# Capacitors in series and parallel

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u22-capacitors-in-series-and-parallel/

This sub-topic explains how to combine capacitances when multiple capacitors are connected in series or parallel circuits. You will learn to derive combination rules, calculate equivalent capacitance, and solve problems for mixed combinations.

**Prerequisites:** [Definition of capacitance](https://www.owlsprep.com/study/cie-9702-u22-capacitance-and-capacitors/); [Kirchhoff's circuit laws](https://www.owlsprep.com/study/cie-9702-u15-circuit-fundamentals/)

## Learning objectives

- Derive combination rules for total capacitance in series and parallel arrangements
- Calculate equivalent capacitance for any mixed combination of capacitors
- Apply charge and voltage rules to solve exam-style circuit problems
- Avoid common confusion with resistor combination rules

## Capacitors in Parallel

When multiple capacitors are connected in parallel, all capacitors share the same potential difference across their plates, equal to the total voltage of the source connected to the combination. Total charge stored by the combination is the sum of the charge stored on each individual capacitor.

**Equivalent Capacitance (Parallel Combination)** — For $n$ capacitors in parallel, equivalent capacitance equals the sum of all individual capacitances, since $Q_{total} = Q_1 + Q_2 + ... + Q_n$ and $V$ is the same for all capacitors.

*Notation:* C_{eq}

*Example:* $C_{eq} = C_1 + C_2 + C_3 + ... + C_n$

**Worked example:** Three capacitors of $2 \ \mu\text{F}$, $3 \ \mu\text{F}$ and $5 \ \mu\text{F}$ are connected in parallel across a $10 \ \text{V}$ battery. Find the equivalent capacitance and total charge stored.

1. Apply the parallel combination rule to find $C_{eq}$:
2. $$C_{eq} = C_1 + C_2 + C_3 = 2 + 3 + 5 = 10 \ \mu\text{F}$$
3. Use $Q = C V$ to calculate total stored charge:
4. $$Q_{total} = C_{eq} V = 10 \times 10^{-6} \times 10 = 100 \times 10^{-6} = 100 \ \mu\text{C}$$

> **note**
>
> Adding more capacitors in parallel *increases* total equivalent capacitance, which is the opposite behaviour to resistors in parallel.

## Capacitors in Series

When capacitors are connected end-to-end in series, the same magnitude of charge is stored on the plates of each capacitor, due to charge conservation. The total potential difference across the combination is the sum of the potential differences across each individual capacitor.

**Derivation:** Derive the equivalent capacitance rule for series capacitors

*Starting from:* Kirchhoff's Voltage Law: $V_{total} = V_1 + V_2 + ... + V_n$, same charge $Q$ on all capacitors, $V_i = \frac{Q}{C_i}$

1. Substitute $V_i = \frac{Q}{C_i}$ into the voltage equation:
2. $$V_{total} = \frac{Q}{C_1} + \frac{Q}{C_2} + ... + \frac{Q}{C_n} = Q \left( \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n} \right)$$
3. By definition, $V_{total} = \frac{Q}{C_{eq}}$, divide both sides by $Q$:
4. $$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$$

*Conclusion:* The reciprocal of the equivalent capacitance of series capacitors equals the sum of the reciprocals of the individual capacitances.

**Worked example:** The same $2 \ \mu\text{F}$, $3 \ \mu\text{F}$, $5 \ \mu\text{F}$ capacitors are connected in series across a $10 \ \text{V}$ battery. Find equivalent capacitance and charge on each capacitor.

1. Calculate the reciprocal sum for the series combination:
2. $$\frac{1}{C_{eq}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{5} = \frac{15 + 10 + 6}{30} = \frac{31}{30} \ \mu\text{F}^{-1}$$
3. Take the reciprocal to find $C_{eq}$:
4. $$C_{eq} = \frac{30}{31} \approx 0.97 \ \mu\text{F}$$
5. Charge is the same on all series capacitors, so $Q = C_{eq} V$:
6. $$Q = 0.97 \times 10^{-6} \times 10 \approx 9.7 \ \mu\text{C}$$

> **warning**
>
> Adding more capacitors in series *decreases* total equivalent capacitance, opposite to the behaviour of resistors in series.

