Potential Divider

Branch note: This page deepens one part of DC Circuits.

Overview

A potential divider is a circuit arrangement that uses resistors in series to divide an input voltage into smaller output voltages.

It is one of the most important DC-circuit ideas because it allows:

• adjustable output voltage
• sensor circuits
• reference voltages
• control systems

This topic supports DC Circuits.

Conditions for the basic rule

$V_{\mathrm{in}}$ means the actual p.d. across the two series resistors. The simple formula initially assumes no load draws current from the output. It is equal to a source emf only for an ideal source or when internal resistance is negligible.

Core Ideas

• A potential divider is a series circuit, so the same current flows through the divider resistors.
• The p.d. across a chosen resistor is proportional to its resistance.
• The output p.d. depends on the two output points, not just on the component name.
• Sensor dividers convert a changing resistance into a changing output p.d.

Exam Relevance

Potential-divider questions often test whether students choose the correct numerator in the divider formula and whether they can predict output trends when a thermistor, LDR, or variable resistor changes value.

Definition

A potential divider is a series-resistor arrangement that produces an output potential difference equal to a chosen fraction of the input voltage.

Why It Matters

Potential dividers are one of the main ways circuits convert:

• a fixed supply into a chosen output voltage
• a changing resistance into a measurable electrical signal
• temperature, light, or position changes into voltage changes

Key Representations

Figure: In an unloaded divider, the same current flows through both resistors. The output p.d. is measured between two stated terminals, so the numerator is the resistance lying between those terminals, not a memorised “top” or “bottom” resistor.

Divider Principle

For resistors connected in series:

• the same current flows through each resistor
• total potential difference equals the sum of voltage drops

$$V_{\text{in}}=V_1+V_2+\cdots$$

Since:

$$V=IR$$

a larger resistance gets a larger share of the voltage.

Two-Resistor Potential Divider

Two resistors $R_1$ and $R_2$ connected in series across input voltage $V_{\text{in}}$.

Total Resistance

$$R_{\text{total}}=R_1+R_2$$

Current

$$I=\frac{V_{\text{in}}}{R_1+R_2}$$

Voltage Across Each Resistor

$$V_1=I{R}_1=V_{\text{in}}\frac{R_1}{R_1+R_2}$$ $$V_2=I{R}_2=V_{\text{in}}\frac{R_2}{R_1+R_2}$$

Output Voltage Relation

If output is taken across $R_2$:

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

This is the standard divider equation.

It also passes two useful limiting checks:

• if $R_2\to 0$, then $V_{\mathrm{out}}\to 0$;
• if $R_2\gg R_1$, then $V_{\mathrm{out}}\to V_{\mathrm{in}}$.

Ratio Form

For two series resistors:

$$V_1:V_2=R_1:R_2$$

Useful for fast mental checks.

Variable Divider

Replace one resistor with a variable resistor.

Changing resistance changes:

$$V_{\text{out}}$$

Common Uses

• volume controls
• brightness adjustment
• tuning knobs
• calibration controls

A two-terminal variable resistor used to control current is a rheostat. A three-terminal component used to select a fraction of a fixed p.d. is also commonly called a potentiometer or variable potential divider. Do not confuse this component use with the null-method potentiometer experiment in the next branch note.

Figure: A sliding contact divides the resistance track into two series parts. For an unloaded output measured from the slider to the lower end, moving the slider changes the selected resistance fraction and hence varies $V_{\mathrm{out}}$ continuously from $0$ to $V_{\mathrm{in}}$.

Sensor Applications

Potential dividers convert changing resistance into changing voltage.

Thermistor Divider

See also Thermistors and LDRs.

For a common NTC thermistor:

• temperature increases
• resistance decreases

Hence output voltage changes with temperature.

Uses

• thermostat circuits
• temperature alarms
• temperature sensing

LDR Divider

For an LDR:

• light intensity increases
• resistance decreases

Hence output voltage changes with light level.

Uses

• automatic street lighting
• camera light sensing
• security systems

Position Matters

Whether output rises or falls depends on where the sensor resistor is placed.

Example

If output is measured across lower resistor:

• lower resistance increases $\Rightarrow$ output increases
• lower resistance decreases $\Rightarrow$ output decreases

Always inspect circuit arrangement.

