The Potential Divider

The idea — two resistors in series across a supply divide the p.d. between them in proportion to their resistances. Taking the output across one of them gives a smaller, controllable p.d.

V_out = V_in × R₂ ÷ (R₁ + R₂)

where R₂ is the resistor the output is taken across.

Example: a 12 V supply across a 4.0 kΩ and a 2.0 kΩ resistor in series. The p.d. across the 2.0 kΩ resistor is 12 × 2.0 ÷ 6.0 = 4.0 V. Note it depends only on the ratio of the resistances — two resistors of 4 Ω and 2 Ω would give exactly the same output.

Why the ratio is what matters — the same current flows through both resistors in series, so their p.d.s are in the same proportion as their resistances. Doubling both resistances halves the current and leaves every p.d. unchanged, though it does reduce the power wasted.

The potentiometer — a single resistive track with a sliding contact, giving a continuously variable output from zero up to the full supply p.d. Volume controls and dimmer switches work this way.

Sensor circuits — replacing one resistor with a thermistor or an LDR makes the output p.d. vary with temperature or light, producing a signal a circuit can act on.

Working out which way the output moves — take it in two steps. First decide what happens to the sensor's resistance, then apply the divider equation. If an LDR is in the position of R₁ and light increases, its resistance falls, so a larger share of the supply appears across R₂ — the output rises. Swap the two components and the output falls instead, which is how the same pair of parts makes either a light-activated or a dark-activated switch.

Loading the output — connecting a device across the output puts it in parallel with R₂, lowering the resistance of that part of the divider and so reducing V_out below the calculated value. The effect is small if the device has a much higher resistance than R₂, which is why it is usually ignored in questions but matters in real circuits.