Skip to content

RC Circuits

Source lecture(s): SC134 Lec8, Lec9

Intuition

A resistor and capacitor together define an exponential timescale—fast for small \(RC\), slow for large.

Formal Definition

Charging: \(q(t) = C\mathcal{E}(1-e^{-t/RC})\) Discharging: \(q(t) = Q_0 e^{-t/RC}\)

Mathematical Formulation

Current: \(i(t) = \frac{\mathcal{E}}{R}e^{-t/RC}\) (charging) Voltage across C: \(V_C(t) = \mathcal{E}(1-e^{-t/RC})\)

Derivation

Kirchhoff loop rule \(\mathcal{E} - iR - q/C = 0\); substitute \(i=dq/dt\); solve separable ODE.

Worked Example

\(\tau=RC=2\,\text{ms}\) reaches \(63\%\) charge in 2 ms.

Common Mistakes

  • Using \(\tau=RC\) for RL circuits by accident (valid there too, but check values).
  • Confusing \(i\) and \(q\) exponential forms.

Quiz

Q1. What fraction of final charge does an RC circuit reach at \(t=\tau\)?

Answer

\(1-e^{-1}\approx 63\%\).