A point soujrce of light is placed 4 m below the surface of water of refractive index `(5)/(3).` The minimum diameter of a disc, which should be placed over the source, on the surface of water to cut off alol light coming out of water is

Options

A. infinite
B. `6 m`
C. `4 m`
D. `3 m`

Answer

To solve the problem, we need to find the minimum diameter of a disc that should be placed over a point source of light located 4 meters below the surface of water with a refractive index of \( \frac{5}{3} \). The disc will cut off all light coming out of the water, which means we need to consider the critical angle for total internal reflection.

### Step-by-Step Solution:

1. **Determine the Critical Angle:**
The critical angle \( \theta_c \) can be calculated using the formula:
\[ \sin \theta_c = \frac{1}{\mu} \]
where \( \mu \) is the refractive index of water. Given \( \mu = \frac{5}{3} \):
\[ \sin \theta_c = \frac{1}{\frac{5}{3}} = \frac{3}{5} \]
Now, we find \( \theta_c \):
\[ \theta_c = \sin^{-1}\left(\frac{3}{5}\right) \]

2. **Calculate the Critical Angle:**
Using a calculator or trigonometric tables, we find:
\[ \theta_c \approx 37^\circ \]

3. **Set Up the Geometry:**
The light rays that emerge from the point source at an angle greater than \( \theta_c \) will undergo total internal reflection. We need to find the radius \( R \) of the circle on the surface of the water that corresponds to this critical angle.

4. **Use Trigonometry to Find the Radius:**
From the point source, we can draw a right triangle where:
- The height from the source to the surface of the water is \( 4 \) m.
- The angle at the point source is \( \theta_c \).

Using the tangent function:
\[ \tan(\theta_c) = \frac{R}{4} \]
We know that:
\[ \tan(37^\circ) = \frac{3}{4} \]
Therefore, we can set up the equation:
\[ \frac{3}{4} = \frac{R}{4} \]

5. **Solve for Radius \( R \):**
Rearranging the equation gives:
\[ R = 3 \text{ m} \]

6. **Calculate the Diameter:**
The diameter \( D \) of the disc is twice the radius:
\[ D = 2R = 2 \times 3 = 6 \text{ m} \]

### Final Answer:
The minimum diameter of the disc that should be placed over the source to cut off all light coming out of the water is **6 meters**.