Question

The balancing length of a potentiometer is at 120 cm. On shunting the cell with a resistance of 4 ohm, the balancing point shifts to a length of 60 cm. The internal resistance of the cell is:

• A. \( 7 \text{ ohm} \)
• B. \( 12 \text{ ohm} \)
• C. \( 3 \text{ ohm} \)
• D. \( 4 \text{ ohm} \)

Hint:

This is a standard method to measure internal resistance; always ensure the potentiometer is balanced before and after adding the shunt resistance.

Answer 1

The Correct Option is D

Step 1: What the balance length measures.
On a potentiometer the balance length is proportional to the voltage being matched. With no shunt the full EMF $\varepsilon$ balances, so $\varepsilon \propto l_1$.
Step 2: Effect of the shunt.
Connecting a resistor $R$ across the cell makes current flow, so only the terminal voltage $V$ is balanced, and $V \propto l_2$.
Step 3: Internal resistance formula.
The internal resistance follows from $\varepsilon = I r + V$ and $V = IR$, giving \[ r = R\left(\frac{\varepsilon}{V} - 1\right) = R\left(\frac{l_1}{l_2} - 1\right). \]
Step 4: Insert the lengths.
Here $l_1 = 120\,\text{cm}$, $l_2 = 60\,\text{cm}$, $R = 4\,\Omega$.
Step 5: Evaluate the bracket.
$\dfrac{l_1}{l_2} - 1 = \dfrac{120}{60} - 1 = 2 - 1 = 1$.
Step 6: Final value.
$r = 4 \times 1 = 4\,\Omega$.
\[ \boxed{4\ \text{ohm}} \]