Potential Divider Principle

1. Definitions (Must Know)

A. Potential divider

A potential divider is a series combination of resistors used to produce a fraction of the input potential difference.

B. Output voltage, Vₒᵤₜ

Vₒᵤₜ is the potential difference across a chosen component (often across R₂).

2. Key Ideas (What Earns Marks)

• For an unloaded two-resistor divider with fixed input Vᵢₙ: Vₒᵤₜ = VᵢₙR₂/R₁ + R₂ if the output is taken across R₂.
• Voltage divides in proportion to resistance because the same current flows through series components: V₁:V₂ = R₁:R₂
• Sensors are often used as one resistor:
  • NTC thermistor: resistance decreases when temperature increases
  • LDR: resistance decreases when light intensity increases

3. Detailed Explanations

A. Derivation of the divider formula

For series resistors:

I = Vᵢₙ/R₁ + R₂

If Vₒᵤₜ is across R₂:

B. How changing resistance changes Vₒᵤₜ

For output across R₂:

• increasing R₂ increases Vₒᵤₜ
• increasing R₁ decreases Vₒᵤₜ

Data table

Sensor as R2 (bottom): Vout/Vin = R2/(1+R2)
Sensor resistance (relative to fixed resistor)	Output fraction, Vout/Vin
0	0.2
1	0.3
1	0.5
2	0.7
4	0.8

Sensor as R1 (top): Vout/Vin = 1/(R1+1)
Sensor resistance (relative to fixed resistor)	Output fraction, Vout/Vin
0	0.8
1	0.7
1	0.5
2	0.3
4	0.2

C. Sensor dividers (NTC thermistor / LDR)

If you replace one resistor with a sensor, Vₒᵤₜ becomes a function of temperature or light.

Typical exam task: choose whether to place the sensor as R₁ or R₂ so that Vₒᵤₜ increases when temperature/light increases.

D. Loading the output

The simple divider formula assumes that the output is unloaded, or that the connected device has a resistance much larger than the divider resistances. If a load RL is connected across R₂, it is in parallel with R₂:

Rₑq = R₂parallel RL = R₂RL/R₂ + RL

Replace R₂ by Rₑq before applying the divider relationship:

Vₒᵤₜ = VᵢₙRₑq/R₁ + Rₑq

Scroll diagram horizontally to read all labels.

For a finite passive load, Rₑq<R₂, so the loaded output is lower than the unloaded output. Current through R₁ now splits between R₂ and RL; it is not correct to treat the same current as flowing through all three resistors.

4. Common Mistakes

• Using the formula but taking Vₒᵤₜ across the wrong resistor.
• Forgetting the divider only holds in this simple form when the output is not significantly loaded by another component.
• Not stating the thermistor/LDR trend (NTC thermistor and LDR both decrease resistance when the stimulus increases).

5. Exam Tips

• If the question is qualitative, use ratios: Vₒᵤₜ propto R₂/R₁ + R₂ and discuss what happens as the sensor resistance changes.
• If the question is quantitative, compute Vₒᵤₜ from the formula after converting units.

6. Worked Examples

Example 1: Simple divider calculationCore

Vᵢₙ = 12 V, R₁ = 3.0 kΩ, R₂ = 1.0 kΩ. Find Vₒᵤₜ across R₂.

Show Answer

Vₒᵤₜ = 121.0/3.0 + 1.0 = 3.0 V

Example 2: Thermistor placement (qualitative)Core

You want Vₒᵤₜ to increase as temperature increases. The circuit is a potential divider with one fixed resistor and one NTC thermistor. Should the thermistor be the top resistor (R₁) or bottom resistor (R₂) if Vₒᵤₜ is across the bottom resistor?

Show Answer

For an NTC thermistor, resistance decreases as temperature increases.

If Vₒᵤₜ is across R₂, we want Vₒᵤₜ = VᵢₙR₂/R₁ + R₂ to increase with temperature.

So we should put the thermistor as R₁ (top resistor). As temperature increases, R₁ decreases, so the denominator decreases and Vₒᵤₜ increases.

Example 3: Output taken across the top resistorCore

Vᵢₙ = 9.0 V, R₁ = 2.0 kΩ, R₂ = 1.0 kΩ. Find the output voltage across R₁.

Show Answer

Across R₁: Vₒᵤₜ = VᵢₙR₁/R₁ + R₂ = 9.02.0/2.0 + 1.0 = 6.0 V

Example 4: Sensor value changes (numerical)Core

A potential divider has Vᵢₙ = 5.0 V, R₁ = 1.0 kΩ, and R₂ is an LDR. The output is taken across R₂.

