Question

Five cells each of emf \(E\) and internal resistance \(r\) send the same amount of current through an external resistance \(R\) whether the cells are connected in parallel or in series. Then the ratio \(\frac{R}{r}\) is:

• A. \(2\)
• B. \(\frac{1}{2}\)
• C. \(\frac{1}{5}\)
• D. \(1\)

Hint:

For the same current in both series and parallel configurations, the external resistance must equal internal resistance.

Answer 1

The Correct Option is D

Step 1: {Current in series combination}
\[ I = \frac{nE}{nr + R} = \frac{5E}{5r + R} \] Step 2: {Current in parallel combination}
\[ I' = \frac{E}{\frac{r}{n} + R} = \frac{5E}{r + 5R} \] Since \(I = I'\), equating both expressions: \[ \frac{5E}{5r + R} = \frac{5E}{r + 5R} \] Solving for \(R\) and \(r\), \[ 5r + R = r + 5R \] \[ 4r = 4R \Rightarrow R = r \] Thus, the ratio \(\frac{R}{r} = 1\).