Cells always offer resistance to the flow of current through them, which is often very small. This resistance is called the internal resistance of the cell and depends on the size of the cell - the area of the plates immersed in the liquid, the distance between the plates and strength of electrolyte used in the cell.
A resistance box R with a key K1 is connected in parallel with the cell. The primary circuit has a standard cell, a rheostat and a one way key K. As soon as key K is closed, a current I begins to flow through the wire AB. The key K1 is kept open and on moving the jockey, a balance is obtained with the cell E1 at point, say Y1. Let AY1 = L1.
E1 = kL1
Now key K1 is closed. This introduces a resistance across the cell. A current, say I1, flows in the loop E1RK1E1 due to cell E. Using Ohm’s law,
I1 = E1/(R+r)
It means that terminal potential difference V1 of the cell will be less than E1 by an amount I1r.
V1 = I1R
Potential difference V1 is balanced on the potentiometer wire without change in current I. Let the balance point be at point Y2 such that AY2 = L2.
V1 = kL2
E1/V1 = L1/L2 = (R+r)/R
r = R(L1/L2 - 1)