In a potentiometer experiment, it is found that no current passes through the galvanometer when the terminals of the cell are connected across 52cm of the potentiometer wire. If the cell is shunted by a resistance of 5Ω, a balance is found when the cell is connected across 40cm of the wire. Find the internal resistance of the cell.

Let E be the emf of the cell and r be its internal resistance.
When the cell is connected directly to the potentiometer, the balance length is 52 cm. Therefore, E is proportional to 52 cm.
E ∝ 52 cm
When the cell is shunted by a resistance of 5Ω, the balance length is 40 cm. The effective emf of the shunted cell is given by:
E' = E * (R/(R+r)) where R is the shunt resistance (5Ω).
E' ∝ 40 cm
Since E ∝ 52 and E' ∝ 40, we can write the ratio:
E/E' = 52/40 = 13/10
Substituting the expression for E':
E / [E * (R/(R+r))] = 13/10
1 / [R/(R+r)] = 13/10
(R+r)/R = 13/10
1 + r/R = 13/10
r/R = 13/10 - 1 = 3/10
r = (3/10) * R
r = (3/10) * 5Ω
r = 1.5Ω
Therefore, the internal resistance of the cell is 1.5Ω.