You have a 9\ \text{V} battery, but the little sensor you want to drive needs exactly 5\ \text{V}. You have a 12\ \text{V} supply, but the microphone input expects a whisper of a volt. How do you tap off a smaller, chosen slice of a voltage you already have — without a fancy converter, using nothing but two resistors?

The answer is the potential divider: two (or more) resistors joined in series across a supply. Because they carry the same current, the supply voltage is shared out between them in proportion to their resistances. Put a wire between the two resistors and you can pick off any fraction of the supply you like — from nearly nothing up to nearly the whole thing. It is one of the most useful little circuits in all of electronics: it sets reference voltages, it is the volume knob on a stereo and the dimmer on a lamp, and — swap one resistor for a thermistor or LDR — it is the heart of every temperature and light sensor.

Where the equation comes from

Put two resistors R_1 and R_2 in series across a supply V_\text{in}. The two series rules do all the work. First, the resistances add, so the current in the single loop is

That same current flows through both resistors. The potential difference across each one is then just V = IR applied to that resistor:

If we take our output from across R_2 — a wire from the point between the resistors, and a wire from the bottom of R_2 — then V_\text{out} = V_2. Notice the two shares add straight back to the whole supply, exactly as Rule 2 demands: V_1 + V_2 = V_\text{in}. Nothing is lost; the supply is simply divided.

Sharing in proportion

The whole idea lives in one clean formula: the fraction of the supply you tap off is the same as the fraction of the total resistance sitting in your output resistor.

Read it as a story. The bigger resistor grabs the bigger share of the push, because the same current squeezing through a larger resistance drops a larger voltage. Two useful limits check the formula at a glance: if R_2 \gg R_1 the fraction approaches 1 and V_\text{out} \to V_\text{in} (nearly all the supply lands on R_2); if R_2 \ll R_1 the fraction approaches 0 and V_\text{out} \to 0. Everything in between is a tunable slice of the supply.

Worked examples

Example 1 — read off an output. A 12\ \text{V} supply sits across R_1 = 3\ \text{k}\Omega and R_2 = 1\ \text{k}\Omega, and we tap the output across R_2. Then

The units of resistance cancel in the ratio, so kΩ against kΩ is fine — you never have to convert. The small resistor took the small share.

Example 2 — design for a target. We have a 9\ \text{V} supply and want V_\text{out} = 5\ \text{V} across R_2. Choosing R_2 = 10\ \text{k}\Omega, what must R_1 be? Rearrange the divider equation for R_1:

Check: 9 \times \frac{10}{8 + 10} = 9 \times \frac{10}{18} = 5\ \text{V}. Exactly the output we wanted. Designing a divider is nothing more than this rearrangement.

Example 3 — the ratio shortcut. Two resistors in series across 20\ \text{V} are in the ratio R_1 : R_2 = 3 : 2. Without knowing the actual ohms, the voltage splits in the same ratio, and the shares must total 20\ \text{V}:

Because only the ratio matters, 3\ \Omega and 2\ \Omega give the same split as 3\ \text{M}\Omega and 2\ \text{M}\Omega. (Bigger resistors just draw a smaller current — handy when you want the divider to waste less energy.)

Play with a divider

Here is the divider itself: the supply V_\text{in} across R_1 (top) and R_2 (bottom) in series, with a voltmeter reading the output V_\text{out} tapped across R_2. Drag the sliders and watch the equation breathe:

• Raise R_2 and its share climbs — the output bar grows towards the dashed full-supply mark.
• Raise R_1 instead and R_2's share shrinks — the same current now drops more of its voltage up top.
• Set R_1 = R_2 and the supply splits exactly in half, whatever the supply is.

The potentiometer: a divider you can turn

So far R_1 and R_2 have been two separate resistors. But a potentiometer ("pot") is a single track of resistance with a sliding contact — a wiper — that you turn or slide. The wiper splits the one track into two parts: everything above it acts as R_1, everything below as R_2. As you turn the knob, resistance moves smoothly from one side to the other, so

sweeps continuously from 0 up to V_\text{in} as you rotate from one end to the other. That is a variable potential divider — an output you dial in by hand. Every volume knob, every light dimmer, every old-style joystick and every sliding fader on a mixing desk is exactly this: a potentiometer picking off an adjustable fraction of a fixed supply, which the rest of the circuit then reads.

Sensor dividers: light and heat that switch things on

The real magic happens when you replace one of the resistors with a thermistor or LDR — a resistor whose value changes with its surroundings. Now the split shifts on its own as the world changes, turning "it got hot" or "it went dark" into a moving output voltage that a switching circuit can act on.

• A temperature alarm. Put a thermistor as R_1 and a fixed resistor as R_2. As it gets hot the thermistor's resistance falls, so it keeps a smaller share and more of the supply appears across the fixed resistor — V_\text{out} rises. Wire the output to a switch and it trips when the temperature crosses a set point.
• A dusk-sensing light. Put an LDR as R_2 and take the output across it. As night falls the LDR's resistance rises, so it grabs a bigger share and V_\text{out} climbs. When it passes the switching level the lamp comes on — then reverses at dawn.

The direction of the switching is entirely in your hands: which component you make the output resistor decides whether "hotter", "colder", "brighter" or "darker" is the event that raises the output. Same equation, four different alarms — just swap what you tap across.

Worked example — a thermistor divider. A 6\ \text{V} supply feeds a thermistor in series with a fixed 2\ \text{k}\Omega resistor, output taken across the fixed resistor. On a cold morning the thermistor reads 10\ \text{k}\Omega; warmed by a fire it drops to 1\ \text{k}\Omega. The output goes from

A single degree of heating has swung the output from 1\ \text{V} to 4\ \text{V} — a big, clean signal for a switch to latch onto. That swing, from a two-resistor circuit, is why potential dividers are everywhere.