Potential divider

Welcome to one of the most useful and commonly examined circuits in A-Level Physics! In this topic, we will look at how we can "divide" a voltage to get exactly the potential difference we need.

What you'll learn:

What a potential divider is and how it shares a source voltage.
How to use the potential divider equation to calculate output voltages.
How to design variable supplies using variable resistors.
How to build sensor circuits that respond to temperature (using thermistors) or light (using LDRs).

What is a potential divider?

A potential divider is exactly what it sounds like: a circuit that takes an input voltage and divides it up.

In its simplest form, a potential divider consists of two resistors connected in series across a power supply. You already know from your earlier studies of series circuits that the total potential difference (p.d.) of the supply is shared between the components.

Definition

Potential divider

A circuit consisting of two or more components connected in series across a power supply, used to supply a specific, chosen fraction of the source potential difference.

If you have a $\text{ V}$ battery, but you only want to supply $\text{ V}$ to a specific component, you can't just plug it directly into the battery. Instead, you can use a potential divider to "tap off" exactly $\text{ V}$ .

Basic potential divider circuit

In the diagram above, the input voltage $VinV_{\text{in}}$ is applied across both resistors, $R_1$ and $R_2$ . The output voltage $VoutV_{\text{out}}$ is taken across just one of them (in this case, $R_2$ ).

Key Idea

The Golden Rule of Potential Dividers

The voltage is shared in the same ratio as the resistances. The resistor with the larger resistance will always take a larger share of the total input voltage.

The Potential Divider Equation

We can calculate the exact output voltage using the potential divider equation.

Since the components are in series, the same current $I$ flows through both. The total resistance is $R_1 + R_2$ , so the current is:

\frac{V_{\text{in}}}{R_1 + R_2}

The output voltage $VoutV_{\text{out}}$ is simply the p.d. across $R_2$ , which is $\times R_2$ . Substituting our expression for current gives us the standard potential divider equation:

V_{\text{out}} = \frac{R_2}{R_1 + R_2} V_{\text{in}}

This formula is incredibly useful. The fraction $R2R1+R2\frac{R_2}{R_1 + R_2}$ simply represents $R_2$ 's "share" of the total resistance.

Example

Calculating the output voltage

A $\text{ V}$ battery is connected in series with a $\Omega$ resistor and a $\Omega$ resistor. An output is taken across the $\Omega$ resistor. Calculate the output voltage.

Identify the variables from the question: We are given $VV_{\text{in}} = 9.0 \text{ V}$ . Let the top resistor be $R1=20ΩR_1 = 20 \Omega$ and the bottom resistor (where we are taking the output) be $R2=40ΩR_2 = 40 \Omega$ .
Write down the potential divider equation:

V_{\text{out}} = \frac{R_2}{R_1 + R_2} V_{\text{in}}

Substitute the values and solve:

V_{\text{out}} = \frac{40}{20 + 40} \times 9.0

V_{\text{out}} = \frac{40}{60} \times 9.0

V_{\text{out}} = \frac{2}{3} \times 9.0 = 6.0 \text{ V}

Tip

Sanity Check

Before doing the math, look at the ratio. $\Omega$ is twice as big as $\Omega$ . Therefore, it should get twice as much voltage. The total voltage ( $\text{ V}$ ) splits into $\text{ V}$ and $\text{ V}$ . Since the output is across the larger resistor, $\text{ V}$ makes perfect sense!

Variable Potential Dividers

Sometimes we don't just want a fixed output voltage; we want to be able to adjust it smoothly (like a volume dial on a stereo, or a dimmer switch).

We can achieve this by replacing the two fixed resistors with a single variable resistor that has a sliding contact (often called a potentiometer, though the AQA spec notes you don't need to know how to use one as a measuring instrument).

Imagine a single long resistive track. By moving a slider up and down this track, you are effectively splitting it into two sections: $R_1$ (above the slider) and $R_2$ (below the slider).

If you move the slider up, $R_1$ decreases and $R_2$ increases. Because $R_2$ now makes up a larger fraction of the total resistance, $VoutV_{\text{out}}$ increases.
If you move the slider down, $R_2$ approaches zero, so $VoutV_{\text{out}}$ drops towards $\text{ V}$ .

This allows you to provide a continuously variable p.d. from $\text{ V}$ all the way up to $VinV_{\text{in}}$ .

