Short on time? This page is the recap. For explanations, Figures 1–3 and the worked examples, use the full study guide.
Recap
- Series capacitors carry the same charge (the isolated inner plates keep a net charge of zero). Their ΔV values add. C_eq is less than the smallest capacitor.
- Parallel capacitors have the same ΔV. Their charges add.
- Charging through R: the current starts at ℰ/R and falls towards zero; Q, ΔV_C and stored energy rise towards their final values.
- Discharging through R: Q, ΔV_C, current and stored energy all fall towards zero. The current in the capacitor’s branch is reversed compared with charging.
- All these changes are fast at first, then slower, and approach steady values asymptotically.
- In this course you describe and sketch these changes and calculate the start and end states. You do not model values at a given time.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| C_eq = C₁ + C₂ + … | parallel; C in F (1 μF = 10⁻⁶ F) | combine parallel capacitors |
| 1/C_eq = 1/C₁ + 1/C₂ + … | series | combine series capacitors |
| τ = RC | Ω × F = s | compare how fast circuits charge or discharge |
| ℰ = IR + Q/C | loop rule at any instant | link current and charge while charging |
| Q = CΔV, U = ½QΔV = ½C(ΔV)² | from Topic 10.6 | final charge and stored energy |
| Moment | Capacitor (uncharged at the start) behaves like | Current in its branch |
|---|---|---|
| Just after the switch closes | a wire (ΔV_C = 0) | largest |
| After a long time (many τ) | a break | zero |
Assumptions behind the numbers
- Batteries, wires and switches are ideal; resistors are ohmic.
- The capacitor starts uncharged when charging is described, unless stated.
- “A long time” means many time constants.
Mistakes to avoid
- Using resistor rules for capacitors. The series and parallel rules are swapped.
- Giving series capacitors different charges. Same Q; the smallest C has the largest ΔV.
- Treating an uncharged capacitor as a break at t = 0. It is a wire at first.
- Saying the capacitor is full after one τ. About 63% only.
- Thinking a bigger emf shortens τ. τ = RC only.
- Assuming the final ΔV_C equals the emf when a resistor shares the loop with a current.
Quick self-check
- Find C_eq for 3.0 μF and 6.0 μF in series, then in parallel. (2.0 μF; 9.0 μF)
- What is τ for 10 kΩ and 100 μF? (1.0 s)
- An uncharged capacitor charges through a resistor from a 12 V battery. Roughly what is ΔV_C after one time constant? (About 7.6 V, 63% of 12 V)
Next: practice questions.