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 of finding the minimum radius of a disc placed on the water surface to stop the light from a source located at a depth of 4 meters in water, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem:**
- We know the refractive index of water (μ) is \( \frac{5}{3} \).
- The depth of the light source in water (h) is 4 meters.
- We need to find the minimum radius (R) of a disc on the water surface that will prevent light from escaping.

2. **Determine the Critical Angle (θc):**
- The critical angle can be calculated using the formula:
\[ \sin(\theta_c) = \frac{1}{\mu} \]
- Substituting the value of μ:
\[ \sin(\theta_c) = \frac{1}{\frac{5}{3}} = \frac{3}{5} \]
- Therefore, the critical angle θc is:
\[ \theta_c = \arcsin\left(\frac{3}{5}\right) \]

3. **Relate the Radius to the Depth and Critical Angle:**
- Using the geometry of the situation, we can relate the radius (R) of the disc to the depth (h) and the critical angle (θc):
\[ R = h \cdot \tan(\theta_c) \]

4. **Calculate tan(θc):**
- From the earlier calculation, we know:
\[ \sin(\theta_c) = \frac{3}{5} \]
- We can find cos(θc) using the Pythagorean identity:
\[ \cos(\theta_c) = \sqrt{1 - \sin^2(\theta_c)} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]
- Now, we can find tan(θc):
\[ \tan(\theta_c) = \frac{\sin(\theta_c)}{\cos(\theta_c)} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

5. **Substitute Values to Find R:**
- Now substituting h = 4 m and tan(θc) = \( \frac{3}{4} \):
\[ R = 4 \cdot \frac{3}{4} = 3 \text{ meters} \]

6. **Conclusion:**
- The minimum radius of the disc that must be placed on the water surface to stop the light from the source is **3 meters**.

### Final Answer:
The minimum radius of the disc is **3 meters**.