Question

Figure shows two combinations of real cells with 6 \(\Omega\) internal resistance. If reading of ammeters are same in both cases, find the value of 'r'. 

• A. 6 \(\Omega\)
• B. 5 \(\Omega\)
• C. 8 \(\Omega\)
• D. 12 \(\Omega\)

Hint:

For a given load \(R\), the current is the same for series and parallel combinations if the load is equal to the internal resistance of the individual cells (\(R = r\)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Current in a circuit with multiple cells is \(I = \frac{E_{eq}}{r_{eq} + R}\).
Step 2: Key Formula or Approach:
1. Cells in series: \(E_{eq} = nE, r_{eq} = nr\).
2. Cells in parallel: \(E_{eq} = E, r_{eq} = r/n\).
Step 3: Detailed Explanation:
Case 1: Two cells in series with external resistance \(6 \Omega\).
\[ I_1 = \frac{E + E}{r + r + 6} = \frac{2E}{2r + 6} \]
Case 2: Two cells in parallel with external resistance \(6 \Omega\).
\[ I_2 = \frac{E}{r/2 + 6} = \frac{2E}{r + 12} \]
It is given that \(I_1 = I_2\):
\[ \frac{2E}{2r + 6} = \frac{2E}{r + 12} \]
\[ 2r + 6 = r + 12 \]
\[ r = 6 \Omega \]
Step 4: Final Answer:
The internal resistance \(r\) is 6 \(\Omega\).