N identical cells whether joined together in series or in parallel, give the same current, when connected to an external resistance of 'R'. The internal resistance of each cell is:-

Options

A. r=nR
B. r=R
C. `r=(R)/(n)`
D. `r=n^(2)R`

Answer

To find the internal resistance of each cell when N identical cells are connected in series or parallel and give the same current through an external resistance R, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Current in Series Connection:**
- When N cells are connected in series, the total voltage (E_total) is the sum of the voltages of each cell. Thus, E_total = N * E, where E is the voltage of one cell.
- The total internal resistance (r_total) in series is the sum of the internal resistances of each cell: r_total = N * r, where r is the internal resistance of one cell.
- The current (I) through the external resistance R can be expressed using Ohm's law:
\[ I = \frac{E_{total}}{R + r_{total}} = \frac{N \cdot E}{R + N \cdot r} \]

2. **Understanding the Current in Parallel Connection:**
- When N cells are connected in parallel, the voltage across each cell remains the same (E).
- The total internal resistance in parallel is given by:
\[ r_{total} = \frac{r}{N} \]
- The current (I) through the external resistance R can be expressed as:
\[ I = \frac{E}{R + r_{total}} = \frac{E}{R + \frac{r}{N}} \]

3. **Setting the Currents Equal:**
- Since the current is the same in both configurations, we can set the two equations for current equal to each other:
\[ \frac{N \cdot E}{R + N \cdot r} = \frac{E}{R + \frac{r}{N}} \]

4. **Cross-Multiplying to Eliminate Fractions:**
- Cross-multiplying gives:
\[ N \cdot E \cdot \left(R + \frac{r}{N}\right) = E \cdot \left(R + N \cdot r\right) \]

5. **Simplifying the Equation:**
- Cancelling E from both sides (assuming E ≠ 0):
\[ N \cdot \left(R + \frac{r}{N}\right) = R + N \cdot r \]
- Expanding the left side:
\[ N \cdot R + r = R + N \cdot r \]

6. **Rearranging the Equation:**
- Bringing all terms involving R and r to one side:
\[ N \cdot R - R = N \cdot r - r \]
- This simplifies to:
\[ (N - 1) \cdot R = (N - 1) \cdot r \]

7. **Dividing by (N - 1):**
- Assuming N ≠ 1, we can divide both sides by (N - 1):
\[ R = r \]

### Conclusion:
The internal resistance of each cell is equal to the external resistance R.

### Final Answer:
The internal resistance of each cell is **R**.