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:
Show Hint
For the same current in both series and parallel configurations, the external resistance must equal internal resistance.
Step 1: Calculate the current in the series combination.
\[
I = \frac{nE}{nr + R} = \frac{5E}{5r + R}
\]
Step 2: Calculate the current in the parallel combination.
\[
I' = \frac{E}{\frac{r}{n} + R} = \frac{5E}{r + 5R}
\]
Equate the expressions for \(I\) and \(I'\) since they are equal:
\[
\frac{5E}{5r + R} = \frac{5E}{r + 5R}
\]
Solve for \(R\) and \(r\):
\[
5r + R = r + 5R
\]
\[
4r = 4R \Rightarrow R = r
\]
Therefore, the ratio \(\frac{R}{r} = 1\).