Discharging RC circuit

Example 3 — Discharging RC circuit

A capacitor of capacitance $C$ charged to voltage $V_0$ discharges through a resistor $R$. Kirchhoff’s law gives: $R\dot Q + \frac{Q}{C} = 0 \quad\Longleftrightarrow\quad \dot Q = -\frac{Q}{RC}.$ It is a separable equation (of the first order!). Solution: $Q(t) = Q_0 e^{-t/\tau}$ with $\tau = RC$ the time constant. The voltage is $V(t) = Q(t)/C = V_0 e^{-t/\tau}$.

After $\tau$ seconds the capacitor retains about $37\%$ ($1/e$) of the initial charge; after $5\tau$ it is practically discharged (less than $1\%$).

Links

Topics: Differential equations
Concepts: Time constant · Decay · First-order linear ODE · Separable variables
Functions: Exponential function
Methods: RC decay
Skills: Modelling · Solving equations
People: Kirchhoff