RC Circuits
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.
A capacitor in a circuit changes over time. Just after the switch closes, an uncharged capacitor acts like a plain wire. After a long time, it’s fully charged and acts like a break in the wire: no current flows through it. The time constant τ = RC sets how fast it gets there.
1. Charging over time
2. Two snapshots solve most problems
At t = 0: replace each uncharged capacitor with a wire and solve the circuit normally.
After a long time: remove each capacitor’s branch (no current), solve the rest, then find the voltage across where the capacitor sits. Q = CΔV.
A bigger R slows the current; a bigger C needs more charge. Either one makes τ longer.
Discharging works the same way in reverse: the charge and current both decay, each falling to 37% after one τ.
3. Example, then try it
A 12 V battery, a 4.0 Ω resistor and an uncharged capacitor are in series. Find the current just after the switch closes and after a long time.
At t = 0 the capacitor is like a wire: I = 12 ÷ 4.0 = 3.0 A.
After a long time: I = 0, and the capacitor has the full 12 V.
A 12 V battery is connected in series with a 2.0 Ω resistor and then a parallel pair: a 4.0 Ω resistor and an uncharged 50 µF capacitor. Find the battery current just after the switch closes and after a long time, and the final charge on the capacitor.
t = 0: the capacitor acts like a wire and bypasses the 4.0 Ω resistor. I = 12 ÷ 2.0 = 6.0 A, all of it into the capacitor.
Long time: no current to the capacitor, so the 2.0 Ω and 4.0 Ω are in series: I = 12 ÷ 6.0 = 2.0 A.
Capacitor voltage: the same as the 4.0 Ω resistor beside it: (2.0)(4.0) = 8.0 V.
Charge: Q = CΔV = (50 × 10⁻⁶)(8.0) = 4.0 × 10⁻⁴ C (400 µC).
Answer: 6.0 A at first, 2.0 A later; 400 µC on the capacitor.
4. Common mistakes
A 2000 Ω resistor is in series with a 500 µF capacitor. What is the time constant?
τ = RC = (2000)(500 × 10⁻⁶) = 1.0 s.
Ready to practice?
4 questions on RC Circuits, each with a hint and an explanation for every wrong answer.