Capacitors in series and parallel

CIE A-Level Physics· 15 min read

1. Capacitors in Parallel★★☆☆☆⏱ 15 min

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.

📘 Definition

Equivalent Capacitance (Parallel Combination)

C_{eq}

For $n$ capacitors in parallel, equivalent capacitance equals the sum of all individual capacitances, since $Q_{t o t a l} = Q_{1} + Q_{2} + ... + Q_{n}$ and $V$ is the same for all capacitors.

Example:

$C_{e q} = C_{1} + C_{2} + C_{3} + ... + C_{n}$

📐 Worked Example

Three capacitors of $2 μ F$ , $3 μ F$ and $5 μ F$ are connected in parallel across a $10 V$ battery. Find the equivalent capacitance and total charge stored.

1
Apply the parallel combination rule to find $C_{e q}$ :
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
$Q_{total} = C_{eq} V = 10 \times 10^{-6} \times 10 = 100 \times 10^{-6} = 100 \ \mu\text{C}$

2. Capacitors in Series★★★☆☆⏱ 20 min

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

Goal:

Derive the equivalent capacitance rule for series capacitors

Starting from:

Kirchhoff's Voltage Law: $V_{t o t a l} = V_{1} + V_{2} + ... + V_{n}$ , 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, $V_{t o t a l} = \frac{Q}{C _{e q}}$ , divide both sides by $Q$ :
4
$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$

Result:

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

📐 Worked Example

The same $2 μ F$ , $3 μ F$ , $5 μ F$ capacitors are connected in series across a $10 V$ 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 $C_{e q}$ :
4
$C_{eq} = \frac{30}{31} \approx 0.97 \ \mu\text{F}$
5
Charge is the same on all series capacitors, so $Q = C_{e q} V$ :
6
$Q = 0.97 \times 10^{-6} \times 10 \approx 9.7 \ \mu\text{C}$

3. Mixed Combinations of Capacitors★★★★☆⏱ 25 min

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 $2 μ F$ and $3 μ F$ capacitor are connected in series. This series combination is connected in parallel with a $4 μ F$ 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 $4 μ F$ capacitor:
4
$C_{eq} = C_{series} + 4 = 1.2 + 4 = 5.2 \ \mu\text{F}$

✓ Quick check

Test your understanding of mixed combinations:

A $4 μ F$ and $6 μ F$ are connected in series, and this combination is connected in parallel with a $5 μ F$ 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}
Reveal answer
7.4 \ \mu\text{F} —
Correct: The series equivalent is $\frac{4 \times 6}{4 + 6} = 2.4 μ F$ , adding the parallel $5 μ F$ gives $7.4 μ F$ .

4. Common Pitfalls

Wrong move:

Stopping at $\frac{1}{C _{e q}}$ and using that value as the equivalent capacitance for series combinations.

Why:

Students often forget the final reciprocal step after summing reciprocals for series.

Correct move:

Always remember to take the reciprocal of your sum to get $C_{e q}$ for series combinations.

Wrong move:

Assuming the same potential difference across all capacitors in series.

Why:

Confuses the properties of series and parallel combinations.

Correct move:

In series, charge is the same across all capacitors; voltage varies inversely with capacitance.

Wrong move:

Using the same combination rules as resistors (sum for series, reciprocal sum for parallel).

Why:

Capacitors have opposite combination rules to resistors, leading to common confusion.

Correct move:

Remember: parallel capacitors sum like series resistors, series capacitors sum like parallel resistors.

Wrong move:

Starting simplifying mixed combinations from the outermost layer first.

Why:

Reverse order of simplification leads to incorrect addition of series and parallel terms.

Correct move:

Always simplify the innermost nested combination first, then work outward to find total equivalent capacitance.

Wrong move:

Assuming total charge is the same across all capacitors in parallel.

Why:

Another common confusion between series and parallel properties.

Correct move:

In parallel, voltage is the same across all capacitors; total charge splits proportional to capacitance.

5. Quick Reference Cheatsheet

Combination Type	Rule for $C_{e q}$	Key Property
Parallel	$C_{e q} = C_{1} + C_{2} + ... + C_{n}$	Same voltage across all capacitors, total charge = sum of individual charges
Series	$\frac{1}{C _{e q}} = \frac{1}{C _{1}} + \frac{1}{C _{2}} + ... + \frac{1}{C _{n}}$	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

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Equivalent capacitance of mixed combination
2023 · 2
Charge and voltage for series capacitors
2021 · 1
Comparison of series/parallel energy storage

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.