Two slits `S_(1)` and `S_(2)` illuminated by a white light source give a white central maxima. A transparent sheet of refractive index 1.25 and thickness `t_(1)` is placed in front of `S_(1)`. Another transparent sheet of refractive index 1.50 and thickness `t_(2)` is placed in front of `S_(2)`. If central maxima is not effected, then ratio of the thickness of the two sheets will be :

Options

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

Answer

To solve the problem, we need to find the ratio of the thickness of two transparent sheets placed in front of two slits, such that the central maxima remains unaffected.

### Step-by-Step Solution:

1. **Understanding Optical Path Length**:
The optical path length (OPL) is given by the product of the refractive index (μ) of the medium and the actual path length (t). The formula for optical path length is:
\[ \text{OPL} = \mu \times t \]

2. **Identifying the Given Values**:
- For the first slit \( S_1 \):
- Refractive index \( \mu_1 = 1.25 \)
- Thickness \( t_1 \)
- For the second slit \( S_2 \):
- Refractive index \( \mu_2 = 1.50 \)
- Thickness \( t_2 \)

3. **Calculating the Optical Path Lengths**:
- The optical path length for the first sheet in front of \( S_1 \):
\[ \text{OPL}_1 = \mu_1 \times t_1 = 1.25 t_1 \]
- The optical path length for the second sheet in front of \( S_2 \):
\[ \text{OPL}_2 = \mu_2 \times t_2 = 1.50 t_2 \]

4. **Condition for Central Maxima**:
For the central maxima to remain unaffected, the optical path lengths must be equal:
\[ \text{OPL}_1 = \text{OPL}_2 \]
Therefore, we have:
\[ 1.25 t_1 = 1.50 t_2 \]

5. **Rearranging the Equation**:
To find the ratio of the thicknesses \( \frac{t_1}{t_2} \):
\[ \frac{t_1}{t_2} = \frac{1.50}{1.25} \]

6. **Simplifying the Ratio**:
\[ \frac{t_1}{t_2} = \frac{1.50 \div 1.25}{1.25 \div 1.25} = \frac{1.50 \div 1.25}{1} = \frac{1.50}{1.25} = \frac{6}{5} = \frac{12}{10} = \frac{2}{1} \]

7. **Final Result**:
The ratio of the thickness of the two sheets is:
\[ \frac{t_1}{t_2} = 2:1 \]

### Conclusion:
Thus, the ratio of the thickness of the two sheets \( t_1 \) and \( t_2 \) is \( 2:1 \).