Question

For two identical cells each having emf \(E\) and internal resistance \(r\), the current through an external resistor of \(6\,\Omega\) is the same when the cells are connected in series as well as in parallel. The value of the internal resistance \(r\) is ________ \(\Omega\).

• A. \(9\)
• B. \(3\)
• C. \(6\)
• D. \(4\)

Hint:

When currents are equal in series and parallel cell combinations, equate the current expressions directly to find internal resistance.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem requires comparing current in series and parallel circuits. For identical cells, there exists a specific external resistance value where the current is invariant to the arrangement.
Step 2: Key Formula or Approach:
Series Current: \(I_s = \frac{nE}{R + nr}\).
Parallel Current: \(I_p = \frac{E}{R + r/n} = \frac{nE}{nR + r}\).
Step 3: Detailed Explanation:
For \(n = 2\) and \(R = 6\):
\[ I_s = \frac{2E}{6 + 2r} \]
\[ I_p = \frac{2E}{2(6) + r} = \frac{2E}{12 + r} \]
Given \(I_s = I_p\):
\[ 6 + 2r = 12 + r \]
\[ 2r - r = 12 - 6 \]
\[ r = 6 \, \Omega \]
Step 4: Final Answer:
The internal resistance \(r\) is \(6 \, \Omega\).