Potential dividers

What you'll learn

How two series components share an input potential difference.
How to use $Vout=R2R1+R2VinV_{\text{out}} = \frac{R_2}{R_1 + R_2}V_{\text{in}}$ and $V1V2=R1R2\frac{V_1}{V_2} = \frac{R_1}{R_2}$ .
How a potentiometer gives a continuously variable output voltage.
How LDRs and thermistors can be used to make light- and temperature-sensing circuits.

The basic idea: sharing voltage in series

Before potential dividers, you need one key rule about series circuits: the same current passes through every component in the loop.

A potential difference is the energy transferred per unit charge between two points in a circuit, measured in volts (V). A resistor with a larger resistance gets a larger share of the supply potential difference, because $V = I R$ and the current is the same through each series resistor.

Definition

Potential divider

A potential divider is a circuit with components in series across an input potential difference, arranged so that a chosen fraction of the input potential difference is obtained as an output potential difference.

In the standard two-resistor divider, the input voltage $VinV_{\text{in}}$ is across both resistors. The output voltage $VoutV_{\text{out}}$ is often measured across the lower resistor, labelled $R_2$ .

Potential divider circuit with two resistors and a potentiometer arrangement

Deriving the potential divider equations

For two resistors in series, the total resistance is:

Rtotal=R1+R2R_{\text{total}} = R_1 + R_2

The current in the circuit is:

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

The output voltage across $R_2$ is then:

Vout=IR2=VinR1+R2R2\begin{aligned} V_{\text{out}} &= IR_2 \\ &= \frac{V_{\text{in}}}{R_1 + R_2}R_2 \end{aligned}

So the potential divider equation is:

Vout=R2R1+R2VinV_{\text{out}} = \frac{R_2}{R_1 + R_2}V_{\text{in}}

Because the current is the same through both resistors, the voltage ratio matches the resistance ratio:

V1V2=R1R2\frac{V_1}{V_2} = \frac{R_1}{R_2}

Key Idea

The voltage share rule

In a series potential divider, a component gets the same fraction of the input voltage as its fraction of the total series resistance.

Example

Calculating the output voltage

A potential divider has $kΩR_1 = 2.2 \text{ k}\Omega$ , $kΩR_2 = 3.3 \text{ k}\Omega$ , and $VV_{\text{in}} = 9.0 \text{ V}$ . Find $VoutV_{\text{out}}$ across $R_2$ .

Choose the correct resistance for the numerator. Since $VoutV_{\text{out}}$ is across $R_2$ , use $R_2$ in the fraction:

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

Substitute the values, keeping both resistances in the same unit:

V_{\text{out}} = \frac{3.3 \text{ k}\Omega}{2.2 \text{ k}\Omega + 3.3 \text{ k}\Omega} \times 9.0 \text{ V}

Calculate the fraction of the input voltage:

V_{\text{out}} = \frac{3.3}{5.5} \times 9.0 \text{ V} = 5.4 \text{ V}

Common Mistake

Using the wrong resistor in the numerator

The numerator is the resistance across which the output voltage is measured. If $VoutV_{\text{out}}$ is across $R_2$ , use $R_2$ . If it is across $R_1$ , use $R_1$ .

Designing a divider for a required output

Sometimes you choose resistor values to give a particular output voltage. A useful rearrangement comes from thinking in ratios.

If the supply is split into $V_1$ across $R_1$ and $V_2$ across $R_2$ , then:

V1V2=R1R2\frac{V_1}{V_2} = \frac{R_1}{R_2}

So, for example, if the top resistor needs twice the voltage of the bottom resistor, it should have twice the resistance.

Example

Choosing a resistor for a required output

A 12.0 V supply is used in a divider. The output is taken across a $\text{ k}\Omega$ lower resistor. Choose the upper resistor so that $VV_{\text{out}} = 4.0 \text{ V}$ .

Work out the voltage across the upper resistor:

V_1 = 12.0 \text{ V} - 4.0 \text{ V} = 8.0 \text{ V}

Use the voltage ratio to find the resistance ratio:

\frac{R_1}{R_2} = \frac{V_1}{V_2} = \frac{8.0 \text{ V}}{4.0 \text{ V}} = 2.0

Calculate the upper resistor:

R_1 = 2.0R_2 = 2.0 \times 2.0 \text{ k}\Omega = 4.0 \text{ k}\Omega

Potentiometers as potential dividers

A potentiometer is a three-terminal variable resistor. It has a resistive track and a sliding contact called a wiper.

