Question

Five identical cells each of internal resistance 1 \(\Omega\) and emf 5 V are connected in series and in parallel with an external resistance 'R'. For what value of 'R', current in series and parallel combination will remain the same ?

• A. 1 \(\Omega\)
• B. 5 \(\Omega\)
• C. 10 \(\Omega\)
• D. 25 \(\Omega\)

Hint:

For n identical cells, the condition for the current to be the same in both series and parallel combinations connected to an external resistor R is when \(R = r\). You can derive this generally: \(\frac{nE}{R+nr} = \frac{E}{R+r/n} \implies n(R+r/n) = R+nr \implies nR+r = R+nr \implies R(n-1) = r(n-1) \implies R=r\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We have five identical cells. We need to find the value of an external resistance 'R' such that the current drawn from the cells is the same whether they are connected in series or in parallel.
Step 2: Key Formula or Approach:
Let n be the number of cells, E be the emf of each cell, and r be the internal resistance of each cell.
1. For series combination, the total emf is \(nE\) and the total internal resistance is \(nr\). The current is: \[ I_{series} = \frac{nE}{R + nr} \] 2. For parallel combination of identical cells, the total emf is \(E\) and the total internal resistance is \(r/n\). The current is: \[ I_{parallel} = \frac{E}{R + r/n} \] We are given that \(I_{series} = I_{parallel}\).
Step 3: Detailed Explanation:
Given values: Number of cells, \(n = 5\)
EMF of each cell, \(E = 5\) V
Internal resistance of each cell, \(r = 1 \, \Omega\)
Now, we set up the equations for the currents.
Current in series combination: \[ I_{series} = \frac{5 \times 5}{R + 5 \times 1} = \frac{25}{R + 5} \] Current in parallel combination: \[ I_{parallel} = \frac{5}{R + 1/5} = \frac{5}{(5R+1)/5} = \frac{25}{5R+1} \] According to the question, \(I_{series} = I_{parallel}\).
\[ \frac{25}{R + 5} = \frac{25}{5R + 1} \] Equating the denominators: \[ R + 5 = 5R + 1 \] \[ 5R - R = 5 - 1 \] \[ 4R = 4 \] \[ R = 1 \, \Omega \] Step 4: Final Answer:
The value of the external resistance R for which the current will be the same in both cases is 1 \(\Omega\).