Question

Which of the following equations gives correct expression for the internal resistance of a cell by using potentiometer?

• A. \( r = R \left( \frac{V}{E} - 1 \right) \)
• B. \( r = R \left( 1 - \frac{E}{V} \right) \)
• C. \( r = R \left( 1 - \frac{V}{E} \right) \)
• D. \( r = R \left( \frac{E}{V} - 1 \right) \)

Hint:

The internal resistance causes a "voltage drop" inside the cell, so the terminal voltage V is always less than the emf E when current flows. The formula reflects this difference.

Answer 1

The Correct Option is D

In a potentiometer experiment, we measure the emf (E) of the cell (open circuit) and the terminal voltage (V) when a known external resistance (R) is connected (closed circuit). The emf corresponds to a balancing length \( l_1 \), and the terminal voltage corresponds to a balancing length \( l_2 \). So, \( E \propto l_1 \) and \( V \propto l_2 \).
The relationship between emf, terminal voltage, and internal resistance (r) is \( E = I(R+r) \) and \( V = IR \).
Dividing the two equations gives \( \frac{E}{V} = \frac{R+r}{R} = 1 + \frac{r}{R} \).
Rearranging for r: \( \frac{r}{R} = \frac{E}{V} - 1 \), which gives \( r = R \left( \frac{E}{V} - 1 \right) \). Since \( E/V = l_1/l_2 \), this is also written as \( r = R \left( \frac{l_1}{l_2} - 1 \right) \).