## Mixed Combinations of Capacitors

Most CIE exam problems involve combinations of capacitors with both series and parallel connections nested within the circuit. To solve these, you work step-by-step, starting with the innermost nested combination and simplifying outward to the total equivalent capacitance.

**Worked example:** A $2 \ \mu\text{F}$ and $3 \ \mu\text{F}$ capacitor are connected in series. This series combination is connected in parallel with a $4 \ \mu\text{F}$ capacitor. Find the total equivalent capacitance of the circuit.

1. First simplify the innermost series combination:
2. $$\frac{1}{C_{series}} = \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \implies C_{series} = \frac{6}{5} = 1.2 \ \mu\text{F}$$
3. Now add this series equivalent to the parallel $4 \ \mu\text{F}$ capacitor:
4. $$C_{eq} = C_{series} + 4 = 1.2 + 4 = 5.2 \ \mu\text{F}$$

**Check your understanding**

Test your understanding of mixed combinations:

1. A $4 \ \mu\text{F}$ and $6 \ \mu\text{F}$ are connected in series, and this combination is connected in parallel with a $5 \ \mu\text{F}$ capacitor. What is the total equivalent capacitance?

   - 3.0 \ \mu\text{F}
   - 7.4 \ \mu\text{F}
   - 15 \ \mu\text{F}
   - 2.4 \ \mu\text{F}

   *Why:* Correct: The series equivalent is $\frac{4 \times 6}{4 + 6} = 2.4 \ \mu\text{F}$, adding the parallel $5 \ \mu\text{F}$ gives $7.4 \ \mu\text{F}$.

## Common pitfalls

- **Wrong:** Stopping at $\frac{1}{C_{eq}}$ and using that value as the equivalent capacitance for series combinations.
  - Why it fails: Students often forget the final reciprocal step after summing reciprocals for series.
  - Correct: Always remember to take the reciprocal of your sum to get $C_{eq}$ for series combinations.
- **Wrong:** Assuming the same potential difference across all capacitors in series.
  - Why it fails: Confuses the properties of series and parallel combinations.
  - Correct: In series, charge is the same across all capacitors; voltage varies inversely with capacitance.
- **Wrong:** Using the same combination rules as resistors (sum for series, reciprocal sum for parallel).
  - Why it fails: Capacitors have opposite combination rules to resistors, leading to common confusion.
  - Correct: Remember: parallel capacitors sum like series resistors, series capacitors sum like parallel resistors.
- **Wrong:** Starting simplifying mixed combinations from the outermost layer first.
  - Why it fails: Reverse order of simplification leads to incorrect addition of series and parallel terms.
  - Correct: Always simplify the innermost nested combination first, then work outward to find total equivalent capacitance.
- **Wrong:** Assuming total charge is the same across all capacitors in parallel.
  - Why it fails: Another common confusion between series and parallel properties.
  - Correct: In parallel, voltage is the same across all capacitors; total charge splits proportional to capacitance.

## Cheatsheet

| Combination Type | Rule for $C_{eq}$ | Key Property |
| --- | --- | --- |
| Parallel | $C_{eq} = C_1 + C_2 + ... + C_n$ | Same voltage across all capacitors, total charge = sum of individual charges |
| Series | $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$ | Same charge on all capacitors, total voltage = sum of individual voltages |
| Mixed | Simplify innermost first, work outwards | Apply the correct rule for each sub-combination step-by-step |

## What's next

Mastering capacitor combination rules is a critical foundation for all further topics in capacitance, including calculating energy stored in capacitor combinations, and solving problems involving charging and discharging of capacitor networks. This sub-topic is frequently combined with Kirchhoff's laws and energy storage to form multi-part exam questions worth 4-8 marks. It also underpins understanding of capacitive reactance in alternating current circuits, a key topic for paper 2 and paper 4 of CIE A-Level Physics.

- [Capacitor charging and discharging](https://www.owlsprep.com/study/cie-9702-u22-capacitor-charging-and-discharging/)
- [Magnetic fields](https://www.owlsprep.com/study/cie-9702-u23-overview/)
- [Magnetic field concepts](https://www.owlsprep.com/study/cie-9702-u23-magnetic-field-concepts/)

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