Sensor Circuits: Thermistors and LDRs

This is where potential dividers get really clever. Instead of a manual slider, what if we use a component whose resistance changes based on the environment?

By replacing one of the resistors with a thermistor or a Light Dependent Resistor (LDR), the output voltage will automatically rise and fall as the temperature or light level changes. We have built a sensor!

Temperature sensor circuit

Using an NTC Thermistor

AQA focuses on Negative Temperature Coefficient (NTC) thermistors.

As temperature increases, the resistance of the thermistor decreases.

Look at the diagram above. The thermistor is in the $R_1$ position, and a fixed resistor is in the $R_2$ position. If the room gets hotter:

The resistance of the thermistor ( $R_1$ ) drops.
The total resistance of the circuit ( $R_1 + R_2$ ) decreases.
The fixed resistor ( $R_2$ ) now represents a larger share of the total resistance.
Therefore, the output voltage $VoutV_{\text{out}}$ (which is across $R_2$ ) increases.

This circuit could be used to switch on a cooling fan when it gets too hot!

Using an LDR

LDRs work similarly, but with light:

As light intensity increases, the resistance of the LDR decreases.

If you place an LDR in the $R_1$ (top) position, $VoutV_{\text{out}}$ across $R_2$ will increase as it gets brighter. If you swap them—putting the LDR in the $R_2$ (bottom) position—then as it gets brighter, $R_2$ 's resistance drops, so its share of the voltage drops, and $VoutV_{\text{out}}$ will decrease. This second setup is perfect for automatically turning on streetlights when it gets dark!

Hint

Ratio thinking

If you ever get confused about whether $VoutV_{\text{out}}$ goes up or down, just ask yourself: "What is happening to the resistance of the component I am measuring across?" If its resistance goes down (relative to the rest of the circuit), its voltage share goes down. If its resistance goes up, its voltage share goes up.

Example

Analyzing a sensor circuit

A potential divider consists of a $\text{ V}$ power supply, an NTC thermistor (top component), and a fixed resistor of $\text{ k}\Omega$ (bottom component). The output is taken across the fixed resistor. At $10∘C10^\circ\text{C}$ , the thermistor has a resistance of $\text{ k}\Omega$ . At $40∘C40^\circ\text{C}$ , its resistance drops to $\text{ k}\Omega$ . Calculate the change in output voltage as the temperature rises from $10∘C10^\circ\text{C}$ to $40∘C40^\circ\text{C}$ .

Calculate $VoutV_{\text{out}}$ at $10∘C10^\circ\text{C}$ : The thermistor $kΩR_1 = 15 \text{ k}\Omega$ . The fixed resistor $kΩR_2 = 5.0 \text{ k}\Omega$ .

V_{\text{out, cold}} = \frac{R_2}{R_1 + R_2} V_{\text{in}}

V_{\text{out, cold}} = \frac{5.0}{15 + 5.0} \times 12

V_{\text{out, cold}} = \frac{5.0}{20} \times 12 = 0.25 \times 12 = 3.0 \text{ V}

Calculate $VoutV_{\text{out}}$ at $40∘C40^\circ\text{C}$ : The thermistor $R_1$ is now $\text{ k}\Omega$ . $R_2$ is still $\text{ k}\Omega$ .

V_{\text{out, hot}} = \frac{5.0}{2.5 + 5.0} \times 12

V_{\text{out, hot}} = \frac{5.0}{7.5} \times 12 = \frac{2}{3} \times 12 = 8.0 \text{ V}

Calculate the change in voltage:

\Delta V = V_{\text{out, hot}} - V_{\text{out, cold}}

\Delta V = 8.0 \text{ V} - 3.0 \text{ V} = +5.0 \text{ V}

Common Mistake

Forgetting to convert units

Watch out for kilo-ohms ( $kΩ\text{k}\Omega$ ) and mega-ohms ( $MΩ\text{M}\Omega$ ). In the ratio $R2R1+R2\frac{R_2}{R_1 + R_2}$ , the powers of 10 will actually cancel out as long as $R_1$ and $R_2$ are in the same units. But it's always safest to convert everything to standard Ohms ( $Ω\Omega$ ) before substituting them into the equation to avoid silly calculation errors.

