# Series and Parallel Circuits

> Two ways to connect components, and the rules for current, voltage and resistance in each.

From HSC Physics on WhizzNotes, free study notes for NSW students: https://whizznotes.com/physics/series-and-parallel/. Updated 2 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The Christmas lights mystery
Every December, Hannah's grandad pulled out his old string of Christmas lights. And every December, one bulb would be broken, and the **whole string** would stay dark. He would spend an hour testing bulbs one by one.

This year Hannah bought him a new string. When one bulb failed, all the others kept shining. Grandad was amazed.

The bulbs were the same. What changed was **how they were connected**. The old string was a **series** circuit. The new one was **parallel**.

**In short:** In a series circuit the parts are in one loop, so the same current flows through each and the voltages add up. In a parallel circuit each part is on its own branch with the full voltage, so the currents add up and one part can work when another is switched off. Homes are wired in parallel.

## Series circuits

In a **series circuit**, components are connected **one after another** in a **single loop**. There is only **one path** for the current.

*Diagram: Series circuit: a cell and two lamps in a single loop*
*Series: one loop, one path.*

**Rules for series circuits**

| Quantity | Rule |
|---|---|
| **Current** | The **same** everywhere: $I_1 = I_2 = I_3$ |
| **Voltage** | **Shared** between components: $V_{\text{total}} = V_1 + V_2 + \dots$ |
| **Resistance** | **Adds up**: $R_{\text{total}} = R_1 + R_2 + \dots$ |

**What happens in real life**
- Adding more bulbs makes **every** bulb **dimmer**, because the voltage is shared and total resistance goes up.
- If one bulb breaks, the loop is broken and **all** bulbs go out.

### Worked example: Two resistors in series
A 12 V battery is connected to a 2 Ω and a 4 Ω resistor in series.

1. Total resistance: $R = 2 + 4 = 6$ Ω
2. Current (same everywhere): $I = \dfrac{V}{R} = \dfrac{12}{6} = 2$ A
3. Voltage across 2 Ω: $V = 2 \times 2 = 4$ V
4. Voltage across 4 Ω: $V = 2 \times 4 = 8$ V

Check: $4 + 8 = 12$ V. The voltages add up to the supply. The bigger resistor takes the bigger share.

## Parallel circuits

In a **parallel circuit**, components are on **separate branches**. The current splits up at a **junction** and joins again afterwards. There is **more than one path**.

*Diagram: Parallel circuit: a cell with two lamps on separate branches*
*Parallel: each lamp has its own branch. Dots mark the junctions.*

**Rules for parallel circuits**

| Quantity | Rule |
|---|---|
| **Current** | **Splits** between branches: $I_{\text{total}} = I_1 + I_2 + \dots$ |
| **Voltage** | The **same** across every branch: $V_1 = V_2 = V_{\text{supply}}$ |
| **Resistance** | Total is **less** than the smallest branch: $\dfrac{1}{R_{\text{total}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dots$ |

**What happens in real life**
- Each bulb gets the **full voltage**, so they all stay **bright**.
- If one bulb breaks, the others **keep working**.
- Each branch can have its own **switch**.
- This is how **houses are wired**. Your TV does not go off when someone turns off the kitchen light.

### Worked example: Two resistors in parallel
A 12 V battery is connected to a 3 Ω and a 6 Ω resistor in parallel.

1. Voltage across each: **12 V**
2. Current in 3 Ω: $I = \dfrac{12}{3} = 4$ A
3. Current in 6 Ω: $I = \dfrac{12}{6} = 2$ A
4. Total current: $4 + 2 = 6$ A
5. Total resistance: $R = \dfrac{12}{6} = 2$ Ω

Check with the formula: $\dfrac{1}{R} = \dfrac{1}{3} + \dfrac{1}{6} = \dfrac{3}{6}$, so $R = 2$ Ω. Smaller than either resistor.

### Exam tip: Shortcut for two resistors in parallel
$$R_{\text{total}} = \frac{R_1 \times R_2}{R_1 + R_2}$$
"Product over sum." For 3 Ω and 6 Ω: $\dfrac{18}{9} = 2$ Ω.

### Watch out: Adding branches lowers resistance
It feels odd, but adding another branch in parallel **reduces** total resistance. You are giving the current another road to travel, so more current can flow overall.

## Series versus parallel at a glance

| | Series | Parallel |
|---|---|---|
| Paths for current | One | More than one |
| Current | Same everywhere | Splits between branches |
| Voltage | Shared | Same on every branch |
| Adding bulbs | All get dimmer | Stay just as bright |
| One bulb breaks | All go out | Others stay on |
| Total resistance | Sum of all | Less than smallest |

**Mind map**

```text
Circuits
  Series
    One loop
    Same current
    Voltage shared
    R = R₁ + R₂
    One breaks, all off
  Parallel
    Branches
    Current splits
    Same voltage
    1/R = 1/R₁ + 1/R₂
    House wiring
  Checks
    Voltages add to supply in series
    Currents add to total in parallel
```

### Year 11: Combination circuits and internal resistance
For mixed circuits, simplify step by step: replace each parallel group with its equivalent resistance, then add series parts.

**Example.** A 4 Ω resistor in series with a parallel pair (6 Ω and 3 Ω). Parallel pair: $\dfrac{6 \times 3}{6 + 3} = 2$ Ω. Total: $4 + 2 = 6$ Ω.

Real batteries have a small **internal resistance** $r$. The voltage at the terminals is less than the battery's emf $\varepsilon$ when current flows:

$$V_{\text{terminal}} = \varepsilon - Ir$$

### Year 12: Power in series and parallel
Using $P = \dfrac{V^2}{R}$ and $P = I^2R$:

- In **series**, the same current flows, so the **larger** resistor dissipates **more** power ($P = I^2R$).
- In **parallel**, the same voltage acts, so the **smaller** resistor dissipates **more** power ($P = V^2/R$).

This is why a 100 W bulb and a 40 W bulb in series behave surprisingly: the 40 W bulb (higher resistance) glows brighter.

## Quick summary

- **Series**: one loop. Same current everywhere. Voltages add up. $R = R_1 + R_2$.
- **Parallel**: branches. Same voltage on each branch. Currents add up. $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$.
- Series bulbs dim as you add more, and one break turns them all off.
- Parallel bulbs stay bright and work independently. Homes use parallel wiring.
- Parallel total resistance is always less than the smallest branch.

## Quiz

Try these questions from the chapter quiz.

1. In a series circuit, the current is:
   - A. The same at every point
   - B. Bigger near the battery
   - C. Split between components
   - D. Zero after the first bulb
   - Answer: A. The same at every point. There is only one path, so the same current flows everywhere.

2. In a parallel circuit, the voltage across each branch is:
   - A. Shared equally
   - B. The same as the supply voltage
   - C. Zero
   - D. Double the supply
   - Answer: B. The same as the supply voltage. Each branch is connected directly across the supply.

3. Resistors of 5 Ω and 7 Ω are in series. The total resistance is:
   - A. 2 Ω
   - B. 2.9 Ω
   - C. 12 Ω
   - D. 35 Ω
   - Answer: C. 12 Ω. Series resistances add: 5 + 7 = 12 Ω.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/series-and-parallel/#quiz