4.4 Capacitors in Circuits

Capacitors resist sudden voltage change

A capacitor stores charge, and its voltage cannot change instantaneously. Just after switching, an uncharged capacitor acts like a wire (0V), and a fully charged capacitor acts like an open circuit (no current flows through it). Between these extremes, charge builds up exponentially as current decays.

RC circuit equations

For a charging capacitor: q(t) = Cε(1 − e^(−t/RC)), and current I(t) = (ε/R)e^(−t/RC). For discharging: q(t) = Q₀e^(−t/RC), I(t) = -(Q₀/RC)e^(−t/RC). The time constant τ = RC sets how fast charge/current change — after one τ, charge is ~63% of max (charging) or ~37% remaining (discharging).

Energy and combinations

Energy stored: U = ½QV = ½CV² = Q²/2C. Capacitors in series share the same charge Q, with 1/C_eq = Σ(1/Cᵢ); in parallel they share the same voltage V, with C_eq = ΣCᵢ. In circuit analysis, apply Kirchhoff's rules combined with these charge/voltage relationships, using derivatives (dq/dt = I) to connect current and charge.

Key terms

Capacitor at t = 0 (uncharged, switch just closed)

Acts like a wire — zero voltage drop, allows current to flow freely.

Capacitor at t → ∞ (fully charged)

Acts like an open circuit — no current flows, voltage across it equals EMF (in simple loops).

Time constant τ

τ = RC; the time for charge/current to change by a factor of e (~63% charged or ~37% remaining).

Energy stored in a capacitor

U = ½QV = ½CV² = Q²/2C

Series vs. parallel capacitors

Series: same Q, 1/C_eq = Σ(1/Cᵢ). Parallel: same V, C_eq = ΣCᵢ.

Try an AP-style question

In steady state, how much current flows through a branch containing a capacitor?

1. None
2. The full circuit current
3. It oscillates
4. Half the circuit current