Potential dividers

In a nutshell

Potential dividers are a type of circuit that can vary the potential difference across an output.

Equations

Description	equation
Ratio of resistances	$\dfrac{V_1}{V_2}=\dfrac{R_1}{R_2}$
Potential divider equation	$V_{out}=\dfrac{R_2}{R_1+R_2} V_{in}$

Variable definitions

quantity name	symbol	derived units	alternate units	si base units
$resistance$	$R$	$\Omega$	$V\ A^{-1}$	$kg\ m^2\ s^{-3}\ A^{-2}$
$potential\ difference$	$V$	$V$	$J\ C^{-1}$	$kg\ m^2 \ s^{-3}\ A^{-1}$

Ratio of resistances

Take a simple circuit as shown below:

Physics; Electrical circuits; KS5 Year 12; Potential dividers

According to Kirchhoff's second law, the potential difference across $R_1$ and $R_2$ (which you can call $V_1$ and $V_2$ ) is equal to the e.m.f. of the power source. These potential differences are split between the resistors according to the ratio of their resistances. Therefore one can write:

$\dfrac{V_1}{V_2}=\dfrac{R_1}{R_2}$

Potential dividers

When you need a specific amount of potential difference for a task but you have a power source with a higher e.m.f., you can use a potential divider. This is a type of circuit that can vary the potential difference across an output by using the ratio of resistances.

Below is a diagram of a simple potential divider:

Physics; Electrical circuits; KS5 Year 12; Potential dividers

By considering the total potential difference $V_{in}$ and the ratio between $R_1$ and $R_2$ , it is possible to know the output voltage $V_{out}$ across $R_2$ through the equation:

$V_{out}=\dfrac{R_2}{R_1+R_2} V_{in}$

You can then change resistors and regulate the output voltage. This can also be done through a potentiometer which is a type of variable resistor with three terminals that allows you to regulate voltage more freely. Potential dividers that use such components are for example used to control volume on a stereo.

Note: $V_{in}$ does not necessarily need to be the e.m.f. of the power source.

Example

A potential divider is composed of a battery and two resistors. The total potential difference is $12\ V$ and the resistors have resistance $5\ \Omega$ and $10\ \Omega$ . What is the voltage output from the $10\ \Omega$ resistor?

Firstly, identify all the variables:

$\begin{aligned}V_{in}&=12\ V \\R_1&=5\ \Omega \\R_2&=10\ \Omega\end{aligned}$ 

Next, write down the equation for potential dividers:

$V_{out}=\dfrac{R_2}{R_1+R_2} V_{in}$ 

Substitute all the numbers into the equation and calculate the output voltage:

$V_{out}=\dfrac{10}{5+10}\times12=8\ V$ 

The output voltage of this potential divider is $\underline{8\ V}$ .