Refractive index of water is `5//3`. A light source is placed in water at a depth of 4m. Then what must be the minimum radius of disc placed on water surface so that the light of source can be stopped?

Options

A. 3m
B. 4m
C. 5m
D. `oo`

Explanation

To solve the problem, we need to find the minimum radius of a disc placed on the water surface so that light from a source at a depth of 4 meters does not escape into the air. We will use the concept of total internal reflection (TIR) and Snell's law.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Refractive index of water, \( \mu_w = \frac{5}{3} \)
- Depth of the light source in water, \( h = 4 \, \text{m} \)

2. **Determine the Critical Angle:**
The critical angle \( \theta_C \) can be calculated using Snell's law:
\[ \mu_w \sin(\theta_C) = \mu_{air} \sin(90^\circ) \]
Since the refractive index of air \( \mu_{air} \) is approximately 1, we have:
\[ \sin(\theta_C) = \frac{1}{\mu_w} = \frac{1}{\frac{5}{3}} = \frac{3}{5} \]
Now, we can find \( \theta_C \):
\[ \theta_C = \sin^{-1}\left(\frac{3}{5}\right) \]

3. **Calculate the Tangent of the Critical Angle:**
Using the relationship between the tangent and sine:
\[ \tan(\theta_C) = \frac{\sin(\theta_C)}{\sqrt{1 - \sin^2(\theta_C)}} \]
First, calculate \( \sin^2(\theta_C) \):
\[ \sin^2(\theta_C) = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]
Then, calculate \( 1 - \sin^2(\theta_C) \):
\[ 1 - \sin^2(\theta_C) = 1 - \frac{9}{25} = \frac{16}{25} \]
Now, calculate \( \tan(\theta_C) \):
\[ \tan(\theta_C) = \frac{\frac{3}{5}}{\sqrt{\frac{16}{25}}} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

4. **Relate the Radius to the Depth:**
The geometry of the situation gives us:
\[ \tan(\theta_C) = \frac{r}{h} \]
Where \( r \) is the radius of the disc and \( h \) is the depth of the light source. Substituting the values:
\[ \frac{3}{4} = \frac{r}{4} \]

5. **Solve for the Radius \( r \):**
Rearranging the equation gives:
\[ r = 4 \cdot \frac{3}{4} = 3 \, \text{m} \]

### Conclusion:
The minimum radius of the disc that must be placed on the water surface to ensure that the light from the source does not escape into the air is \( r = 3 \, \text{m} \).