Resistors in Series and Parallel

Combining resistors

Real circuits often contain several resistors. To understand them, we need to know how resistors behave when connected in series or in parallel, and how to find their total, or equivalent resistance.

Resistors in series

In series, resistors are connected end to end. Current has only one path.

Rules

Adding resistors in series always increases total resistance, like adding more narrow sections to a pipe.

Example

A 12 V battery is connected to resistors of 2 Ω, 4 Ω and 6 Ω in series.

1. R_total = 2 + 4 + 6 = 12 Ω
2. Current: I = V ÷ R = 12 ÷ 12 = 1 A (same through each)
3. Voltage across each: V = I × R
   • 2 Ω: 2 V
   • 4 Ω: 4 V
   • 6 Ω: 6 V
4. Check: 2 + 4 + 6 = 12 V ✓

The biggest resistor gets the biggest share of voltage.

Voltage dividers

A series circuit used to split voltage is called a voltage divider. It’s used in sensors: a thermistor and a fixed resistor in series change their voltage share as temperature changes.

Resistors in parallel

In parallel, resistors are connected side by side, each across the same two points. Current splits between the branches at a junction.

Rules

Adding resistors in parallel always decreases total resistance, like opening more checkout lanes in a store. The total is always less than the smallest single resistor.

Shortcut for two resistors

R_total = (R₁ × R₂) ÷ (R₁ + R₂)

("product over sum")

For identical resistors: R_total = R ÷ number of resistors. Two 10 Ω resistors in parallel = 5 Ω.

Example

A 12 V battery is connected to a 6 Ω and a 3 Ω resistor in parallel.

1. R_total = (6 × 3) ÷ (6 + 3) = 18 ÷ 9 = 2 Ω
2. Total current: I = 12 ÷ 2 = 6 A
3. Each branch has 12 V:
   • 6 Ω branch: I = 12 ÷ 6 = 2 A
   • 3 Ω branch: I = 12 ÷ 3 = 4 A
4. Check: 2 + 4 = 6 A ✓

More current flows through the smaller resistor. A parallel circuit acts as a current divider.

Comparing the two

Combination circuits

Many circuits mix series and parallel parts. Solve them step by step:

1. Find the equivalent resistance of each parallel group.
2. Add series resistances.
3. Use Ohm’s law to find the total current.
4. Work back to find the voltage and current for each part.

Example

A 4 Ω resistor in series with a parallel pair of 12 Ω and 6 Ω, connected to 16 V.

1. Parallel pair: (12 × 6) ÷ (12 + 6) = 72 ÷ 18 = 4 Ω
2. Total: 4 + 4 = 8 Ω
3. Total current: 16 ÷ 8 = 2 A
4. Voltage across the 4 Ω series resistor: 2 × 4 = 8 V
5. Voltage across the parallel pair: 16 − 8 = 8 V
6. Currents: 8 ÷ 12 = 0.67 A and 8 ÷ 6 = 1.33 A (total 2 A ✓)

Real-world uses

• Homes wire outlets and lights in parallel so each gets the full 120 V and works independently.
• Dimmer switches use variable resistance in series with a light.
• Holiday lights may combine series sections connected in parallel.

Common mistakes

• Adding parallel resistances like series ones.
• Forgetting to flip the result after adding 1/R values.
• Thinking current is "used up" along a series circuit.

Try it yourself

1. Find the total resistance of 5 Ω, 10 Ω and 15 Ω in series.
2. Find the total resistance of 10 Ω and 40 Ω in parallel.
3. Three 30 Ω resistors are in parallel across 9 V. Find the total resistance and total current.
4. Explain why adding more appliances to one power bar increases the current from the wall.

(Answers: 30 Ω; 8 Ω; 10 Ω and 0.9 A; each appliance is a new parallel branch, lowering total resistance and increasing total current)

Quick recap

• Series resistances add; current is the same throughout.
• Parallel resistances combine to less than the smallest; voltage is the same across each branch.
• Series circuits divide voltage; parallel circuits divide current.
• Solve combination circuits step by step.
• Homes use parallel wiring.