= Voltage divider
{title2=$V_{\mathrm{out}}=V_sR_b/(R_a+R_b)$}
{wiki}
A voltage divider uses two <resistors> connected in series to obtain a fraction of an applied <electric potential difference>. Let the upper <electrical resistance> be $R_1>0$, the lower one $R_2>0$, and the input be $V_{\mathrm{in}}=V_s$ relative to a common <electrical reference potential>. The output is the junction's <electric potential difference> relative to that same reference, across $R_2$. The familiar ratio assumes zero output <electric current>: no external <electrical load> draws from the junction. In a steady <direct current> model with fixed <resistors> obeying <Ohm's law> and an ideal <voltage source>,
$$
I=\frac{V_{\mathrm{in}}}{R_1+R_2},\qquad
V_0=IR_2=V_{\mathrm{in}}\frac{R_2}{R_1+R_2}.
$$
Here $V_0$ is the <open-circuit voltage>; $R_1,R_2$ correspond to $R_a,R_b$ in the header formula. With no junction branch, <Kirchhoff's current law> makes the same <electric current> pass through both <resistors>, and <Ohm's law> gives the two voltage drops. For $V_s\ne0$, division gives $V_0/V_s=R_2/(R_1+R_2)$; if $V_s=0$, the predicted output is zero but this ratio is undefined.
An original stipulated example has $R_1=R_2=10\,\mathrm{k\Omega}$ and $V_{\mathrm{in}}=10\,\mathrm V$. With zero output <electric current>, $I=0.5\,\mathrm{mA}$ and $V_0=5\,\mathrm V$. An attached $10\,\mathrm{k\Omega}$ <electrical load> instead gives $V_{\mathrm{out}}=10/3\,\mathrm V$; a $1\,\mathrm{M\Omega}$ load gives $V_{\mathrm{out}}=1000/201\,\mathrm V=4.975124378\ldots\,\mathrm V$. These are exact model predictions, not measured readings or specifications of a particular <multimeter> or <oscilloscope>.
\Image[/voltage-divider-load-and-readout-comparison.png]
{title=The same voltage divider has three different predicted outputs as the attached load changes}
{description=Original circuit and readout comparison for ideal $10\,\mathrm V$ input and two $10\,\mathrm{k\Omega}$ resistors. Each finite $R_L$ connects across $R_2$; the displayed numbers are model predictions with ideal readout, not experimental data.}
{source=voltage-divider-load-and-readout-comparison.py}
The divider is useful for scaling a voltage or converting a change in a sensor's <electrical resistance> into an <electric potential difference>. Its beauty is also its trap: multiplying both <electrical resistances> by the same positive factor preserves the unloaded ratio, but changes the effect of any fixed <electrical load>. A divider therefore supplies a predictable ratio under specified loading conditions; it is not a source that maintains that voltage for arbitrary output <electric currents>. A <Wheatstone bridge> compares two divider junctions, while <Thévenin's theorem> describes the divider's response to many different loads.
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