RC circuits — charging, discharging, time constant τ = RC

1. Setting up from first principles

WHY a differential equation? Because the unknown ($Q$) and its rate of change ($I = dQ/dt$) appear together. We use Kirchhoff's voltage law (KVL): around any loop, the voltages sum to zero.

2. Charging — deriving the law from scratch

A battery (emf $\mathcal{E}$), resistor $R$, and uncharged capacitor $C$ in series. Switch closes at $t=0$.

KVL around the loop (battery rise minus drops = 0): $\mathcal{E} - IR - \frac{Q}{C} = 0$

Why this step? The battery pushes ($+\mathcal{E}$); the resistor drops $IR$; the capacitor holds back $Q/C$. Everything must balance.

Substitute $I = \dfrac{dQ}{dt}$: $\mathcal{E} = R\frac{dQ}{dt} + \frac{Q}{C}$

Rearrange to isolate the rate: $\frac{dQ}{dt} = \frac{\mathcal{E}}{R} - \frac{Q}{RC} = \frac{1}{RC}\big(\mathcal{E}C - Q\big)$

Why this step? Writing it as $\frac{1}{RC}(Q_{\max} - Q)$ shows the rate is proportional to how far we still have to go. Define $Q_{\max} = \mathcal{E}C$ (final charge, when $V_C = \mathcal{E}$ and current stops).

Separate variables and integrate: $\int_0^Q \frac{dQ'}{Q_{\max} - Q'} = \int_0^t \frac{dt'}{RC}$ $-\ln(Q_{\max}-Q')\Big|_0^Q = \frac{t}{RC}$ $-\ln\!\left(\frac{Q_{\max}-Q}{Q_{\max}}\right) = \frac{t}{RC} \implies \frac{Q_{\max}-Q}{Q_{\max}} = e^{-t/RC}$

3. Discharging — deriving from scratch

Now the battery is removed; the charged capacitor ($Q_0$) drives current through $R$. KVL (no source): $\frac{Q}{C} + IR = 0 \implies R\frac{dQ}{dt} = -\frac{Q}{C}$

Why the minus sign? The capacitor is losing charge, so $dQ/dt < 0$.

$\frac{dQ}{Q} = -\frac{dt}{RC} \implies \ln\frac{Q}{Q_0} = -\frac{t}{RC}$

4. What τ actually means

5. Worked examples

6. Steel-manned mistakes

7. The 80/20 core

8. Active-recall flashcards

9. Connections

• Capacitors and Capacitance — source of $V_C = Q/C$.
• Ohm's Law and Resistance — gives $V_R = IR$.
• Kirchhoff's Voltage Law — the loop equation we started from.
• Exponential Decay and Differential Equations — same math as radioactive decay, Newton's Law of Cooling.
• Energy Stored in a Capacitor — half the battery energy is dissipated in $R$ during charging.
• LR Circuits — the inductive analogue with $\tau = L/R$.

Concept Map

Hinglish (regional understanding)