In a potentiometer experiment two cells of e.m.f. E1 and E2 are used in series and in conjunction and the balancing length is found to be 58 cm of the wire. If the polarity of E2 is reversed, then the balancing length becomes 29 cm . The ratio `(E_(1))/(E_(2))` of the e.m.f. of the two cells is

Options

A. `1 : 1`
B. `3:1`
C. `5:1`
D. `7:1`

Explanation

To find the ratio of the e.m.f. of the two cells \( \frac{E_1}{E_2} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Potentiometer Setup**:
- In a potentiometer, the potential drop per unit length of the wire is constant. This means that the potential difference across a length \( x \) of the wire is proportional to \( x \).

2. **First Condition**:
- When the cells \( E_1 \) and \( E_2 \) are connected in series with the same polarity, the total e.m.f. is \( E_1 + E_2 \).
- The balancing length is given as 58 cm. Therefore, we can write the equation for this condition:
\[ E_1 + E_2 = k \cdot 58 \quad (1) \]
where \( k \) is the potential gradient (potential drop per unit length).

3. **Second Condition**:
- When the polarity of \( E_2 \) is reversed, the total e.m.f. becomes \( E_1 - E_2 \).
- The new balancing length is 29 cm. Thus, we can write:
\[ E_1 - E_2 = k \cdot 29 \quad (2) \]

4. **Setting Up the Equations**:
- We now have two equations:
\[ E_1 + E_2 = k \cdot 58 \quad (1) \]
\[ E_1 - E_2 = k \cdot 29 \quad (2) \]

5. **Eliminating \( k \)**:
- To eliminate \( k \), we can divide equation (1) by equation (2):
\[ \frac{E_1 + E_2}{E_1 - E_2} = \frac{58}{29} = 2 \]

6. **Cross Multiplying**:
- Cross multiplying gives us:
\[ E_1 + E_2 = 2(E_1 - E_2) \]
- Expanding this gives:
\[ E_1 + E_2 = 2E_1 - 2E_2 \]

7. **Rearranging the Equation**:
- Rearranging the terms leads to:
\[ E_2 + 2E_2 = 2E_1 - E_1 \]
\[ 3E_2 = E_1 \]

8. **Finding the Ratio**:
- Therefore, we can express the ratio \( \frac{E_1}{E_2} \):
\[ \frac{E_1}{E_2} = \frac{3E_2}{E_2} = 3 \]

### Final Answer:
The ratio of the e.m.f. of the two cells is:
\[ \frac{E_1}{E_2} = 3 \]