Cambridge Syllabus Notes

4.3.2 Series and Parallel Circuits

Objective: Know how to construct and use series and parallel circuits. 🚀

Series Circuits – The “One‑Way Street” Analogy

Imagine a single lane road where all cars must travel one after another. In a series circuit, the current flows through each component one after the other – just like cars on a single lane. The total resistance is the sum of all individual resistances: $$R_{\text{total}} = R_1 + R_2 + R_3 + \dots$$ The voltage supplied by the battery is divided across each component: $$V_{\text{total}} = V_1 + V_2 + V_3 + \dots$$ The current is the same through every component.

Parallel Circuits – The “Multi‑Lane Highway” Analogy

Picture a highway with several lanes, each lane carrying its own cars. In a parallel circuit, the current splits into different paths, each path containing a component. The voltage across each component is the same: $$V_{\text{total}} = V_1 = V_2 = V_3 = \dots$$ The total resistance is found using: $$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$$ The total current is the sum of the currents in each branch.

Key Differences at a Glance

Current: same in series, splits in parallel.
Voltage: splits in series, same across each in parallel.
Resistance: additive in series, decreases in parallel.
Failure: one component fails → whole series circuit stops; parallel circuit still works.

Calculating with Series Circuits

List all resistances.
Sum them to get $R_{\text{total}}$.
Use Ohm’s law $I = \frac{V}{R_{\text{total}}}$ to find current.
Find voltage drop across each resistor: $V_i = I \times R_i$.

Calculating with Parallel Circuits

List all resistances.
Compute reciprocal sum: $\frac{1}{R_{\text{total}}} = \sum \frac{1}{R_i}$.
Find $R_{\text{total}}$ by taking reciprocal.
Use Ohm’s law $I = \frac{V}{R_{\text{total}}}$ for total current.
Find current in each branch: $I_i = \frac{V}{R_i}$.

Example 1 – Series Circuit

Three resistors: $R_1 = 4\,\Omega$, $R_2 = 6\,\Omega$, $R_3 = 10\,\Omega$. Battery voltage $V = 12\,\text{V}$.

$R_{\text{total}} = 4 + 6 + 10 = 20\,\Omega$
Current: $I = \frac{12}{20} = 0.60\,\text{A}$
Voltage drops: $V_1 = 0.60 \times 4 = 2.4\,\text{V}$ $V_2 = 0.60 \times 6 = 3.6\,\text{V}$ $V_3 = 0.60 \times 10 = 6.0\,\text{V}$

Example 2 – Parallel Circuit

Two resistors: $R_1 = 8\,\Omega$, $R_2 = 12\,\Omega$. Battery voltage $V = 9\,\text{V}$.

$\frac{1}{R_{\text{total}}} = \frac{1}{8} + \frac{1}{12} = 0.125 + 0.0833 = 0.2083$
$R_{\text{total}} = \frac{1}{0.2083} \approx 4.80\,\Omega$
Total current: $I = \frac{9}{4.80} \approx 1.88\,\text{A}$
Branch currents: $I_1 = \frac{9}{8} = 1.125\,\text{A}$ $I_2 = \frac{9}{12} = 0.75\,\text{A}$

Exam Tips 📚

Always check whether the circuit is series or parallel before calculating.
For series, remember “add resistances”. For parallel, remember “add reciprocals”.
Use Ohm’s law in the form that matches the known values (e.g., $I = V/R$ or $V = IR$).
When a component fails, the whole series circuit stops; a parallel circuit keeps running.
Practice converting between total resistance and individual resistances to build confidence.

Quick Reference Table

Circuit Type	Total Resistance	Voltage Distribution	Current Distribution
Series	$R_{\text{total}} = \sum R_i$	$V_{\text{total}} = \sum V_i$	$I$ same through all components
Parallel	$\displaystyle \frac{1}{R_{\text{total}}} = \sum \frac{1}{R_i}$	$V_{\text{total}} = V_i$ for all branches	$I_{\text{total}} = \sum I_i$

Circuit Type	Total Resistance	Voltage Distribution	Current Distribution
Series	\(R_{\text{total}} = \sum R_i\)	\(V_{\text{total}} = \sum V_i\)	\(I\) same through all components
Parallel	\(\displaystyle \frac{1}{R_{\text{total}}} = \sum \frac{1}{R_i}\)	\(V_{\text{total}} = V_i\) for all branches	\(I_{\text{total}} = \sum I_i\)

Know how to construct and use series and parallel circuits