Voltage Dividers

Drop two resistors in series across a battery, tap the wire between them, and you've built a voltage divider — the most common way to make a smaller voltage out of a bigger one. It's Ohm's Law wrapped in a naming convention, but naming it is half the battle, because you stop deriving and start recognising.

The formula

For two resistors in series — $R_1$ (top) and $R_2$ (bottom) — across a source $V_{in}$, the voltage measured across $R_2$ is:

That's the whole topic in one line. Every derivation below just relabels this formula.

Where it comes from

In a series loop, current is $I = V_{in} / (R_1 + R_2)$. The voltage across $R_2$ is then $V_{out} = I \cdot R_2$. Substitute and the $R_2 / (R_1 + R_2)$ ratio pops out.

That ratio is the divider ratio. It's always between 0 and 1 — you can only divide $V_{in}$ down, never amplify it (resistors don't add energy). When $R_2 = R_1$ the ratio is ½ and $V_{out}$ is half the input. When $R_2 \gg R_1$ the ratio approaches 1 and most of the source sits across $R_2$. When $R_1 \gg R_2$ the ratio approaches 0 and only a sliver of $V_{in}$ reaches the output.

Picking the resistor pair

For a given divider ratio, there are infinitely many$(R_1, R_2)$ pairs that work — $(1\,k, 1\,k)$, $(10\,k, 10\,k)$, $(1\,M, 1\,M)$ all give a ½ divider. Two considerations pick the sweet spot:

• Current budget. Smaller resistors draw more current $(I = V_{in}/(R_1+R_2))$. That wastes power and can overheat components. In a 12 V divider, a 10 Ω pair pulls 600 mA and burns 7.2 W of heat; a 10 kΩ pair pulls 0.6 mA and burns 7.2 mW.
• Loading effect. Larger resistors are more sensitive to anything you connect downstream. If a load has resistance comparable to $R_2$, the divider output sags because the load sits in parallel with $R_2$. Rule of thumb: make $R_2$ at least 10× smaller than the load it drives.

“Big enough to save power, small enough to hold the output steady” — that's the design tension in one sentence.

Common uses

• Reference voltage. Generate a specific bias for a transistor, an ADC reference, or an op-amp input from a single higher rail.
• Sensor readout. A thermistor or photoresistor paired with a fixed resistor forms a divider whose output tracks the sensor's resistance — cheap analogue measurement with one wire.
• Level shifting. Safely bring a 5 V logic signal down to 3.3 V for a pin that can't tolerate the higher voltage.