Capacitors & RC Circuits

You’ll be able to

Relate the charge, voltage, and capacitance of a capacitor
Describe how a capacitor charges and discharges through a resistor over time
Calculate the time constant of an RC circuit and interpret it

What a capacitor does

A capacitor stores charge and energy in the electric field between two conducting plates. The charge it holds is proportional to the voltage across it: Q = CV, where C is the capacitance in farads (F). A bigger capacitance holds more charge at the same voltage. Unlike a resistor, a capacitor does not dissipate energy — it banks it, then releases it, making capacitors ideal for smoothing voltages, timing, and delivering quick bursts of current (like a camera flash).

Charging and discharging through a resistor

Put a capacitor and resistor in series with a battery — an RC circuit — and the charging is not instant. At the first instant the uncharged capacitor offers no opposition and behaves like a plain wire, so current is maximum. As charge builds, the capacitor’s voltage rises and opposes the battery, so the current tapers off. After a long time the capacitor is fully charged, its voltage equals the battery’s, and current stops — the capacitor now acts like a broken (open) wire. Discharging runs the same story in reverse.

Capacitor charge and RC time constant

Q = C · V · τ = R · C

The time constant τ (tau), in seconds, sets the pace of charging: after one τ the capacitor reaches about 63% of full charge, and after ~5τ it is essentially fully charged. Larger R or C means slower charging.

Worked example

A 2.0 µF capacitor is charged by a 10 V battery through a 1000 Ω resistor. Find the final charge stored and the circuit’s time constant.

1.Final charge (fully charged, capacitor at 10 V): Q = CV = (2.0 × 10⁻⁶ F)(10 V).
2.Multiply: Q = 2.0 × 10⁻⁵ C = 20 µC.
3.Time constant: τ = RC = (1000 Ω)(2.0 × 10⁻⁶ F).
4.Multiply: τ = 2.0 × 10⁻³ s = 2.0 ms.

Answer: The capacitor ends up holding 20 µC, and the time constant is 2.0 ms (so it is essentially fully charged after about 10 ms).

Checkpoint

A 2.0 µF capacitor is fully charged to 10 V. How much charge is stored on it?

Watch out

At the instant an uncharged capacitor is connected, treat it as a wire (zero voltage, maximum current). Once fully charged, treat it as an open switch (no current flows through that branch). Confusing these two limits is the classic RC mistake.

Checkpoint

In an RC circuit, a 1000 Ω resistor is in series with a 2.0 µF capacitor. What is the time constant?

On the exam

For RC problems, answer three questions: what happens at t = 0 (capacitor = wire), what happens as t → ∞ (capacitor = open, fully charged with Q = CV), and how fast (τ = RC). Those three checkpoints capture nearly every exam item.

Answer the 2 checkpoints as you read.