Derive a relationship between emf, terminal voltage and internal resistance.

Hint:All voltage sources are having two major components – an internal resistance r and a source of electrical energy that has a characteristic of electromotive force (emf). So basically, the emf is the potential difference of a source when there is no current in it or not current is flowing through it. And second thing is that the internal resistance r of the voltage source affects the output voltage when current flows in it.

Formula used:
The voltage output of a source or device is said to be its terminal voltage V and it is given as $V=emf-Ir$
Where $I$= electric current and measured (+) ve when flowing away from the positive terminal of the voltage source.

Complete answer:
Let $V$be the terminal voltage and $E$be the emf. Having a load $R$ and current flowing $I$and internal resistance $r$ can be calculated as –
$I=\dfrac{E}{R+r}$
And the potential difference across the terminal voltage calculated as:
$V=I.R=\dfrac{ER}{R+r}$

Additional Information:
The terminal voltage of a battery is less than the electromotive force(emf) when it is discharging because of the voltage drop across the internal resistance of the device. If current is passing through the battery “backwards” from positive pole to negative pole then the terminal voltage can be calculated as the submission of emf of the battery and voltage drop across the internal resistance.

Note:
Don’t get confused between emf and terminal voltage as the electromotive force(emf) is the potential difference of a battery when no current is flowing from the source in case of open circuit while terminal voltage is the voltage output across the terminal of a device. In short, EMF is the cause and Voltage or potential difference is the effect.