A monochromatic light passes through a glass slab `(mu = (3)/(2))` of thickness 90 cm in time `t_(1)`. If it takes a time `t_(2)` to travel the same distance through water `(mu = (4)/(3))`. The value of `(t_(1) - t_(2))` is

Options

A. `5 xx 10^(-8) s`
B. `5 xx 10^(-10)s`
C. `2.5 xx 10^(-10) s`
D. `2.5 xx 10^(-8) s`

Answer

To solve the problem, we need to calculate the time taken by light to travel through a glass slab and through water, and then find the difference between these two times.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Thickness of the glass slab, \( d = 90 \, \text{cm} = 0.90 \, \text{m} \)
- Refractive index of glass, \( \mu_g = \frac{3}{2} \)
- Refractive index of water, \( \mu_w = \frac{4}{3} \)
- Speed of light in vacuum, \( c = 3 \times 10^8 \, \text{m/s} \)

2. **Calculate the Speed of Light in Glass:**
\[ v_g = \frac{c}{\mu_g} = \frac{3 \times 10^8 \, \text{m/s}}{\frac{3}{2}} = 2 \times 10^8 \, \text{m/s} \]

3. **Calculate the Time Taken in Glass (t1):**
\[ t_1 = \frac{d}{v_g} = \frac{0.90 \, \text{m}}{2 \times 10^8 \, \text{m/s}} = 4.5 \times 10^{-9} \, \text{s} \]

4. **Calculate the Speed of Light in Water:**
\[ v_w = \frac{c}{\mu_w} = \frac{3 \times 10^8 \, \text{m/s}}{\frac{4}{3}} = \frac{9 \times 10^8}{4} = 7.5 \times 10^7 \, \text{m/s} \]

5. **Calculate the Time Taken in Water (t2):**
\[ t_2 = \frac{d}{v_w} = \frac{0.90 \, \text{m}}{7.5 \times 10^7 \, \text{m/s}} = 1.2 \times 10^{-9} \, \text{s} \]

6. **Calculate the Difference in Time (t1 - t2):**
\[ t_1 - t_2 = 4.5 \times 10^{-9} \, \text{s} - 1.2 \times 10^{-9} \, \text{s} = 3.3 \times 10^{-9} \, \text{s} \]

### Final Answer:
\[ t_1 - t_2 = 3.3 \times 10^{-9} \, \text{s} \]