Resistor Voltage Divider✓ verified

Understanding the Parameters

• $V_{in}$ drives a single current through the two series resistors.

• $R_1$ and $R_2$ set the division ratio only through their proportion $R_2/(R_1+R_2)$ — doubling both leaves $V_{out}$ unchanged but halves the current (and the power drawn).

• $V_{out}$ is the unloaded output; drawing current from the node reduces it (see the loading note).

Derivation (from Ohm's Law + Kirchhoff)

In a series connection the same current flows through both resistors. By Kirchhoff's voltage law the input divides across them, and by Ohm's Law that shared current is

$$I = \frac{V_{in}}{R_1 + R_2}.$$

The output is the voltage across $R_2$, again by Ohm's Law:

$$V_{out} = I R_2 = V_{in}\,\frac{R_2}{R_1 + R_2}.$$

So the divider is simply Ohm's Law applied twice; the fraction $R_2/(R_1+R_2)$ is the share of the total resistance that $R_2$ occupies. See Ohm's Law solve for current and Kirchhoff's Laws.

Loading effect (why "unloaded" matters). The formula assumes no current is drawn from the output node. A real load $R_L$ appears in parallel with $R_2$, replacing $R_2$ by $R_2 \parallel R_L$ and lowering $V_{out}$. The divider behaves well only when $R_L \gg R_2$; its Thévenin output resistance is $R_1 \parallel R_2$.

History

The voltage divider is a direct corollary of Ohm's Law (1827) and Kirchhoff's circuit laws (1845), not a separately "discovered" result. Its adjustable form — the potentiometer — became a standard laboratory and control component in the late 19th century and remains ubiquitous as a volume/position control and reference‑voltage source.

Related Concepts: Ohm's Law solve for current, Kirchhoff's Laws, Watt's Law

Notes: Registry calculator resistor-divider (human‑verified). Output is the open‑circuit value; account for load and source resistance in precision use.