Question

Two identical cells gave the same current through an external resistance of \( 2 \)$\omega$ regardless of whether the cells are grouped in series or parallel. The internal resistance of the cells is:

• A. \( 1 \) $\omega$
• B. \( 0.5 \) $\omega$
• C. \( 1.5 \) $\omega$
• D. \( 2.0 \) $\omega$

Hint:

When identical cells produce the same current in series and parallel, use the formula: \[ R = 2r. \] This condition allows quick identification of the internal resistance.

Answer 1

The Correct Option is D

Step 1: Understanding the Given Condition - The current remains the same whether the cells are connected in series or parallel. - Let the EMF of each cell be \( E \) and the internal resistance of each cell be \( r \). - The external resistance is \( R = 2 \)$\omega$.
Step 2: Equating Current in Series and Parallel Cases Case 1: Cells in Series \[ I_{\text{series}} = \frac{2E}{R + 2r}. \] Case 2: Cells in Parallel \[ I_{\text{parallel}} = \frac{E}{R + \frac{2r}{2}} = \frac{E}{R + r}. \] Since both currents are equal, we equate: \[ \frac{2E}{R + 2r} = \frac{E}{R + r}. \]
Step 3: Solving for \( r \) Cancel \( E \) from both sides: \[ \frac{2}{R + 2r} = \frac{1}{R + r}. \] Cross multiplying: \[ 2(R + r) = R + 2r. \] Expanding: \[ 2R + 2r = R + 2r. \] Cancel \( 2r \) from both sides: \[ 2R = R. \] \[ R = 2r. \] Since \( R = 2 \)$\omega$, we get: \[ 2 = 2r. \] \[ r = 2 \text{ $\omega$}. \] Thus, the correct answer is: \[ \boxed{2.0 \text{ $\omega$}}. \]