Figure 3.34 shows a potentiometer circuit for comparison of two resistances. The balance point with a standard resistor R = 10.0 Ω is found to be 58.3 cm, while that with the unknown resistance X is 68.5 cm. Determine the value of X. What might you do if you failed to find a balance point with the given cell of emf ε ?

Resistance of the standard resistor, R = 10.0 Ω

Balance point for this resistance, l₁ = 58.3 cm

Current in the potentiometer wire = i

Hence, potential drop across R, E₁ = iR

Resistance of the unknown resistor = X

Balance point for this resistor, l₂ = 68.5 cm

Hence, potential drop across X, E₂ = iX

The relation connecting emf and balance point is,

E₁/E₂ =l₁/l₂

=> iR/iX =l₁/l₂

X= (l₁/l₂ ) x R

= 68.5/58.3 x 10 = 11.749 Ω

Therefore, the value of the unknown resistance, X, is 11.75Ω.

If we fail to find a balance point with the given cell of emf, ε , then the potential drop across R and X must be reduced by putting a resistance in series with it. Only if the potential drop across R or X is smaller than the potential drop across the potentiometer wire AB, a balance point is obtained.