When the supply is connected across the two ends of the track, the wiper taps off a chosen fraction of the input voltage. Moving the wiper changes the ratio of the resistance above and below it, so $VoutV_{\text{out}}$ can vary smoothly from about 0 V up to about $VinV_{\text{in}}$ .

Definition

Potentiometer

A potentiometer is a variable potential divider with three terminals: two ends of a resistive track and one sliding wiper contact.

Tip

Potentiometer versus variable resistor

Using all three terminals makes a potentiometer act as a potential divider. Using only two terminals makes it act like a variable resistor controlling current.

Variable components in potential dividers

Potential dividers become especially useful when one resistor changes with the environment.

An LDR, or light-dependent resistor, has a resistance that decreases as light intensity increases.

A thermistor is a resistor whose resistance changes with temperature. At A Level, the usual thermistor is an NTC thermistor: its resistance decreases as temperature increases.

Potential divider sensor circuits using an LDR and a thermistor

The output depends on where the sensor is placed and where $VoutV_{\text{out}}$ is measured.

If the variable component is the lower resistor

If $VoutV_{\text{out}}$ is measured across the lower component, then:

Vout=RlowerRupper+RlowerVinV_{\text{out}} = \frac{R_{\text{lower}}}{R_{\text{upper}} + R_{\text{lower}}}V_{\text{in}}

So if the lower resistance decreases, $VoutV_{\text{out}}$ decreases.

For an LDR placed as the lower resistor, brighter light gives lower resistance, so the output voltage decreases.

If the variable component is the upper resistor

If the sensor is the upper resistor but $VoutV_{\text{out}}$ is measured across the fixed lower resistor, then a decrease in the sensor resistance makes the fixed resistor a larger fraction of the total resistance.

So for an NTC thermistor placed above a fixed lower resistor, higher temperature gives lower thermistor resistance, so $VoutV_{\text{out}}$ across the fixed resistor increases.

Example

Predicting how a sensor output changes

An NTC thermistor is the upper resistor in a potential divider. A fixed resistor is the lower resistor, and $VoutV_{\text{out}}$ is measured across the fixed resistor. What happens to $VoutV_{\text{out}}$ when the temperature increases?

Identify how the thermistor changes: for an NTC thermistor, increasing temperature decreases its resistance.
Compare the fixed lower resistor with the new total resistance. The lower resistor stays the same, but the total $Rthermistor+RfixedR_{\text{thermistor}} + R_{\text{fixed}}$ decreases.
Apply the divider fraction:

V_{\text{out}} = \frac{R_{\text{fixed}}}{R_{\text{thermistor}} + R_{\text{fixed}}}V_{\text{in}}

Since the denominator decreases while the numerator stays the same, $VoutV_{\text{out}}$ increases.

Common Mistake

Assuming every sensor output rises

An LDR or thermistor resistance may decrease, but the output voltage might increase or decrease depending on whether the sensor is above or below the output point.

Choosing component positions for sensing circuits

For sensing circuits, decide what you want the output voltage to do.

If you want $VoutV_{\text{out}}$ to increase when light intensity increases, place the LDR as the upper resistor and measure the output across the fixed lower resistor.

If you want $VoutV_{\text{out}}$ to decrease when light intensity increases, place the LDR as the lower resistor and measure the output across it.

The same logic applies to an NTC thermistor:

Thermistor lower, output across thermistor: higher temperature gives lower $VoutV_{\text{out}}$ .
Thermistor upper, output across fixed lower resistor: higher temperature gives higher $VoutV_{\text{out}}$ .

A good fixed resistor value is often similar to the sensor resistance near the temperature or light level where you want the circuit to be most responsive. This makes the output voltage change noticeably around the useful range.

Practical investigation of a potential divider

To investigate a potential divider, you normally build the circuit, vary one physical quantity, and measure the output potential difference with a voltmeter or data logger.

For a thermistor investigation:

Use a low-voltage direct-current supply.
Put the thermistor in a water bath with a thermometer.
Stir the water and allow the thermistor reading to settle before recording.
Measure $VoutV_{\text{out}}$ at different temperatures.
Keep $VinV_{\text{in}}$ constant throughout.

For an LDR investigation:

Use a lamp or phone torch at measured distances, or use a light meter if available.
Shield the LDR from background light as much as possible.
Keep the orientation of the LDR and lamp fixed.
Measure $VoutV_{\text{out}}$ for different light intensities.

