3.4 Capacitors in Circuits

Charging and steady-state behavior

When a capacitor is connected in a circuit with a resistor and battery (RC circuit), current flows as charge builds on the plates. Initially the uncharged capacitor acts like a wire (short), allowing max current. At steady state, the capacitor is fully charged and acts like an open switch, blocking current — voltage across it equals the EMF (in a simple loop).

Series and parallel capacitors

For capacitors in series, charge Q is the same on each, and 1/C_eq = 1/C₁ + 1/C₂ + ... (equivalent capacitance decreases). In parallel, voltage V is the same across each, and C_eq = C₁ + C₂ + ... (equivalent capacitance increases). These rules are opposite to resistor combination rules.

Energy and RC time behavior

Energy stored: U = ½QV = ½CV² = Q²/2C. In charging/discharging RC circuits, charge and voltage change exponentially with time constant τ = RC, approaching final values asymptotically. Larger R or C means slower charging/discharging, since more resistance limits current and more capacitance requires more charge to reach the same voltage.

Key terms

Uncharged capacitor at t = 0 in a circuit

Acts like a wire (short circuit) — allows maximum current initially.

Fully charged capacitor at steady state

Acts like an open switch — blocks current flow through that branch.

Series capacitors rule

Same charge Q on each; 1/C_eq = 1/C₁ + 1/C₂ + ... (equivalent capacitance decreases).

Parallel capacitors rule

Same voltage V across each; C_eq = C₁ + C₂ + ... (equivalent capacitance increases).

Time constant τ

τ = RC; determines the rate of exponential charging/discharging in an RC circuit.

Try an AP-style question

What is the equivalent capacitance of 3 μF and 6 μF in series?

1. 9 μF
2. 2 μF
3. 4.5 μF
4. 18 μF