Find Vₒᵤₜ when:

1. R₂ = 4.0 kΩ (dark)
2. R₂ = 1.0 kΩ (bright)

Show Answer

Use Vₒᵤₜ = VᵢₙR₂/R₁ + R₂.

1. R₂ = 4.0 kΩ: Vₒᵤₜ = 5.04.0/1.0 + 4.0 = 4.0 V

2. R₂ = 1.0 kΩ: Vₒᵤₜ = 5.01.0/1.0 + 1.0 = 2.5 V

Example 5: LDR placement for “Vout increases with light”Core

An LDR’s resistance decreases when light intensity increases. You want Vₒᵤₜ to increase with light.

Should the LDR be R₁ (top) or R₂ (bottom) if Vₒᵤₜ is across the bottom resistor?

Show Answer

If Vₒᵤₜ is across R₂, then: Vₒᵤₜpropto R₂/R₁ + R₂

We want Vₒᵤₜ to increase when light increases (so LDR resistance decreases).

Put the LDR as R₁: as light increases, R₁ decreases, the denominator decreases, so Vₒᵤₜ increases.

Example 6: Loaded divider calculationCore

A divider has Vᵢₙ = 12 V, R₁ = 2.0 kΩ and R₂ = 4.0 kΩ. A load RL = 4.0 kΩ is connected across R₂. Find the unloaded and loaded output voltages.

Show Answer

Unloaded:

Vₒᵤₜ = 124.0/2.0 + 4.0 = 8.0 V

With the load connected:

Rₑq = R₂parallel RL = (4.0)(4.0)/4.0 + 4.0 = 2.0 kΩ

Therefore:

Vₒᵤₜ = 122.0/2.0 + 2.0 = 6.0 V

The result is reasonable: the parallel load reduces the effective lower-branch resistance, so the output falls from 8.0 V to 6.0 V.

7. Mind Stretchers

Mind stretcher 1: Voltmeter loadingExtension

A 9.0 V divider uses R₁ = R₂ = 10 kΩ. A voltmeter with resistance 10 kΩ is connected across R₂. Compare the ideal unloaded output with the voltmeter reading.

Show Answer

Without the meter:

Vₒᵤₜ = 9.010/10 + 10 = 4.5 V

The meter is in parallel with R₂:

Rₑq = 10parallel10 = 5.0 kΩ

So the reading is:

Vₒᵤₜ = 9.05.0/10 + 5.0 = 3.0 V

The meter resistance is not large compared with the divider resistances, so it significantly loads the output.

Mind stretcher 2: When does the simple divider formula fail?Extension

Give one practical condition under which Vₒᵤₜ = VᵢₙR₂/R₁ + R₂ does not predict the measured output accurately, and explain why.

Show Answer

If the output is connected to a device with comparable or low resistance (a heavy load), that device is effectively in parallel with R₂.

This changes the effective resistance across the output and therefore changes the divider ratio, so the simple two-resistor formula no longer applies.

Mind stretcher 3: Variable-divider terminologyExtension

A three-terminal potentiometer used as a variable divider has a resistive track connected across the supply and a movable slider that selects Vₒᵤₜ. Do not confuse this with a slide-wire null-method potentiometer, which is a different measurement circuit.

Mind stretcher 4: Simulation Bridge: Potential Divider LabExtension

Concept Explorer: Potential Divider Lab

Vary supply, resistor values, sensor placement, and output loading to track voltage trends and checkpoint your divider reasoning.

BetaO LevelA LevelElectricityBest for: O Level practical electricity

• Divider Ratio
• Sensor Placement
• Trend Analysis
• Source and Branch Current

Open the full interactive simulation on its own page

Use the standalone simulation page for the live controls, SVG scene, run modes, and scoring flow.

Open Interactive Simulation

The lesson stays lightweight and links out to the dedicated simulation page.

Apply the principle in the Potential Divider Lab. Open More controls, enable Output loading, and compare the unloaded output with different load resistances.

8. Practice, Quiz and Next Step

Close your notes and use Potential Divider Principle in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: An unfamiliar data set or physical system requires you to apply Potential Divider Principle while stating the model, regime and assumptions.

1. Retrieve: define potential divider principle in your own words, including units, sign or conditions where relevant.
2. Represent: Choose and label an appropriate diagram, graph, table or symbolic model; derive or justify the relationship used.
3. Apply: Reach a conclusion, then evaluate it using units, uncertainty, a limiting case and one practical or modelling limitation.

Check the response before looking back

• The model, regime, coordinates and assumptions are explicit.
• The derivation or multi-step reasoning is visible rather than implied.
• The conclusion is tested against units, data quality and a limiting case.
• A practical control, uncertainty or model limitation is evaluated where applicable.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theA-Level Physics course hub orpractice browser for an independent re-test.