The internal resistance of a cell by potentiometer is given by

Explanation

Principle : A cell of emf E and internal resistance r, which is connected to an external resistance R, has its terminal potential difference V less than its emf E, and
`E/V=(R+r)/R=1+r/R" "("where "R rarr oo, V rarr E)`
`:. R=(E-V)/V R`
Working : A battery of stable emf E' is used to set up a potential gradient `V_(AB)//L` along the potentiometer wire, where `V_(AB) equiv p.d`. across total length L of the wire AB. The positive terminal of the cell of emf E and internal resistance r is connected to the higher potential terminal A of the potentiometer, the negative terminal is connected through a centre-zero galvanometer to a pencil jockey. A resistance box R with a plug key K in series is counted across the cell.

Firstly, key K is kept open, then, effectively, `R=oo`. The kockey is tapped on the potentiometer wire to locate the null point D. Let the null length `AD=l`, so that
`E=(V_(AB)//L)l`
With the same potential gradient, and a small resistance R in the resistance box, key K is closed. The new null length `AD'=l_(1)` for the terminal p.d. V is found : `V=(V_(AB)//L)l_(1)`
`:. E/V=l/l_(1)` or `(E-V)/V=(l-l_(1))/l_(1)=l/l_(1)-1`
Now, `r=(E-V)/V R" ":. r=R(l/l_(1)-1)`
R, l and `l_(1)` being known, e can be calculated. The experiment is repeated either with different potential gradients or with different values of R.