Discharge equation and exponential solution

The discharge circuit

We start with a capacitor charged to a voltage U0 (for example, U0 = E after a full charge), which is connected to a resistor R on its own, without a power source. The capacitor will discharge through the resistor.

Formulating the equation

Applying the loop rule, with i directed in the direction of discharge, we obtain: 0 = Ri(t) + uc(t), where i(t) = -C(duc/dt) (the capacitor discharges, its charge decreases)

We obtain: RC(duc/dt) + uc(t) = 0, or tau*(duc/dt) + uc = 0

This is the same homogeneous equation as for the charging process, but without the right-hand side (E=0 here).

Solution

With the initial condition uc(0) = U0, the solution is: uc(t) = U0*exp(-t/tau)

and the current (using the convention above) is: i(t) = (U₀/R)*exp(-t/τ), which decays to 0 just like uc.

Comparison: charging/discharging

Quantity	Charging	Discharging
uc(t)	E*(1-exp(-t/tau))	U0*exp(-t/tau)
Behaviour	increasing, saturates at E	decreasing, tends towards 0
tau	R*C	R*C (identical)

Common pitfall

The time constant τ = R*C is the SAME for both charging and discharging in the same RC circuit: it depends only on the components, not on the initial condition or the direction of the process. Do not assume that discharging is faster or slower than the corresponding charging if R and C are identical.