Basic principle

A potential divider circuit is typically composed of two or more resistors connected in series across a voltage source.
The input voltage ($V_{in}$) from the power source is divided across these resistors.
The voltage across each resistor is proportional to its resistance. This is based on the principle that in a series circuit, the current is the same through all the components, and the voltage drop across each component is dictated by Ohm's law (V=IR).

Deriving the potential divider equation

Consider a basic potential divider with two resistors, $R_{1}$ and $R_{2}$, connected in series to a source voltage . The output voltage $V_{out}$ is measured across $R_{2}$.
The same current (I) flows through both resistors because they are in series.
The total resistance of the circuit is $ R_{total} = R_1 + R_2 $.
Using Ohm’s law, the current in the circuit is $ I = \frac{V_{in}}{R_{total}} $, which means $ I = \frac{V_{in}}{R_1 + R_2} $.
The output voltage ($V_{1out}$), which is the voltage across $R_{2}$, can be expressed as $ V_{\text{out}} = I \times R_2 $.
Substituting the value of I, we get the potential divider equation: $V_{\text{out}} = \left(\frac{R_2}{R_1 + R_2}\right) \times V_{\text{in}}$

Fixed Potential Divider: This type uses two or more fixed resistors. The output voltage ($V_{out}$) is a specific fraction of the input voltage and is constant.
Variable Potential Divider: A variable resistor (rheostat or potentiometer) can be used. By adjusting the position of the sliding contact, the output voltage can be changed.
A variable potential divider changes $V_{out}$ by changing the value of $R_{2}$ and $R_{1}$