF capacitor give τ = 1 second, because 100 μF equals 0.0001 F. A larger resistance or capacitance increases the time constant and makes the response slower. A smaller value makes it faster.

Discharging and Other Initial Voltages

The same RC time constant applies when a charged capacitor discharges through a resistor. If its initial voltage is V₀ and there is no continuing source, then V_C(t) = V₀e^−t/(RC). After one time constant, about 36.8% of the starting voltage remains; after five, about 0.7% remains.

If a capacitor starts at a nonzero voltage and is connected through a resistor to a constant final voltage V_F, the more general expression is V_C(t) = V_F + (V₀ − V_F)e^−t/(RC). The calculator uses the special charging case V₀ = 0 and V_F = V_S; use the general expression when the initial condition differs.

Units, Assumptions, and Common Mistakes

Use consistent SI units in the formula: resistance in ohms, capacitance in farads, time in seconds, voltage in volts, and current in amperes. Common capacitance conversions are 1 μF = 0.000001 F and 1 nF = 0.000000001 F. Resistance may also be stated in kilohms: 1 kΩ = 1,000 Ω. The SI formula itself does not require separate US customary units; electrical component values are ordinarily given in these units.

• Do not enter microfarads as though they were farads. Convert the value first; confusing μF with F changes the predicted time by a factor of one million.
• Do not mistake the time constant for the time to reach exactly the supply voltage. It marks the 63.2% point for charging from zero.
• Check the initial condition and circuit topology. A changing supply, additional resistors or capacitors, or a capacitor that does not start discharged can require a different equation.
• Real capacitors have tolerances, leakage, and other non-ideal behavior, and a source may have internal resistance. Those effects can make measured results differ from the ideal calculation.

For an ideal series RC circuit, the current formula above follows from the voltage across the resistor divided by R. At the moment charging begins, the capacitor behaves like a short circuit in this idealized model; after a long time on DC, it behaves like an open circuit. These are limiting descriptions, not claims about every real circuit or transient.

Frequently Asked Questions

What is the formula for a capacitor charging through a resistor?
For an initially uncharged capacitor connected to a constant supply, V_C(t) = V_S(1 − e^−t/(RC)), where τ = RC.

How do I calculate the RC time constant?
Multiply resistance in ohms by capacitance in farads: τ = RC. The result is in seconds.

How long does it take a capacitor to charge?
There is no finite time at which the ideal exponential reaches exactly its final voltage. It reaches about 95% after 3τ and about 99.3% after 5τ.

What is the capacitor voltage after one time constant?
Starting from zero and charging toward a constant supply, it is about 63.2% of the supply voltage after one τ.

What is the capacitor discharge formula?
For discharge through a resistor from initial voltage V₀, V_C(t) = V₀e^−t/(RC).

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in r | Resistance | ohm | 10000 | 0 | 1000000000000
in c | Capacitance | F | 0.0001 | 0
in v | Supply voltage | V | 5 | 0
in t | Elapsed time | s | 1 | 0
out vc | Capacitor voltage while charging | V | v * (1 - exp(-t / (r * c))) | 3
out tau | Time constant | s | r * c | 3
out i | Charging current at elapsed time | A | (v / r) * exp(-t / (r * c)) | 6