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 \( \left( \frac{R}{r} \right) \) is

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

Hint:

For \(n\) identical cells, the current delivered to an external resistor \(R\) is the same in series and parallel combinations if and only if \(R = r\).

Answer 1

The Correct Option is D

Concept: The current (\(I\)) in a circuit containing cells depends on the total electromotive force (EMF) and the total resistance of the circuit.
• Cells in Series: For \(n\) identical cells, the total EMF is \(nE\) and the total internal resistance is \(nr\). The current is \( I_s = \frac{nE}{R + nr} \).
• Cells in Parallel: For \(n\) identical cells, the total EMF remains \(E\) and the total internal resistance is \(r/n\). The current is \( I_p = \frac{E}{R + (r/n)} \).

Step 1: Formulate equations for both cases.
Given \(n = 5\). In series: \[ I_s = \frac{5E}{R + 5r} \] In parallel: \[ I_p = \frac{E}{R + \frac{r}{5}} = \frac{5E}{5R + r} \]

Step 2: Equate the currents and solve for the ratio.
According to the problem, \( I_s = I_p \): \[ \frac{5E}{R + 5r} = \frac{5E}{5R + r} \] This implies the denominators must be equal: \[ R + 5r = 5R + r \] Rearranging terms: \[ 4r = 4R \implies R = r \] Thus, the ratio \(\frac{R}{r} = 1\).