Result

Discharging

q (t) = Q e^{- t / R C}

Statement

Charge and current both decay exponentially with the same time constant.

Symbols and units

$q (t)$	charge left at time t	C
$Q$	charge at t = 0	C
$R C$	time constant τ	s

When it applies

Series R and C with no source, starting from charge Q.

Don't use when

Assumes a series R and C with no source, starting from a known initial charge — an actively driven circuit needs a different differential equation.

Derivation

With the source removed, the loop rule has only two terms:

\frac{q}{C} + R \frac{d q}{d t} = 0

Separate variables:

\frac{d q}{q} = - \frac{d t}{R C}

Integrate from the initial charge Q:

ln \frac{q}{Q} = - \frac{t}{R C}

Exponentiate:

q (t) = Q e^{- t / R C}

The current is its derivative — negative, since charge is leaving:

I (t) = - \frac{Q}{R C} e^{- t / R C}

Worked examples

Bleeder resistor

470 µF charged to 325 V, drained through 100 kΩ:

τ = 47 s

Time to fall to 50 V:

t = τ ln \frac{325}{50} = 88 s

Service instructions specify minutes, not seconds, before opening a power supply.

Watch out for

Half the charge is gone after τ ln 2 ≈ 0.69RC. A charged capacitor left in equipment can hold a dangerous voltage for a long time if R is large — bleeder resistors exist for that reason.

Connects to

Charging Stored energy