What is a potentiometer? Explain the principle of a potentiometer.

(N/A) potentiometer is a device used to measure potential difference or to compare electromotive forces $(EMF)$ of cells. It is also used to measure internal resistance of a cell.
Principle:
As shown in the figure,a battery with $EMF$ $\varepsilon$ and internal resistance $r$ is connected in series with a resistance box $R$ and a uniform wire $AB$ of length $L$ and resistance per unit length $\rho$.
The total resistance of the circuit is $R_{total} = R + L\rho + r$.
The current flowing through the wire $AB$ is $I = \frac{\varepsilon}{R + L\rho + r}$.
If the length of a segment $AC$ of the wire is $l$,then the resistance of segment $AC$ is $R_{AC} = \rho l$.
The potential difference $V$ across the segment $AC$ is given by:
$V = I \cdot R_{AC} = I \rho l$
Substituting the value of $I$:
$V = \left( \frac{\varepsilon \rho}{R + L\rho + r} \right) l$
Since $\varepsilon, \rho, R, L,$ and $r$ are constants for a given setup,we can write $V = \phi l$,where $\phi = \frac{\varepsilon \rho}{R + L\rho + r}$ is known as the potential gradient.
Principle Statement: The potential difference across any segment of a uniform wire carrying a constant current is directly proportional to the length of that segment.
The unit of potential gradient is $V \cdot m^{-1}$ and its dimensional formula is $[M^1 L^1 T^{-3} A^{-1}]$.