# Voltage Dividers

## Extra Credit: Proof

If you haven’t yet gotten your fill of voltage dividers, in this section we’ll evaluate how Ohm’s law is applied to produce the voltage divider equation. This is a fun exercise, but not super-important to understanding what voltage dividers do. If you’re interested, prepare for some fun times with Ohm’s law and algebra.

### Evaluating the circuit

So, what if you wanted to measure the voltage at V_{out}? How could Ohm’s law be applied to create a formula to calculate the voltage there? Let’s assume that we know the values of V_{in}, R_{1}, and R_{2}, so let’s get our V_{out} equation in terms of those values.

Let’s start by drawing out the currents in the circuit–I_{1} and I_{2}–which we’ll call the currents across the respective resistors.

Our goal is to calculate V_{out}, what if we applied Ohm’s law to that voltage? Easy enough, there’s just one resistor and one current involved:

Sweet! We know R_{2}’s value, but what about I_{2}? That’s an unknown value, but we do know a little something about it. We can assume (and this turns out to be a big assumption) that **I _{1} is equivalent to I_{2}**. Alright, but does that help us? Hold that thought. Our circuit now looks like this, where

*I*equals both I

_{1}and I

_{2}.

What do we know about V_{in}? Well, V_{in} is the voltage across both resistors R_{1} and R_{2}. Those resistors are in series. Series resistors add up to one value, so we could say:

And, for a moment, we can simplify the circuit to:

Ohm’s law at its most basic! V_{in} = I * R. Which, if we turn that *R* back into *R _{1} + R_{2}*, can also be written as:

And since I is equivalent to I_{2}, plug that into our V_{out} equation to get:

And that, my friends, is the voltage divider equation! The output voltage is a fraction of the input voltage, and that fraction is R_{2} divided by the sum of R_{1} and R_{2}.