Figure: For output from the centre node to $0\mathrm{V}$, $V_{\mathrm{out}}=V_{\mathrm{in}}{R}_{\mathrm{lower}}/(R_{\mathrm{upper}}+R_{\mathrm{lower}})$. An NTC thermistor or LDR decreases in resistance when its stimulus increases. If it is below the output node, $V_{\mathrm{out}}$ falls; if it is above, $V_{\mathrm{out}}$ rises.

The fixed resistor is essential: together, the fixed resistor and sensor form the series pair that turns a resistance change into a p.d. change.

Loaded and Unloaded Divider

Enrichment/application: loading

Loading is useful for real measurement circuits but is not the central statement of the potential-divider syllabus outcome.

Unloaded Divider

No device draws current from output.

Standard divider formula applies directly.

Loaded Divider

Output connected to another component drawing current.

Then:

• effective resistance changes
• the original unloaded-divider formula no longer applies directly
• if the load is connected across the output resistor, the output p.d. is usually lower than the unloaded value

If a load $R_L$ is placed across $R_2$, replace the lower arm by

$$R_{\mathrm{lower},\mathrm{eff}}=R_2\parallel R_L.$$

Then

$$V_{\mathrm{out}}=V_{\mathrm{in}}\frac{R_2\parallel R_L}{R_1+(R_2\parallel R_L)}.$$

Because $R_2\parallel R_L<R_2$, loading lowers the output in this arrangement. Many core questions assume an unloaded divider unless stated otherwise.

Loaded-divider example

Let $R_1=6.0\mathrm{k\Omega }$, $R_2=6.0\mathrm{k\Omega }$, $R_L=6.0\mathrm{k\Omega }$, and $V_{\mathrm{in}}=12\mathrm{V}$. Unloaded, $V_{\mathrm{out}}=6.0\mathrm{V}$. Loaded,

$$R_2\parallel R_L=3.0\mathrm{k\Omega },$$ $$V_{\mathrm{out}}=12\frac{3.0}{6.0+3.0}=4.0\mathrm{V}.$$

Worked Example 1

Two resistors $2\Omega$ and $4\Omega$ connected across $12V$ supply.

Output taken across $4\Omega$ resistor.

$$V_{\text{out}}=12\frac{4}{2+4}$$ $$V_{\text{out}}=8.0V$$

Worked Example 2

A thermistor forms the lower resistor in a divider.

Temperature rises, thermistor resistance decreases.

Result

Output across thermistor:

$$V_{\text{out}}=V_{\text{in}}\frac{R}{R_{\text{top}}+R}$$

As $R$ decreases:

$$V_{\text{out}}\text{ decreases}$$

If the same thermistor were the upper resistor and the output were still measured across the lower fixed resistor, the decrease in upper resistance would instead make $V_{\mathrm{out}}$ increase.

Worked Example 3: Find an Unknown Resistance

An unloaded divider has $V_{\mathrm{in}}=10.0\mathrm{V}$, upper resistance $R_1=3.0\mathrm{k\Omega }$, and $V_{\mathrm{out}}=4.0\mathrm{V}$ across the lower resistor $R_2$.

$$4.0=10.0\frac{R_2}{3.0+R_2},$$

where resistances are in $\mathrm{k\Omega }$. Hence

$$R_2=2.0\mathrm{k\Omega }.$$

The answer is sensible because the output is less than half the input, so the lower resistance should be smaller than the upper resistance.

Common Exam Question Types

1. Calculate Output Voltage

Use divider formula directly.

2. Predict Direction of Change

Determine how resistance changes.

3. Design Sensor Circuit

Choose resistor positions so output increases or decreases as required.

4. Determine Unknown Resistance

Rearrange formula algebraically.

Common Mistakes

1. Using Parallel Formula

Potential dividers use series resistors.

2. Wrong Numerator

Use the resistor across which output is measured.

3. Assuming Equal Split

Voltage divides equally only if resistances are equal.

4. Ignoring Sensor Position

Top and bottom placement changes behaviour.

5. Forgetting Units

Use volts for output.

Quick Comparison Table

Situation	Output Across Lower Resistor
Lower resistance increases	increases
Lower resistance decreases	decreases
Upper resistance increases	decreases
Upper resistance decreases	increases

Links

• DC Circuits
• Current Electricity Fundamentals
• Potentiometer
• Circuit Fault Finding
• DC Circuits Common Exam Traps

Summary

A potential divider converts a fixed input voltage into a controlled output voltage.

Key ideas:

• same current in series
• voltage share proportional to resistance
• changing resistance changes output voltage
• useful for sensors and controls

Master this topic well because it appears frequently in H2 circuit questions.