Series and parallel resistor combinations

Physics· Unit 10: D.C. Circuits· 20 min read

1. Series Resistor Combinations★★☆☆☆⏱ 5 min

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📘 Definition

Series combination of resistors

Resistors connected in a single end-to-end path, so the same current flows through every resistor, and total potential difference equals the sum of potential differences across individual resistors.

Example:

Three 2 Ω resistors connected in a line between a battery's positive and negative terminals

🔬 Derivation

Goal:

Derive equivalent resistance for n resistors in series

Starting from:

Kirchhoff's Voltage Law and Ohm's Law

1
For resistors $R_{1}, R_{2}, ... R_{n}$ in series, current $I$ is the same through all. By KVL:
2
$V = V_1 + V_2 + ... + V_n$
3
Substitute Ohm's law $V = I R$ for each term:
4
$I R_{eq} = I R_1 + I R_2 + ... + I R_n$
5
Divide both sides by common current $I$ :
6
$R_{eq} = R_1 + R_2 + ... + R_n$

Result:

Total equivalent resistance of series resistors is the sum of all individual resistances.

📐 Worked Example

Three resistors of 2 Ω, 3 Ω and 5 Ω are connected in series to a 10 V battery. Calculate total equivalent resistance and the current drawn from the battery.

1
Use the series resistance rule to find $R_{e q}$ :
2
$R_{eq} = 2 + 3 + 5 = 10 \ \Omega$
3
Apply Ohm's law to find total current $I$ :
4
$I = \frac{V}{R_{eq}} = \frac{10}{10} = 1.0 \ A$

Exam tip:

Use the voltage divider rule $V_{i} = V_{t o t a l} \times \frac{R _{i}}{R _{e q}}$ to quickly find voltage across any individual series resistor, this saves time in multiple choice questions.

2. Parallel Resistor Combinations★★★☆☆⏱ 7 min

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📘 Definition

Parallel combination of resistors

Resistors connected across the same two nodes, so the same potential difference acts across every resistor, and total current equals the sum of currents through individual resistors.

🔬 Derivation

Goal:

Derive equivalent resistance for n resistors in parallel

Starting from:

Kirchhoff's Current Law and Ohm's Law

1
For resistors $R_{1}, R_{2}, ... R_{n}$ in parallel, potential difference $V$ is the same across all. By KCL:
2
$I = I_1 + I_2 + ... + I_n$
3
Substitute Ohm's law $I = V / R$ :
4
$\frac{V}{R_{eq}} = \frac{V}{R_1} + \frac{V}{R_2} + ... + \frac{V}{R_n}$
5
Divide by common potential difference $V$ :
6
$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$

Result:

The reciprocal of equivalent resistance equals the sum of reciprocals of individual resistances. For two resistors, this simplifies to $R_{e q} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}$ .

📐 Worked Example

A 4 Ω and a 6 Ω resistor are connected in parallel across a 12 V supply. Calculate equivalent resistance and total current from the supply.

1
Apply the reciprocal rule for parallel resistance:
2
$\frac{1}{R_{eq}} = \frac{1}{4} + \frac{1}{6} = \frac{3 + 2}{12} = \frac{5}{12}$
3
Take the reciprocal to find $R_{e q}$ :
4
$R_{eq} = \frac{12}{5} = 2.4 \ \Omega$
5
Calculate total current using Ohm's law:
6
$I = \frac{V}{R_{eq}} = \frac{12}{2.4} = 5.0 \ A$

3. Mixed Resistor Networks★★★★☆⏱ 8 min

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Most CIE exam problems have combinations of series and parallel resistors in the same network. To solve these, you simplify step-by-step, starting from the innermost combination and working outwards.

Identify the innermost purely series or parallel combination that contains no other nested combinations
Calculate the equivalent resistance for this innermost combination
Replace the original combination with its equivalent resistance on the circuit diagram
Repeat the process until you get a single equivalent resistance for the entire network

📐 Worked Example

Calculate the total equivalent resistance of this network: a 2 Ω resistor is in series with a parallel combination of 3 Ω and 6 Ω resistors.