You can then plot a graph of output voltage against temperature or light intensity. The graph may not be a straight line, because LDRs and thermistors do not usually have linear resistance changes.

Common Mistake

Loading the potential divider

The simple divider equations assume the output is measured by a very high-resistance device, such as a voltmeter, so that negligible current is drawn from the output. If a low-resistance load is connected across the output, it changes the effective resistance and the output voltage.

Example

Finding resistance from a measured output

A thermistor is the lower component in a divider with a $\text{ k}\Omega$ fixed upper resistor. The supply is $\text{ V}$ . At one temperature, $VoutV_{\text{out}}$ across the thermistor is $\text{ V}$ . Find the thermistor resistance.

Start from the divider equation with the thermistor as the lower resistor:

V_{\text{out}} = \frac{R_T}{R_{\text{fixed}} + R_T}V_{\text{in}}

Substitute the known values:

\text{ V} = \frac{R_T}{4.7 \text{ k}\Omega + R_T} \times 6.0 \text{ V}

Divide both sides by $\text{ V}$ and rearrange:

\frac{1}{3} = \frac{R_T}{4.7 \text{ k}\Omega + R_T}

\text{ k}\Omega + R_T = 3R_T

Solve for the thermistor resistance:

\text{ k}\Omega = 2R_T

R_T = 2.35 \text{ k}\Omega

To two significant figures, $kΩR_T = 2.4 \text{ k}\Omega$ .

Final checklist for the method

When you see a potential divider question:

Mark the input voltage across the whole series combination.
Mark exactly where the output voltage is measured.
Put the output resistance in the numerator.
Put the total series resistance in the denominator.
Think physically: larger share of resistance means larger share of voltage.

Exam technique

In the exam

Draw or annotate the circuit before calculating; most mistakes come from choosing the wrong output component.
For LDRs and thermistors, state the resistance change first, then use the divider equation to decide the voltage change.
In practical questions, mention controlling $VinV_{\text{in}}$ , using a voltmeter or data logger, repeating readings, and reducing unwanted heating or background light.

Self review

Check yourself

If $R_2$ is doubled while $R_1$ and $VinV_{\text{in}}$ stay fixed, what happens to $VoutV_{\text{out}}$ across $R_2$ ?
How would you arrange an LDR divider so that the output voltage rises in brighter light?
Why does connecting a low-resistance load across the output change the divider behaviour?

What you'll learn

How two series components share an input potential difference.
How to use $Vout=R2R1+R2VinV_{\text{out}} = \frac{R_2}{R_1 + R_2}V_{\text{in}}$ and $V1V2=R1R2\frac{V_1}{V_2} = \frac{R_1}{R_2}$ .
How a potentiometer gives a continuously variable output voltage.
How LDRs and thermistors can be used to make light- and temperature-sensing circuits.

The basic idea: sharing voltage in series

Before potential dividers, you need one key rule about series circuits: the same current passes through every component in the loop.

Definition

Potential divider

Potential divider circuit with two resistors and a potentiometer arrangement

Deriving the potential divider equations

For two resistors in series, the total resistance is:

Rtotal=R1+R2R_{\text{total}} = R_1 + R_2

The current in the circuit is:

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

The output voltage across $R_2$ is then:

Vout=IR2=VinR1+R2R2\begin{aligned} V_{\text{out}} &= IR_2 \\ &= \frac{V_{\text{in}}}{R_1 + R_2}R_2 \end{aligned}

So the potential divider equation is:

Vout=R2R1+R2VinV_{\text{out}} = \frac{R_2}{R_1 + R_2}V_{\text{in}}

Because the current is the same through both resistors, the voltage ratio matches the resistance ratio:

V1V2=R1R2\frac{V_1}{V_2} = \frac{R_1}{R_2}

Key Idea

The voltage share rule

In a series potential divider, a component gets the same fraction of the input voltage as its fraction of the total series resistance.

Example

Calculating the output voltage

A potential divider has $kΩR_1 = 2.2 \text{ k}\Omega$ , $kΩR_2 = 3.3 \text{ k}\Omega$ , and $VV_{\text{in}} = 9.0 \text{ V}$ . Find $VoutV_{\text{out}}$ across $R_2$ .

