What is internal resistance of a cell? Derive an expression for it.

Explanation

Internal resistance of a cell : Resistance offered by the electrolyte of a cell when current flows through it, is known as internal resistance of cell.
Consider a cell of e.m.f. E connected to an external resistance R and a one way key as shown in figure.

Let r = internal resistance of the cell
A voltmeter is connected across the erminals of the cell.
When the circuit is open, then note the reading of the voltmeter which gives us the (E) emf of the cell.
Now close the key and let current I is flowing through the circuit.
Let V= Reading of the voltmeter in closed circuit
I = Terminal potential difference
`:.` R and r in series
`:.` Total resistance of the circuit = (R + r)
`:.` Current flowing through the circuit,
`I =(E)/(R+r)` (Using Ohm.s law) `" "...(1)`
The reading V of the voltmeter (in closed circuit) is less than the reading E (in open circuit) by an amount Ir, which is the fall of potential inside the cell. So Ir is also called lost volt and clearly e.m.f of cell is greater than terminal potential difference.
Thus, V = E – Ir
`V=E-(E/(R+r))r" "` [From eqn. (1)]
`V(R+r)=E(R+r)-Er`
`VR+Vr=ER+Er-Er`
`Vr = ER - VR`
or `Vr = (E-V) R`
or `r=((E-V)/(V))R`
Knowing E, V and R, we can calculate the internal resistance (r) of a cell.