Exam technique

In the exam

Always check where $VoutV_{\text{out}}$ is measured. Do not blindly assume $VoutV_{\text{out}}$ is always across the bottom component. Read the text and look at the diagram. If the output is across the top resistor $R_1$ , the numerator in your formula must be $R_1$ .
Look out for loaded dividers. The potential divider equation assumes that no current flows out of the $VoutV_{\text{out}}$ terminals. If the question states you connect a voltmeter with a finite resistance, or another component, across $VoutV_{\text{out}}$ , that component is now in parallel with $R_2$ . You must first calculate the new combined resistance of $R_2$ and the component before finding the voltage!
Use the phrase "share of the total resistance". When asked to explain why an output voltage changes in a sensor circuit, examiners love it when you state that "the component's resistance drops, so its share of the total resistance drops, leading to a smaller share of the supply p.d.".

Self review

Check yourself

If a potential divider is made of two identical resistors, what is the output voltage across one of them compared to the input voltage?
An LDR and a fixed resistor are in series. The output is taken across the LDR. Does the output voltage increase or decrease as the room gets darker?
Write down the potential divider equation from memory. What does the denominator ( $R_1 + R_2$ ) represent?

Welcome to one of the most useful and commonly examined circuits in A-Level Physics! In this topic, we will look at how we can "divide" a voltage to get exactly the potential difference we need.

What you'll learn:

What a potential divider is and how it shares a source voltage.
How to use the potential divider equation to calculate output voltages.
How to design variable supplies using variable resistors.
How to build sensor circuits that respond to temperature (using thermistors) or light (using LDRs).

What is a potential divider?

A potential divider is exactly what it sounds like: a circuit that takes an input voltage and divides it up.

Definition

Potential divider

A circuit consisting of two or more components connected in series across a power supply, used to supply a specific, chosen fraction of the source potential difference.

Basic potential divider circuit

Key Idea

The Golden Rule of Potential Dividers

The voltage is shared in the same ratio as the resistances. The resistor with the larger resistance will always take a larger share of the total input voltage.

The Potential Divider Equation

We can calculate the exact output voltage using the potential divider equation.

Since the components are in series, the same current $I$ flows through both. The total resistance is $R_1 + R_2$ , so the current is:

\frac{V_{\text{in}}}{R_1 + R_2}

The output voltage $VoutV_{\text{out}}$ is simply the p.d. across $R_2$ , which is $\times R_2$ . Substituting our expression for current gives us the standard potential divider equation:

V_{\text{out}} = \frac{R_2}{R_1 + R_2} V_{\text{in}}

This formula is incredibly useful. The fraction $R2R1+R2\frac{R_2}{R_1 + R_2}$ simply represents $R_2$ 's "share" of the total resistance.

Example

Calculating the output voltage

A $\text{ V}$ battery is connected in series with a $\Omega$ resistor and a $\Omega$ resistor. An output is taken across the $\Omega$ resistor. Calculate the output voltage.

Identify the variables from the question: We are given $VV_{\text{in}} = 9.0 \text{ V}$ . Let the top resistor be $R1=20ΩR_1 = 20 \Omega$ and the bottom resistor (where we are taking the output) be $R2=40ΩR_2 = 40 \Omega$ .
Write down the potential divider equation:

V_{\text{out}} = \frac{R_2}{R_1 + R_2} V_{\text{in}}

Substitute the values and solve:

V_{\text{out}} = \frac{40}{20 + 40} \times 9.0

V_{\text{out}} = \frac{40}{60} \times 9.0

V_{\text{out}} = \frac{2}{3} \times 9.0 = 6.0 \text{ V}

Tip

Sanity Check

Variable Potential Dividers

Sometimes we don't just want a fixed output voltage; we want to be able to adjust it smoothly (like a volume dial on a stereo, or a dimmer switch).

Imagine a single long resistive track. By moving a slider up and down this track, you are effectively splitting it into two sections: $R_1$ (above the slider) and $R_2$ (below the slider).

If you move the slider up, $R_1$ decreases and $R_2$ increases. Because $R_2$ now makes up a larger fraction of the total resistance, $VoutV_{\text{out}}$ increases.
If you move the slider down, $R_2$ approaches zero, so $VoutV_{\text{out}}$ drops towards $\text{ V}$ .

This allows you to provide a continuously variable p.d. from $\text{ V}$ all the way up to $VinV_{\text{in}}$ .

Sensor Circuits: Thermistors and LDRs

This is where potential dividers get really clever. Instead of a manual slider, what if we use a component whose resistance changes based on the environment?

