Two cells when connected in series are balanced on 6 m on a potentiometer. If the polarity one of these cell is reversed, they balance on 2m. The ratio of e.m.f of the two cells.

Options

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

Answer

To solve the problem, we need to find the ratio of the electromotive forces (e.m.f) of two cells based on their balancing lengths on a potentiometer when connected in series.

### Step-by-step Solution:

1. **Understand the Setup**:
- When two cells (let's call them E1 and E2) are connected in series, their total e.m.f is the sum of their individual e.m.f values:
\[ E_{total} = E_1 + E_2 \]
- This total e.m.f balances on a length \( L_1 \) of the potentiometer.

2. **Write the First Equation**:
- According to the problem, when both cells are connected in the same direction, they balance on 6 m:
\[ E_1 + E_2 = K \cdot L_1 \quad \text{(Equation 1)} \]
- Here, \( L_1 = 6 \) m.

3. **Reverse the Polarity of One Cell**:
- If we reverse the polarity of one of the cells (let's say E1), the effective e.m.f becomes:
\[ E_{effective} = E_1 - E_2 \]
- This balances on a length \( L_2 \) of the potentiometer.

4. **Write the Second Equation**:
- When the polarity of E1 is reversed, they balance on 2 m:
\[ E_1 - E_2 = K \cdot L_2 \quad \text{(Equation 2)} \]
- Here, \( L_2 = 2 \) m.

5. **Set Up the Ratio of Equations**:
- Now we can set up the ratio of the two equations:
\[ \frac{E_1 + E_2}{E_1 - E_2} = \frac{K \cdot L_1}{K \cdot L_2} \]
- The \( K \) cancels out:
\[ \frac{E_1 + E_2}{E_1 - E_2} = \frac{L_1}{L_2} \]

6. **Substitute the Values**:
- Substitute \( L_1 = 6 \) m and \( L_2 = 2 \) m into the equation:
\[ \frac{E_1 + E_2}{E_1 - E_2} = \frac{6}{2} = 3 \]

7. **Cross Multiply to Solve for the Ratio**:
- Cross multiplying gives:
\[ E_1 + E_2 = 3(E_1 - E_2) \]
- Expanding this gives:
\[ E_1 + E_2 = 3E_1 - 3E_2 \]
- Rearranging terms leads to:
\[ 3E_2 + E_2 = 3E_1 - E_1 \]
\[ 4E_2 = 2E_1 \]

8. **Find the Ratio**:
- Dividing both sides by \( E_2 \) gives:
\[ \frac{E_1}{E_2} = \frac{4}{2} = 2 \]
- Therefore, the ratio of the e.m.f of the two cells is:
\[ E_1 : E_2 = 2 : 1 \]

### Final Answer:
The ratio of the e.m.f of the two cells is \( 2 : 1 \).