Potential Dividers

A potential divider (also called a voltage divider) is a simple circuit arrangement in which two or more resistors are connected in series across a supply voltage. Because resistors in series share the total voltage in proportion to their resistances, you can 'tap off' any fraction of the supply voltage from the junction between the resistors. This makes the potential divider one of the most practically useful circuits in electronics, appearing in everything from volume controls on audio equipment to the sensing circuits that make your smartphone screen respond to ambient light. The key insight is that voltage divides in the same ratio as resistance. If one resistor is twice as large as the other, it takes twice the share of the voltage. This proportionality means you can calculate exactly what output voltage you will get for any chosen pair of resistors — no guesswork required. The output is taken across one of the resistors (usually the lower one), and that voltage depends both on the resistor values and on the total supply voltage provided by the source. Potential dividers become especially powerful when one of the resistors is replaced by a sensor — a component whose resistance changes with a physical quantity such as temperature (thermistor) or light intensity (LDR, light-dependent resistor). As the physical quantity changes, the sensor's resistance changes, which shifts the output voltage. That changing voltage can then trigger alarms, control heaters, or feed data into a microcontroller.

Key takeaways
  • Series resistors share current equally
  • Each resistor claims its share of voltage
  • Deriving the output voltage formula
  • Sensor applications — the thermistor example
  • Loading effects

Series resistors share current equally

When two resistors R₁ and R₂ are connected in series, the same current I flows through both. There is nowhere else for the current to go — no branches, no shortcuts. This shared current is the foundation of the divider's behaviour.

Each resistor claims its share of voltage

By Ohm's law, the voltage across each resistor equals I × R. Since I is the same for both, the resistor with the larger resistance gets the larger share of the total supply voltage V_in. The total voltage is split: V₁ = I·R₁ and V₂ = I·R₂, and V₁ + V₂ = V_in.

Deriving the output voltage formula

The current in the series circuit is I = V_in / (R₁ + R₂). The output voltage V_out is taken across R₂, so V_out = I · R₂ = V_in × R₂ / (R₁ + R₂). This is the potential divider equation. Notice that if R₂ is very small compared with R₁, V_out is nearly zero; if R₂ ≫ R₁, V_out approaches V_in. You control the fraction simply by choosing the ratio R₂ : (R₁ + R₂).

Sensor applications — the thermistor example

Replace R₂ with a thermistor (NTC type — resistance falls as temperature rises). At low temperature the thermistor's resistance is high, so it claims a large share of V_in, and V_out is high. As temperature rises, its resistance falls, and V_out decreases. A comparator or microcontroller monitoring V_out can therefore detect temperature changes accurately. The fixed resistor R₁ is chosen to maximise the sensitivity of V_out over the temperature range of interest — typically R₁ is set equal to the thermistor's resistance at the midpoint of the expected range.

Loading effects

In a real circuit, connecting a load (e.g. a voltmeter or the next stage of a circuit) across R₂ places another resistance in parallel with R₂. This reduces the effective lower resistance and therefore lowers V_out below the theoretical prediction. To minimise loading, the load resistance should be much larger than R₂, or the divider resistors should be chosen to be much smaller than the load — a practical compromise since very small resistors waste more power from the supply.

The Key Equation

Vout=Vin×R2R1+R2V_{\text{out}} = V_{\text{in}} \times \frac{R_2}{R_1 + R_2}
  • V_{\text{out}} — Output voltage (Volt (V)): The voltage measured across the lower resistor R₂; the 'tapped' fraction of the supply.
  • V_{\text{in}} — Input (supply) voltage (Volt (V)): The total EMF or voltage provided by the source across the whole series combination.
  • R_1 — Upper resistor (Ohm (Ω)): The resistor connected between the positive terminal of the supply and the output junction.
  • R_2 — Lower resistor (Ohm (Ω)): The resistor across which V_out is measured; may be a fixed resistor or a sensor component.

Worked Example

Problem. A 12 V battery is connected to a potential divider made of a 3300 Ω fixed resistor (R₁) in series with an LDR (R₂). In dim light the LDR has a resistance of 8200 Ω. (a) Calculate the output voltage V_out across the LDR in dim light. (b) In bright sunlight the LDR drops to 150 Ω. Calculate the new V_out and explain whether this circuit could be used to detect the transition from darkness to daylight, assuming a trigger threshold of 1 V.

Vout,dim=12×820011500≈8.55 V,Vout,bright=12×1503450≈0.52 VV_{\text{out,dim}} = 12 \times \frac{8200}{11500} \approx 8.55\text{ V}, \quad V_{\text{out,bright}} = 12 \times \frac{150}{3450} \approx 0.52\text{ V}

Answer. V_out (dim) ≈ 8.55 V; V_out (bright) ≈ 0.52 V. The circuit successfully distinguishes between the two conditions across the 1 V threshold.

Figure
Diagram
Potential Divider CircuitV_in12 VR₁3300 ΩR₂(LDR/sensor)VmeterV_outV₁ across R₁V₂ = V_out+−
Origins · 1 minCharles Wheatstone and how this was discovered
CW
Charles Wheatstone
1802–1875 · British

Wheatstone is best known for popularising the Wheatstone bridge — a precision resistance-measurement network that is itself a balanced pair of potential dividers. His systematic work on resistive circuits in the 1840s, including careful analysis of how voltage distributes across series resistances, underpins the modern understanding of potential dividers. He also made fundamental contributions to telegraphy, stereoscopy, and the measurement of electrical resistance. Wheatstone was famously shy about public speaking. When he was due to lecture at the Royal Institution in London, he reportedly fled the building just minutes before taking the stage, leaving his friend Michael Faraday to deliver the lecture in his place!

  • Ohm's Law — The potential divider equation is derived directly from Ohm's law: V = IR applied to each resistor in the series combination.
  • Series Circuits — A potential divider is a series circuit; understanding that current is constant and voltages add in series is essential to deriving and using the divider equation.
  • Parallel Circuits — Loading effects occur when a load is connected in parallel with R₂, altering the effective resistance and reducing V_out below the ideal prediction.
  • Thermistors and LDRs — Sensors such as thermistors and LDRs are placed in potential divider circuits so that their changing resistance produces a measurable changing voltage output.
  • Electromotive Force and Internal Resistance — Internal resistance acts like an upper resistor in a potential divider, causing terminal voltage to drop below EMF when current flows — a direct application of divider principles.