Light takes 8 min 20 sec to reach from sun on the earth. If the whole atmosphere is filled with water, the light will take the time `(._(a)mu_(w)=4//3)`

Options

A. 8 min 20 sec
B. 8 min
C. 6 min 11 sec
D. 11 min 6 sec

Answer

To solve the problem, we need to determine how long it would take for light to travel from the Sun to the Earth if the atmosphere were filled with water, given that the refractive index of water (μ) is \( \frac{4}{3} \).

### Step-by-Step Solution:

1. **Convert Time to Seconds**:
The time taken for light to travel from the Sun to the Earth is given as 8 minutes and 20 seconds.
\[ \text{Total time in seconds} = 8 \times 60 + 20 = 500 \text{ seconds} \]

**Hint**: Always convert time into a single unit (seconds) for easier calculations.

2. **Understand the Relationship of Speed and Refractive Index**:
The speed of light in a medium can be expressed in terms of the speed of light in a vacuum (Vc) and the refractive index (μ):
\[ V_m = \frac{V_c}{\mu} \]
where \( V_m \) is the speed of light in the medium (water in this case).

3. **Calculate the Speed of Light in Water**:
Given that \( \mu = \frac{4}{3} \):
\[ V_m = \frac{V_c}{\frac{4}{3}} = \frac{3}{4} V_c \]

**Hint**: Remember that the refractive index indicates how much the speed of light is reduced in a medium compared to a vacuum.

4. **Set Up the Time Relationship**:
Since the distance (D) from the Sun to the Earth remains constant, we can relate the times using the speeds:
\[ \frac{D}{V_c} = t \quad \text{and} \quad \frac{D}{V_m} = t_1 \]
From this, we can derive:
\[ t_1 = \frac{D}{V_m} = \frac{D}{\frac{3}{4} V_c} = \frac{4}{3} \frac{D}{V_c} = \frac{4}{3} t \]

5. **Calculate the New Time**:
Now substituting the known value of \( t \):
\[ t_1 = \frac{4}{3} \times 500 \text{ seconds} = \frac{2000}{3} \text{ seconds} \approx 666.67 \text{ seconds} \]

**Hint**: When multiplying fractions, ensure you maintain the correct order of operations.

6. **Convert Seconds Back to Minutes and Seconds**:
To convert 666.67 seconds into minutes:
\[ 666.67 \text{ seconds} = 11 \text{ minutes} + 6.67 \text{ seconds} \approx 11 \text{ minutes} \, 7 \text{ seconds} \]

**Hint**: When converting seconds to minutes, divide by 60 and take the integer part as minutes.

### Final Answer:
The time taken for light to reach Earth through water is approximately **11 minutes and 7 seconds**.