1
First simplify the innermost parallel combination of 3 Ω and 6 Ω:
2
$\frac{1}{R_{parallel}} = \frac{1}{3} + \frac{1}{6} = \frac{2 +1}{6} = \frac{1}{2}$
3
The equivalent resistance of the parallel section is $R_{p a r a l l e l} = 2 Ω$ .
4
This 2 Ω equivalent is in series with the 2 Ω resistor, so add the resistances:
5
$R_{total} = 2 + 2 = 4 \ \Omega$

✓ Quick check

Check your understanding of simplification order

A network has a 10 Ω resistor in series with a parallel branch. The parallel branch contains a 2 Ω resistor in series with two parallel 4 Ω resistors. What do you calculate first?
- The two 4 Ω resistors in parallel
- The 2 Ω resistor and the parallel equivalent in series
- The whole parallel branch with the 10 Ω resistor

Exam tip:

Always redraw the circuit after each simplification step to avoid mixing up series and parallel connections. CIE examiners award method marks for correct working even if your final answer is wrong.

4. Common Pitfalls

Wrong move:

Adding resistances directly for parallel combinations instead of summing reciprocals

Why:

Confusing series and parallel rules leads to an equivalent resistance that is far too large

Correct move:

Remember the core rule: series = sum of resistances, parallel = sum of reciprocals

Wrong move:

Stopping after summing reciprocals for parallel resistance and forgetting to take the final reciprocal

Why:

This leaves you with a resistance value that is far too small, costing easy exam marks

Correct move:

Always double-check: after calculating $\sum 1/ R_{i}$ , take the reciprocal to get $R_{e q}$

Wrong move:

Simplifying outer combinations before innermost nested combinations

Why:

This leads to incorrect grouping of series and parallel resistors, producing wrong results

Correct move:

Always work from the inside out: simplify the deepest nested group first, then move outwards

Wrong move:

Claiming total voltage is the sum of voltages across parallel resistors

Why:

Confusing voltage and current rules for parallel combinations

Correct move:

Voltage is equal across all parallel resistors; total current is the sum of individual currents

Wrong move:

Assuming any two resistors with equal current are always in series

Why:

Equal current can occur in non-series resistors in balanced networks, this is not the definition of series

Correct move:

A series combination is defined by sharing a single current path with no branches between the resistors

5. Quick Reference Cheatsheet

Combination Type	Current Rule	Voltage Rule	Equivalent Resistance
Series	Same current through all	$V_{t o t a l} = \sum V_{i}$	$R_{e q} = \sum R_{i}$
Parallel	$I_{t o t a l} = \sum I_{i}$	Same voltage across all	$\frac{1}{R _{e q}} = \sum \frac{1}{R _{i}}$
Two resistors parallel	$I_{1} / I_{2} = R_{2} / R_{1}$	Same voltage across both	$R_{e q} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}$
n equal resistors R parallel	Total current = n × I per resistor	Same voltage	$R_{e q} = R / n$

6. Frequently Asked

Why is parallel equivalent resistance always smaller than the smallest resistor?

Adding a resistor in parallel creates an extra path for current, increasing total current for the same applied voltage. By Ohm's law ( $R = V / I$ ), higher total current gives lower total resistance.

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 · 11
Calculate parallel equivalent resistance
2023 · 22
Total resistance for mixed network
2024 · 13
Voltage across series resistor

Going deeper

syllabusOfficial CIE 9702 Physics SyllabusFull syllabus reference for DC circuits

What's Next

Understanding series and parallel resistor combinations is the foundation for all more complex DC circuit topics, including potential dividers, internal resistance of batteries, and Kirchhoff's laws for multi-loop circuits. This subtopic appears in almost every CIE A-Level Physics paper 1 and paper 2, so mastering step-by-step simplification for mixed networks will earn you consistent easy marks. Next, you will apply these combination rules to potential dividers, a common exam topic for sensor and measurement circuit problems, before moving on to more complex multi-loop circuits. Building a solid understanding of combination rules now will make all subsequent DC circuit topics far easier to master.