The voltage divider

Series resistors share the voltage

Put two resistors in series across a battery: one path, from the battery's top (+) through the top resistor R1, then the bottom resistor R2, down to ground (0 V) and back. The same current flows through both, and each one takes a share of the battery's voltage. Each share is the current times that resistor (V = I × R), so the bigger resistor takes the bigger share, and the shares add up to the battery's voltage.

Tap the middle: a voltage divider

The wire between R1 and R2 sits partway between the battery's top and ground. Connect an output wire there and you have a voltage divider: the output, V out, is the voltage across the bottom resistor, measured from ground.

• Two equal resistors split the voltage in half: a 9 V battery gives 4.5 V.
• Make the bottom resistor twice the top one (R and 2R) and it takes two of the three shares: 9 V gives 6 V.

The rule

The output equals the input, times the bottom resistor, divided by both resistors added together:

V out = V in × R2 ÷ (R1 + R2)

Check the 6 V example: 9 × 2R ÷ (R + 2R) = 9 × 2 ÷ 3 = 6 V. Because R2 ÷ (R1 + R2) is always less than 1, a divider can only give a voltage between 0 V and the input. It can never raise the voltage.

Potentiometers: an adjustable divider

A potentiometer is one resistor with a sliding contact, the wiper, that can move along it. The two ends connect across the input, and the wiper is the tap. Turn the knob, the wiper slides, and the split between "top" and "bottom" changes, so the output moves anywhere from 0 V to the input. Many volume knobs work this way (some newer volume controls are digital instead).

Try it yourself

If you have a 9 V battery, two equal resistors (say 10 kΩ each) and a multimeter, connect the resistors end to end across the battery and measure from the battery's minus terminal to the middle joint. You should read about half the battery's voltage. Swap the bottom resistor for one twice as big and measure again. A fresh 9 V battery often reads a little above 9 V, so your numbers will scale with it.

Good to know

• The formula V out = V in × R2 ÷ (R1 + R2) uses R2 as the resistor nearest ground.
• Equal resistors give exactly half the input.
• "Any lower voltage" means any voltage between 0 V and the input; a divider can't raise voltage.
• The formula assumes nothing (or very little current) is drawn from the tap. Connecting a load changes the output, so dividers are used for signals and reference levels, not to power devices like motors or phones; that needs a voltage regulator.
• Potentiometers are often wired as voltage dividers in volume controls; some volume controls are digital.
• Battery voltages only: never experiment with mains (wall outlet) power.

Answer to the comment question

Twelve volts in, ten ohms on top, twenty on the bottom. What's the output? 8 volts. V out = 12 × 20 ÷ (10 + 20) = 12 × 20 ÷ 30 = 8 V. The bottom resistor is twice the top one, so it takes two of the three shares: two thirds of 12 V is 8 V.