When two identical cells are connected either in series or in parallel across a 4 ohm resistor, they send the same current through it. The internal resistance of the cell in ohm is

Options

A. `1.2`
B. `2`
C. `4`
D. `4.8`

Answer

To solve the problem, we need to analyze the two configurations of the cells (series and parallel) and set up equations based on Ohm's Law and the properties of the circuits.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the electromotive force (emf) of each cell be \( E \).
- Let the internal resistance of each cell be \( r \).
- The external resistance (the resistor across which the cells are connected) is given as \( R = 4 \, \Omega \).

2. **Series Connection**:
- When the two cells are connected in series, the total emf is \( 2E \) and the total internal resistance is \( 2r \).
- The total resistance in the circuit is \( R + 2r \).
- Using Ohm's Law, the current \( I_s \) in the series circuit can be expressed as:
\[ I_s = \frac{2E}{R + 2r} = \frac{2E}{4 + 2r} \]

3. **Parallel Connection**:
- When the two cells are connected in parallel, the total emf remains \( E \) and the total internal resistance becomes \( \frac{r}{2} \).
- The total resistance in this circuit is \( R + \frac{r}{2} \).
- The current \( I_p \) in the parallel circuit can be expressed as:
\[ I_p = \frac{E}{R + \frac{r}{2}} = \frac{E}{4 + \frac{r}{2}} \]

4. **Equating Currents**:
- According to the problem, the currents through the resistor in both configurations are the same, so we set \( I_s = I_p \):
\[ \frac{2E}{4 + 2r} = \frac{E}{4 + \frac{r}{2}} \]

5. **Simplifying the Equation**:
- We can cancel \( E \) from both sides (assuming \( E \neq 0 \)):
\[ \frac{2}{4 + 2r} = \frac{1}{4 + \frac{r}{2}} \]
- Cross-multiplying gives:
\[ 2(4 + \frac{r}{2}) = 1(4 + 2r) \]
- Expanding both sides:
\[ 8 + r = 4 + 2r \]

6. **Solving for \( r \)**:
- Rearranging the equation:
\[ 8 - 4 = 2r - r \]
\[ 4 = r \]

7. **Conclusion**:
- The internal resistance \( r \) of each cell is \( 4 \, \Omega \).

### Final Answer:
The internal resistance of the cell is \( 4 \, \Omega \).

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