Resistors in Series Equation, Derivation & Rearrangements

Core idea

Overview

In a series circuit, resistors are connected end-to-end along a single path, forcing the same electrical current to flow through each component sequentially. The total or equivalent resistance of the network is calculated by summing the individual resistance values of every component in the chain.

When to use: Apply this formula when components are arranged in a single loop without any branching paths. It assumes that the current remains constant through all components and that the wiring resistance between resistors is negligible.

Why it matters: This principle is fundamental for designing voltage dividers and controlling current levels in electronic devices. It allows engineers to create specific resistance values for a circuit by combining standard, commercially available resistor components.

Symbols

Variables

R_T = Total Resistance, R_1 = Resistor 1, R_2 = Resistor 2, R_3 = Resistor 3

R_{T}

Total Resistance

Ω

R_{1}

Resistor 1

Ω

R_{2}

Resistor 2

Ω

R_{3}

Resistor 3

Ω

Walkthrough

Derivation

Derivation of Resistors in Series

Proves that total resistance in a series circuit is the sum of individual resistances using Kirchhoff's Laws.

The connecting wires have zero resistance.
The components obey Ohm's Law.

1

Apply Kirchhoff's Voltage Law:

In a series circuit, the total potential difference is shared across the components.

V_{total} = V_{1} + V_{2} + V_{3}

2

Apply Ohm's Law:

Substitute V = IR. Because it's a series circuit, the current I is the same through all components.

I R_{total} = I R_{1} + I R_{2} + I R_{3}

3

Cancel the Current:

Divide the entire equation by the common current I.

R_{total} = R_{1} + R_{2} + R_{3}

Result

R_{total} = R_{1} + R_{2} + R_{3}

Source: AQA A-Level Physics — Current Electricity

Free formulas

Rearrangements

Solve for $R_{T}$

Make Rt the subject

R_{T} = R_{1} + R_{2} + R_{3}

Rt is already the subject of the formula.

Difficulty: 1/5

Solve for $R_{1}$

Make R1 the subject

R_{1} = R_{T} - R_{2} - R_{3}

Start from the formula for total resistance in a series circuit. To make R1 the subject, subtract R2 and R3 from both sides.

Difficulty: 2/5

Solve for $R_{2}$

Make R2 the subject

R_{2} = R_{T} - R_{1} - R_{3}

Start from the formula for total resistance in a series circuit. To make R2 the subject, subtract R1 and R3 from both sides.

Difficulty: 2/5

Solve for $R_{3}$

Make R3 the subject

R_{3} = R_{T} - R_{1} - R_{2}

Start from the formula for total resistance in a series circuit. To make R3 the subject, subtract R1 and R2 from both sides.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line with a slope of 1, where the total resistance increases at a constant rate as R1 grows from a y-intercept equal to the sum of R2 and R3. For a physics student, this linear relationship means that increasing the value of R1 adds directly to the total resistance, where small x-values represent a circuit with minimal additional resistance and large x-values represent a circuit dominated by the first resistor. The most important feature is that the slope remains constant, meaning that any c

Graph type: linear

Why it behaves this way

Intuition

Visualize a single-lane road where each resistor is an individual speed bump, with the total resistance being the cumulative effect of all bumps slowing down the traffic (current).

R_{T}

Total equivalent electrical resistance of the series circuit

The overall opposition to current flow presented by all components combined, as if it were a single resistor.

R_{n}

Individual electrical resistance of a single component (e.g., R_1, R_2, R_3)

How much a specific component impedes the flow of electrical current passing through it.

Free study cues

Insight

Canonical usage

All resistance values ( $R_{1}$ , $R_{2}$ , $R_{3}$ , etc.) must be expressed in the same unit (typically Ohms) to yield a total resistance ( $R_{T}$ ) in that same unit.

Common confusion

Mixing units (e.g., Ohms and kilo-Ohms) without proper conversion before summation.

Unit systems

$R_{T}, R_{1}, R_{2}, R_{3}$ Ohm (Ω) · All individual resistance values must be in the same unit (e.g., Ohms) for direct summation to yield the total resistance in that unit.

One free problem

Practice Problem

A simple circuit is constructed by connecting three resistors in a single loop with values of 12 Ω, 33 Ω, and 45 Ω. What is the total equivalent resistance of the circuit?

Resistor 112 Ω

Resistor 233 Ω

Resistor 345 Ω

Solve for: $R t$

Hint: The total resistance is found by summing the three individual resistances directly.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating total resistance in a string of LEDs.

Study smarter

Tips

Total resistance always increases as more resistors are added in series.
The total resistance will always be larger than the highest individual resistor value in the chain.
Ensure all resistance values use the same units, such as Ohms or kilo-Ohms, before performing the addition.

Avoid these traps

Common Mistakes

Using the parallel formula.
Forgetting to add all resistors.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Proves that total resistance in a series circuit is the sum of individual resistances using Kirchhoff's Laws.

Apply this formula when components are arranged in a single loop without any branching paths. It assumes that the current remains constant through all components and that the wiring resistance between resistors is negligible.

This principle is fundamental for designing voltage dividers and controlling current levels in electronic devices. It allows engineers to create specific resistance values for a circuit by combining standard, commercially available resistor components.

Using the parallel formula. Forgetting to add all resistors.

Calculating total resistance in a string of LEDs.

Total resistance always increases as more resistors are added in series. The total resistance will always be larger than the highest individual resistor value in the chain. Ensure all resistance values use the same units, such as Ohms or kilo-Ohms, before performing the addition.

References

Sources

Halliday, Resnick, and Walker, Fundamentals of Physics
Wikipedia: Series and parallel circuits
Halliday, Resnick, Walker, Fundamentals of Physics, 11th ed.
Wikipedia: Electrical resistance
Halliday, Resnick, Walker, Fundamentals of Physics, 10th ed.
Bird, Stewart, Lightfoot, Transport Phenomena, 2nd ed.
IUPAC Gold Book, 'Ohm's law'
IUPAC Gold Book, 'Ohmic conductor'

Resistors in Series

Overview

Variables

Derivation

Apply Kirchhoff's Voltage Law:

Apply Ohm's Law:

Cancel the Current:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Resistors in Parallel

Frequently Asked Questions

Sources