Choose the correct resistance for the numerator. Since $VoutV_{\text{out}}$ is across $R_2$ , use $R_2$ in the fraction:

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

Substitute the values, keeping both resistances in the same unit:

V_{\text{out}} = \frac{3.3 \text{ k}\Omega}{2.2 \text{ k}\Omega + 3.3 \text{ k}\Omega} \times 9.0 \text{ V}

Calculate the fraction of the input voltage:

V_{\text{out}} = \frac{3.3}{5.5} \times 9.0 \text{ V} = 5.4 \text{ V}

Common Mistake

Using the wrong resistor in the numerator

The numerator is the resistance across which the output voltage is measured. If $VoutV_{\text{out}}$ is across $R_2$ , use $R_2$ . If it is across $R_1$ , use $R_1$ .

Designing a divider for a required output

Sometimes you choose resistor values to give a particular output voltage. A useful rearrangement comes from thinking in ratios.

If the supply is split into $V_1$ across $R_1$ and $V_2$ across $R_2$ , then:

V1V2=R1R2\frac{V_1}{V_2} = \frac{R_1}{R_2}

So, for example, if the top resistor needs twice the voltage of the bottom resistor, it should have twice the resistance.

Example

Choosing a resistor for a required output

A 12.0 V supply is used in a divider. The output is taken across a $\text{ k}\Omega$ lower resistor. Choose the upper resistor so that $VV_{\text{out}} = 4.0 \text{ V}$ .

Work out the voltage across the upper resistor:

V_1 = 12.0 \text{ V} - 4.0 \text{ V} = 8.0 \text{ V}

Use the voltage ratio to find the resistance ratio:

\frac{R_1}{R_2} = \frac{V_1}{V_2} = \frac{8.0 \text{ V}}{4.0 \text{ V}} = 2.0

Calculate the upper resistor:

R_1 = 2.0R_2 = 2.0 \times 2.0 \text{ k}\Omega = 4.0 \text{ k}\Omega

Potentiometers as potential dividers

A potentiometer is a three-terminal variable resistor. It has a resistive track and a sliding contact called a wiper.

Definition

Potentiometer

A potentiometer is a variable potential divider with three terminals: two ends of a resistive track and one sliding wiper contact.

Tip

Potentiometer versus variable resistor

Using all three terminals makes a potentiometer act as a potential divider. Using only two terminals makes it act like a variable resistor controlling current.

Variable components in potential dividers

Potential dividers become especially useful when one resistor changes with the environment.

An LDR, or light-dependent resistor, has a resistance that decreases as light intensity increases.

A thermistor is a resistor whose resistance changes with temperature. At A Level, the usual thermistor is an NTC thermistor: its resistance decreases as temperature increases.

Potential divider sensor circuits using an LDR and a thermistor

The output depends on where the sensor is placed and where $VoutV_{\text{out}}$ is measured.

If the variable component is the lower resistor

If $VoutV_{\text{out}}$ is measured across the lower component, then:

Vout=RlowerRupper+RlowerVinV_{\text{out}} = \frac{R_{\text{lower}}}{R_{\text{upper}} + R_{\text{lower}}}V_{\text{in}}

So if the lower resistance decreases, $VoutV_{\text{out}}$ decreases.

For an LDR placed as the lower resistor, brighter light gives lower resistance, so the output voltage decreases.

If the variable component is the upper resistor

So for an NTC thermistor placed above a fixed lower resistor, higher temperature gives lower thermistor resistance, so $VoutV_{\text{out}}$ across the fixed resistor increases.

Example

Predicting how a sensor output changes

Identify how the thermistor changes: for an NTC thermistor, increasing temperature decreases its resistance.
Compare the fixed lower resistor with the new total resistance. The lower resistor stays the same, but the total $Rthermistor+RfixedR_{\text{thermistor}} + R_{\text{fixed}}$ decreases.
Apply the divider fraction:

V_{\text{out}} = \frac{R_{\text{fixed}}}{R_{\text{thermistor}} + R_{\text{fixed}}}V_{\text{in}}

Since the denominator decreases while the numerator stays the same, $VoutV_{\text{out}}$ increases.

Common Mistake

Assuming every sensor output rises

An LDR or thermistor resistance may decrease, but the output voltage might increase or decrease depending on whether the sensor is above or below the output point.

Choosing component positions for sensing circuits

For sensing circuits, decide what you want the output voltage to do.

If you want $VoutV_{\text{out}}$ to increase when light intensity increases, place the LDR as the upper resistor and measure the output across the fixed lower resistor.

If you want $VoutV_{\text{out}}$ to decrease when light intensity increases, place the LDR as the lower resistor and measure the output across it.

