When two cells of different emf's are connected in series to an external resistance the current is 5A. When the poles of one cell are interchanged, the current is 3A. The ratio of emf's of two cell is

Options

A. A) `2 :1`
B. B) `1 :3`
C. C) `4 :1`
D. D) `5:1`

Answer

To solve the problem, we need to analyze the two scenarios given in the question and derive the relationship between the EMFs of the two cells.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let \( E_1 \) be the EMF of the first cell.
- Let \( E_2 \) be the EMF of the second cell.
- Let \( R \) be the external resistance.

2. **First Scenario:**
- When the two cells are connected in series with the same polarity, the total EMF is \( E_1 + E_2 \).
- The current \( I_1 \) in the circuit is given as 5 A.
- According to Ohm's law, we can write:
\[ I_1 = \frac{E_1 + E_2}{R} \]
- Substituting the known current:
\[ 5 = \frac{E_1 + E_2}{R} \quad \text{(1)} \]

3. **Second Scenario:**
- When the polarity of the first cell is reversed, the total EMF becomes \( E_1 - E_2 \).
- The current \( I_2 \) in this case is given as 3 A.
- Again, applying Ohm's law:
\[ I_2 = \frac{E_1 - E_2}{R} \]
- Substituting the known current:
\[ 3 = \frac{E_1 - E_2}{R} \quad \text{(2)} \]

4. **Eliminate R:**
- From equation (1):
\[ R = \frac{E_1 + E_2}{5} \]
- From equation (2):
\[ R = \frac{E_1 - E_2}{3} \]
- Setting these two expressions for \( R \) equal to each other:
\[ \frac{E_1 + E_2}{5} = \frac{E_1 - E_2}{3} \]

5. **Cross-Multiply:**
- Cross-multiplying gives:
\[ 3(E_1 + E_2) = 5(E_1 - E_2) \]
- Expanding both sides:
\[ 3E_1 + 3E_2 = 5E_1 - 5E_2 \]

6. **Rearranging Terms:**
- Bringing all terms involving \( E_1 \) to one side and terms involving \( E_2 \) to the other side:
\[ 3E_1 - 5E_1 = -5E_2 - 3E_2 \]
- Simplifying gives:
\[ -2E_1 = -8E_2 \]
- Dividing both sides by -2:
\[ E_1 = 4E_2 \]

7. **Finding the Ratio:**
- Thus, the ratio of the EMFs is:
\[ \frac{E_1}{E_2} = 4 \]
- Therefore, the ratio of the EMFs of the two cells is:
\[ E_1 : E_2 = 4 : 1 \]

### Final Answer:
The ratio of the EMFs of the two cells is \( 4 : 1 \).