The time taken by a monochromatic light to travel a certain distance in air in `9 xx 10^(-6)` seconds. The time taken by the light ray to travel the same distance in water of refractive index `(4)/(3)` is

Options

A. 12s
B. `12mu s`
C. `(27)/(4)mu s`
D. `(27)/(2)mu s`

Answer

To solve the problem, we need to find the time taken by a monochromatic light to travel the same distance in water, given the time taken in air and the refractive index of water.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Time taken by light in air, \( t_{air} = 9 \times 10^{-6} \) seconds.
- Refractive index of water, \( n_{water} = \frac{4}{3} \).
- Speed of light in air, \( c_{air} = 3 \times 10^{8} \) m/s.

2. **Calculate the Distance Traveled in Air:**
- The formula relating distance, speed, and time is:
\[ d = c_{air} \times t_{air} \]
- Substituting the values:
\[ d = (3 \times 10^{8} \, \text{m/s}) \times (9 \times 10^{-6} \, \text{s}) = 27 \times 10^{2} \, \text{m} = 2700 \, \text{m} \]

3. **Calculate the Speed of Light in Water:**
- The speed of light in any medium is given by:
\[ c_{medium} = \frac{c_{air}}{n_{medium}} \]
- For water:
\[ c_{water} = \frac{3 \times 10^{8} \, \text{m/s}}{\frac{4}{3}} = 3 \times 10^{8} \times \frac{3}{4} = \frac{9}{4} \times 10^{8} \, \text{m/s} \]

4. **Calculate the Time Taken in Water:**
- Using the formula for time:
\[ t_{water} = \frac{d}{c_{water}} \]
- Substituting the values:
\[ t_{water} = \frac{27 \times 10^{2} \, \text{m}}{\frac{9}{4} \times 10^{8} \, \text{m/s}} = \frac{27 \times 10^{2} \times 4}{9 \times 10^{8}} = \frac{108 \times 10^{2}}{9 \times 10^{8}} = \frac{12 \times 10^{-6}}{1} \, \text{s} = 12 \times 10^{-6} \, \text{s} \]

5. **Final Result:**
- The time taken by the light ray to travel the same distance in water is:
\[ t_{water} = 12 \, \mu s \]