Temperature sensor circuit

Using an NTC Thermistor

AQA focuses on Negative Temperature Coefficient (NTC) thermistors.

As temperature increases, the resistance of the thermistor decreases.

Look at the diagram above. The thermistor is in the $R_1$ position, and a fixed resistor is in the $R_2$ position. If the room gets hotter:

The resistance of the thermistor ( $R_1$ ) drops.
The total resistance of the circuit ( $R_1 + R_2$ ) decreases.
The fixed resistor ( $R_2$ ) now represents a larger share of the total resistance.
Therefore, the output voltage $VoutV_{\text{out}}$ (which is across $R_2$ ) increases.

This circuit could be used to switch on a cooling fan when it gets too hot!

Using an LDR

LDRs work similarly, but with light:

As light intensity increases, the resistance of the LDR decreases.

Hint

Ratio thinking

Example

Analyzing a sensor circuit

Calculate $VoutV_{\text{out}}$ at $10∘C10^\circ\text{C}$ : The thermistor $kΩR_1 = 15 \text{ k}\Omega$ . The fixed resistor $kΩR_2 = 5.0 \text{ k}\Omega$ .

V_{\text{out, cold}} = \frac{R_2}{R_1 + R_2} V_{\text{in}}

V_{\text{out, cold}} = \frac{5.0}{15 + 5.0} \times 12

V_{\text{out, cold}} = \frac{5.0}{20} \times 12 = 0.25 \times 12 = 3.0 \text{ V}

Calculate $VoutV_{\text{out}}$ at $40∘C40^\circ\text{C}$ : The thermistor $R_1$ is now $\text{ k}\Omega$ . $R_2$ is still $\text{ k}\Omega$ .

V_{\text{out, hot}} = \frac{5.0}{2.5 + 5.0} \times 12

V_{\text{out, hot}} = \frac{5.0}{7.5} \times 12 = \frac{2}{3} \times 12 = 8.0 \text{ V}

Calculate the change in voltage:

\Delta V = V_{\text{out, hot}} - V_{\text{out, cold}}

\Delta V = 8.0 \text{ V} - 3.0 \text{ V} = +5.0 \text{ V}

Common Mistake

Forgetting to convert units

Exam technique

In the exam

Always check where $VoutV_{\text{out}}$ is measured. Do not blindly assume $VoutV_{\text{out}}$ is always across the bottom component. Read the text and look at the diagram. If the output is across the top resistor $R_1$ , the numerator in your formula must be $R_1$ .
Look out for loaded dividers. The potential divider equation assumes that no current flows out of the $VoutV_{\text{out}}$ terminals. If the question states you connect a voltmeter with a finite resistance, or another component, across $VoutV_{\text{out}}$ , that component is now in parallel with $R_2$ . You must first calculate the new combined resistance of $R_2$ and the component before finding the voltage!
Use the phrase "share of the total resistance". When asked to explain why an output voltage changes in a sensor circuit, examiners love it when you state that "the component's resistance drops, so its share of the total resistance drops, leading to a smaller share of the supply p.d.".

Self review

Check yourself

If a potential divider is made of two identical resistors, what is the output voltage across one of them compared to the input voltage?
An LDR and a fixed resistor are in series. The output is taken across the LDR. Does the output voltage increase or decrease as the room gets darker?
Write down the potential divider equation from memory. What does the denominator ( $R_1 + R_2$ ) represent?

What is a potential divider?

The Potential Divider Equation

Variable Potential Dividers

Sensor Circuits: Thermistors and LDRs

Using an NTC Thermistor

Using an LDR

Review Potential divider by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Potential divider

What is a potential divider?

The Potential Divider Equation

Variable Potential Dividers

Sensor Circuits: Thermistors and LDRs

Using an NTC Thermistor

Using an LDR

Review Potential divider by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Potential divider

What is a potential divider?

The Potential Divider Equation

Variable Potential Dividers

Sensor Circuits: Thermistors and LDRs

Using an NTC Thermistor

Using an LDR

Review Potential divider by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Potential divider Revision Guide

Potential divider

What is a potential divider?

The Potential Divider Equation

Variable Potential Dividers

Sensor Circuits: Thermistors and LDRs

Using an NTC Thermistor

Using an LDR

Review Potential divider by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Potential divider Revision Guide