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

Fixed and variable potential-divider circuits

The upper circuit has resistors R1 and R2 in series across a supply, with output voltage measured across the lower resistor R2. The lower circuit has a three-terminal potentiometer across a supply, with output measured from its slider to the zero-volt end.

View figure data

Potential-divider topology

Circuit	Connection
Fixed divider	R₁ and R₂ are in series; output is across R₂
Variable divider	The potentiometer track is across the supply; output is taken from its wiper

Potential-divider topology

Circuit	Connection
Fixed divider	R₁ and R₂ are in series; output is across R₂
Variable divider	The potentiometer track is across the supply; output is taken from its wiper

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

Potential divider: choosing sensor position (scaled)

Two divider curves showing how Vout/Vin changes with sensor resistance depending on whether the sensor is placed at R1 (top) or R2 (bottom).

Scroll across the graph to read all labels.

Open full-size graph

View figure data

Values for Potential divider: choosing sensor position (scaled)

Sensor resistance (relative to fixed resistor)	Sensor as R2 (bottom): Vout/Vin = R2/(1+R2)	Sensor as R1 (top): Vout/Vin = 1/(R1+1)
0.25	0.2	0.8
0.5	0.333	0.667
1	0.5	0.5
2	0.667	0.333
4	0.8	0.2

Values for Potential divider: choosing sensor position (scaled)

Sensor resistance (relative to fixed resistor)	Sensor as R2 (bottom): Vout/Vin = R2/(1+R2)	Sensor as R1 (top): Vout/Vin = 1/(R1+1)
0.25	0.2	0.8
0.5	0.333	0.667
1	0.5	0.5
2	0.667	0.333
4	0.8	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 R_L is connected across R₂, it is in parallel with R₂:

R_eq = R₂∥ R_L = R₂R_L/(R₂ + R_L)

Replace R₂ by R_eq before applying the divider relationship:

Vₒᵤₜ = VᵢₙR_eq/(R₁ + R_eq)

For a finite passive load, R_eq < R₂, so the loaded output is lower than the unloaded output. Current through R₁ now splits between R₂ and R_L; 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 ∝ 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

7. Mind Stretchers