The same logic applies to an NTC thermistor:

Thermistor lower, output across thermistor: higher temperature gives lower $VoutV_{\text{out}}$ .
Thermistor upper, output across fixed lower resistor: higher temperature gives higher $VoutV_{\text{out}}$ .

Practical investigation of a potential divider

To investigate a potential divider, you normally build the circuit, vary one physical quantity, and measure the output potential difference with a voltmeter or data logger.

For a thermistor investigation:

Use a low-voltage direct-current supply.
Put the thermistor in a water bath with a thermometer.
Stir the water and allow the thermistor reading to settle before recording.
Measure $VoutV_{\text{out}}$ at different temperatures.
Keep $VinV_{\text{in}}$ constant throughout.

For an LDR investigation:

Use a lamp or phone torch at measured distances, or use a light meter if available.
Shield the LDR from background light as much as possible.
Keep the orientation of the LDR and lamp fixed.
Measure $VoutV_{\text{out}}$ for different light intensities.

You can then plot a graph of output voltage against temperature or light intensity. The graph may not be a straight line, because LDRs and thermistors do not usually have linear resistance changes.

Common Mistake

Loading the potential divider

Example

Finding resistance from a measured output

Start from the divider equation with the thermistor as the lower resistor:

V_{\text{out}} = \frac{R_T}{R_{\text{fixed}} + R_T}V_{\text{in}}

Substitute the known values:

\text{ V} = \frac{R_T}{4.7 \text{ k}\Omega + R_T} \times 6.0 \text{ V}

Divide both sides by $\text{ V}$ and rearrange:

\frac{1}{3} = \frac{R_T}{4.7 \text{ k}\Omega + R_T}

\text{ k}\Omega + R_T = 3R_T

Solve for the thermistor resistance:

\text{ k}\Omega = 2R_T

R_T = 2.35 \text{ k}\Omega

To two significant figures, $kΩR_T = 2.4 \text{ k}\Omega$ .

Final checklist for the method

When you see a potential divider question:

Mark the input voltage across the whole series combination.
Mark exactly where the output voltage is measured.
Put the output resistance in the numerator.
Put the total series resistance in the denominator.
Think physically: larger share of resistance means larger share of voltage.

Exam technique

In the exam

Draw or annotate the circuit before calculating; most mistakes come from choosing the wrong output component.
For LDRs and thermistors, state the resistance change first, then use the divider equation to decide the voltage change.
In practical questions, mention controlling $VinV_{\text{in}}$ , using a voltmeter or data logger, repeating readings, and reducing unwanted heating or background light.

Self review

Check yourself

If $R_2$ is doubled while $R_1$ and $VinV_{\text{in}}$ stay fixed, what happens to $VoutV_{\text{out}}$ across $R_2$ ?
How would you arrange an LDR divider so that the output voltage rises in brighter light?
Why does connecting a low-resistance load across the output change the divider behaviour?

What you'll learn

The basic idea: sharing voltage in series

Deriving the potential divider equations

Designing a divider for a required output

Potentiometers as potential dividers

Variable components in potential dividers

If the variable component is the lower resistor

If the variable component is the upper resistor

Choosing component positions for sensing circuits

Practical investigation of a potential divider

Final checklist for the method

Review Potential dividers by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Potential dividers

What you'll learn

The basic idea: sharing voltage in series

Deriving the potential divider equations

Designing a divider for a required output

Potentiometers as potential dividers

Variable components in potential dividers

If the variable component is the lower resistor

If the variable component is the upper resistor

Choosing component positions for sensing circuits

Practical investigation of a potential divider

Final checklist for the method

Review Potential dividers by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Potential dividers

What you'll learn

The basic idea: sharing voltage in series

Deriving the potential divider equations

Designing a divider for a required output

Potentiometers as potential dividers

Variable components in potential dividers

If the variable component is the lower resistor

If the variable component is the upper resistor

Choosing component positions for sensing circuits

Practical investigation of a potential divider

Final checklist for the method

Review Potential dividers by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Potential dividers Revision Guide

Potential dividers

What you'll learn

The basic idea: sharing voltage in series

Deriving the potential divider equations

Designing a divider for a required output

Potentiometers as potential dividers

Variable components in potential dividers

If the variable component is the lower resistor

If the variable component is the upper resistor

Choosing component positions for sensing circuits

Practical investigation of a potential divider

Final checklist for the method

Review Potential dividers by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Potential dividers Revision Guide