To measure the internal resistance of a battery, the potentiometer is used. For R = 10 Ω, the balance point is observed at ℓ = 500 cm and for R = 1Ω the balance point is observed at ℓ = 400 cm. The internal resistance of the battery is approximately :

Concept:

The internal resistance of a battery can be measured using a potentiometer by comparing the potential drops across known resistances in an external circuit. The balance point of the potentiometer corresponds to the condition where the potential drop across the external resistor equals the potential drop across the potentiometer wire.

Equations used,

• The potential drop across the potentiometer wire:
  • Vpot = kl,
  • where:
  • Vpot is the potential difference across the potentiometer wire,
  • k is the potential gradient (volts per cm),
  • l is the length at the balance point.
• Emf and Internal Resistance:
  • The internal resistance r of the battery can be found using the relationship:
  • Vterm = E - I r
  • Where:
  • E is the emf of the battery,
  • Vterm is the terminal voltage across the external resistor,
  • I is the current through the circuit,
  • r is the internal resistance.

Calculation:

Let the potential gradient be λ.

\( i \times 10 = \lambda \times 500 = \varepsilon - ir_s \)

\( \Rightarrow 500\lambda = \varepsilon - 50\lambda r_s \)

Also,

\( i' \times 1 = \lambda \times 400 = \varepsilon - i' r_s \)

\( \Rightarrow 400\lambda = \varepsilon - 400\lambda r_s \)

\( \therefore 100\lambda = 350\lambda r_s \Rightarrow r_s = \frac{10}{35} \approx 0.3\Omega \)

Hence, the correct answer is (4) 0.3 Ω.