Potential dividers

​​In a nutshell

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

Equations

Variable definitions

Ratio of resistances

Take a simple circuit as shown below:

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:

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 $$\underline{8\ V}$$.