Capacitor Voltage Formula: Charge and RC Circuits

The capacitor voltage formula is V = Q/C: voltage equals stored charge divided by capacitance. For a capacitor charging or discharging through a resistor, voltage also changes with time according to an exponential RC formula; enter charge, capacitance, and elapsed time to calculate the relevant voltage.

Formula

The calculator uses the voltage of an initially uncharged capacitor as it charges from a constant DC source:

V_C(t) = (Q/C)[1 − exp(−t/(RC))]

• V_C is the capacitor voltage at elapsed time t, in volts (V).
• Q is the charge supplied by the source, in coulombs (C).
• C is capacitance, in farads (F).
• R is series resistance, in ohms (Ω).
• t is elapsed time, in seconds (s).
• exp(x) means e raised to the power x; e is approximately 2.71828.

The second output is the time constant, τ = RC, in seconds. The formula assumes the source voltage is constant, the resistor and capacitor are ideal, and the capacitor begins with zero voltage. Here, Q/C is the final voltage implied by the chosen charge and capacitance. For a known source voltage V_S, the usual charging formula is V_C(t) = V_S[1 − exp(−t/RC)].

Worked Example

Using the calculator’s defaults, Q = 0.00001 C, C = 0.000001 F, t = 0.001 s, and R = 1000 Ω:

1. Calculate the final voltage implied by charge and capacitance: Q/C = 0.00001/0.000001 = 10 V.
2. Calculate the time constant: τ = RC = 1000 × 0.000001 = 0.001 s.
3. Substitute the elapsed time: V_C = 10[1 − exp(−0.001/0.001)] = 10(1 − exp(−1)).
4. Since exp(−1) is approximately 0.367879, V_C ≈ 6.321 V. Rounded to three decimal places, the calculator gives 6.321 V.

The time constant output is 0.001 s. After one time constant, an initially uncharged capacitor reaches about 63.2% of its final voltage.

What the Basic Formula Means

The fundamental relationship is Q = CV, rearranged as V = Q/C. A capacitor stores separated electric charge, and its capacitance describes how much charge corresponds to a given voltage. For example, if a 2 μF capacitor holds 10 μC, then V = 10 μC / 2 μF = 5 V. The micro-units cancel because both quantities use the same prefix.

Use this direct formula when the stored charge and capacitance are known. It does not describe how quickly voltage changes. To find voltage during a circuit’s charging or discharging process, time and the circuit arrangement matter as well.

Charging and Discharging Over Time

In a simple series RC circuit, the resistor limits current as the capacitor charges. As capacitor voltage rises, the voltage difference driving current falls, so charging slows. With a constant source V_S and initial voltage zero, the voltage approaches V_S exponentially rather than reaching it instantly.

For a capacitor discharging through a resistor from initial voltage V₀, use V_C(t) = V₀ exp(−t/RC). This result falls toward zero. If the capacitor starts at a nonzero voltage and charges toward a different source voltage, the general ideal RC expression is V_C(t) = V_S + (V₀ − V_S) exp(−t/RC).

At one time constant, charging reaches about 63.2% of the voltage change toward its final value; discharging retains about 36.8% of its starting voltage. At five time constants, the remaining difference from the final value is about 0.67%. These are practical approximations, not exact completion points: an ideal exponential approaches its endpoint asymptotically.

Units and Practical Use

Use consistent SI units in the equations: charge in coulombs, capacitance in farads, resistance in ohms, time in seconds, and voltage in volts. Common component values are often printed with prefixes, so convert them before calculating:

• 1 μF = 0.000001 F; 1 nF = 0.000000001 F.
• 1 kΩ = 1000 Ω; 1 MΩ = 1,000,000 Ω.
• 1 μC = 0.000001 C; 1 ms = 0.001 s.

For instance, 4.7 kΩ and 100 μF give τ = 4700 × 0.0001 = 0.47 s. The time constant has units of seconds because ohms multiplied by farads reduce dimensionally to seconds. SI is the standard system for this calculation; converting a measured time from milliseconds or a capacitance from microfarads is usually more reliable than mixing units in the formula.

The relationship is useful for understanding delay circuits, signal coupling, filtering, and the rise or decay of voltage in a simple circuit. It predicts voltage, not whether a particular component or circuit is suitable for an application. Real capacitors have tolerances and nonideal behavior, and actual results can differ from the ideal model.

Common Mistakes and Limits

• Confusing charge with current: Q is measured in coulombs, while current I is measured in amperes. If current is constant, charge transferred is Q = It; for changing current, charge is the integral of current over time.
• Forgetting the initial condition: The calculator assumes an initially uncharged capacitor. Use the general expression when its starting voltage is not zero.
• Mixing prefixes: Putting 100 for a 100 μF capacitor while treating the value as farads creates a million-fold error. Convert to 0.0001 F.
• Assuming every circuit is a single RC circuit: Multiple components, changing sources, leakage, or nonlinear devices can require a different model.
• Treating “fully charged” as a precise time: The ideal exponential gets progressively closer to its final value but never reaches it at a finite time.

The calculator’s time-dependent result specifically models charging from zero toward Q/C. If the input charge represents something other than the total charge corresponding to the source’s final voltage, that interpretation must be checked before using the result.

Frequently Asked Questions

What is the simplest formula for capacitor voltage?
Use V = Q/C when charge Q and capacitance C are known. Voltage is in volts, charge in coulombs, and capacitance in farads.

How do I calculate capacitor voltage after a time?
For an initially uncharged capacitor charging through a resistor from a constant source, use V_C = V_S[1 − exp(−t/RC)]. For discharge from V₀, use V_C = V₀ exp(−t/RC).

How long does a capacitor take to charge?
It depends on resistance and capacitance. Their product, RC, is the time constant: after one time constant, the capacitor has completed about 63.2% of its voltage change, and after five it is about 99.3% of the way to its final value.

What happens to capacitor voltage when capacitance increases?
At fixed charge, V = Q/C means increasing capacitance lowers voltage. In an RC circuit, increasing capacitance also increases the time constant if resistance stays unchanged, so the voltage changes more slowly.

Can capacitor voltage be negative?
Yes. Voltage polarity depends on the chosen reference direction and how charge is arranged. The equations work with signed voltage and charge when signs are used consistently; the calculator is set up for nonnegative charging inputs.

Is capacitor voltage the same as the source voltage?
Not during the transient in a charging circuit. It starts at the initial capacitor voltage and approaches the source voltage over time. In the ideal steady-state DC model, it approaches the source voltage after sufficient time.

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in q | Charge | C | 0.00001 | 0
in c | Capacitance | F | 0.000001 | 0
in t | Time | s | 0.001 | 0
in r | Resistance | ohm | 1000 | 0
out v | Capacitor voltage | V | q / c * (1 - exp(-t / (r * c))) | 3
out tau | Time constant | s | r * c | 3