Two cells of emf `E_(1)` and `E_(2) ( E_(1) gt E_(2))` are connected in series to potentiometer for balancing length 625 cm . When polarity of `E_(2)` is reversed then balancing length becomes 125 cm. Then the ratio `(E_(1))/(E_(2))` is

Options

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

Answer

To solve the problem, we need to analyze the situation with the two cells connected in series to a potentiometer and how the balancing lengths change with the reversal of the polarity of one of the cells.

### Step-by-Step Solution:

1. **Understanding the Initial Setup**:
- When the two cells \( E_1 \) and \( E_2 \) are connected in series, the total EMF in the circuit is \( E_1 + E_2 \).
- The balancing length on the potentiometer is given as 625 cm.

2. **Using the Balancing Length**:
- The potential difference across the potentiometer wire can be expressed in terms of the balancing length and the total EMF:
\[ V = k \cdot L \]
where \( V \) is the total potential difference, \( k \) is the potential gradient, and \( L \) is the balancing length.
- For the initial condition:
\[ E_1 + E_2 = k \cdot 625 \]

3. **Reversing the Polarity of \( E_2 \)**:
- When the polarity of \( E_2 \) is reversed, the effective EMF becomes \( E_1 - E_2 \).
- The new balancing length is given as 125 cm.

4. **Setting Up the Equation for the New Condition**:
- For the new condition:
\[ E_1 - E_2 = k \cdot 125 \]

5. **Forming the System of Equations**:
- We now have two equations:
1. \( E_1 + E_2 = k \cdot 625 \) (1)
2. \( E_1 - E_2 = k \cdot 125 \) (2)

6. **Solving the System of Equations**:
- We can add equations (1) and (2):
\[ (E_1 + E_2) + (E_1 - E_2) = k \cdot 625 + k \cdot 125 \]
\[ 2E_1 = k \cdot 750 \]
\[ E_1 = \frac{k \cdot 750}{2} = k \cdot 375 \]
- Now, subtract equation (2) from equation (1):
\[ (E_1 + E_2) - (E_1 - E_2) = k \cdot 625 - k \cdot 125 \]
\[ 2E_2 = k \cdot 500 \]
\[ E_2 = \frac{k \cdot 500}{2} = k \cdot 250 \]

7. **Finding the Ratio \( \frac{E_1}{E_2} \)**:
- Now, we can find the ratio:
\[ \frac{E_1}{E_2} = \frac{k \cdot 375}{k \cdot 250} = \frac{375}{250} = \frac{3}{2} \]

### Final Answer:
The ratio \( \frac{E_1}{E_2} \) is \( \frac{3}{2} \).