DC Circuits

Source lecture(s): SC134 Lec8

Intuition

Kirchhoff's rules enforce charge and energy conservation around loops and junctions.

Formal Definition

Loop rule: \(\sum \mathcal{E} - \sum IR = 0\). Junction rule: \(\sum I_{\text{in}} = \sum I_{\text{out}}\).

Mathematical Formulation

Series: \(R_{\text{eq}} = \sum R_i\) Parallel: \(1/R_{\text{eq}} = \sum 1/R_i\) RC charge: \(q(t) = C\mathcal{E}(1-e^{-t/RC})\) Time constant: \(\tau = RC\)

Derivation

Apply Kirchhoff's loop rule to the RC circuit; solve first-order linear ODE for \(q(t)\).

Worked Example

\(R=2\,\text{k}\Omega\), \(C=1\,\mu\text{F}\): \(\tau=2\,\text{ms}\).

Common Mistakes

Forgetting battery polarity in loop rule.
Setting \(V\) across capacitor to zero immediately after connection.

Quiz

Q1. Does current through series resistors differ?

Answer

No—same current everywhere.