The refractive index of a material of a prism of angles `45^(@)-45^(@)-90^(@)` is `1.5.` The path of the ray of light incident normally on the hypotenuse side is shown in

Options

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D.

Explanation

To solve the problem, we need to analyze the path of a ray of light incident normally on the hypotenuse of a prism with angles 45°, 45°, and 90° and a refractive index of 1.5. We will also check for total internal reflection (TIR) conditions.

### Step-by-Step Solution:

1. **Identify the Prism and Its Properties:**
- The prism has angles of 45°, 45°, and 90°.
- The refractive index (μ) of the prism material is given as 1.5.

2. **Determine the Critical Angle (C):**
- The critical angle can be calculated using the formula:
\[ \sin C = \frac{1}{\mu} \]
- Substituting the value of μ:
\[ \sin C = \frac{1}{1.5} = \frac{2}{3} \]
- Therefore, the critical angle \( C \) is:
\[ C = \sin^{-1}\left(\frac{2}{3}\right) \]

3. **Calculate the Value of the Critical Angle:**
- Using a calculator or trigonometric tables, we find:
\[ C \approx 41.81^\circ \]
- This value is less than 45°, which is the angle of incidence on the prism's hypotenuse.

4. **Analyze the Incident Ray:**
- Since the ray is incident normally on the hypotenuse, the angle of incidence \( i \) is 0°.
- The angle of refraction \( r \) can be calculated using Snell's law:
\[ n_1 \sin i = n_2 \sin r \]
- Here, \( n_1 = 1 \) (air), \( n_2 = 1.5 \), and \( i = 0° \):
\[ 1 \cdot \sin(0) = 1.5 \cdot \sin r \implies \sin r = 0 \implies r = 0° \]
- The ray will pass straight into the prism without bending.

5. **Refraction at the Other Surface:**
- When the ray reaches the second surface of the prism (the 45° angle), the angle of incidence is 45°.
- We need to check if this angle is greater than the critical angle:
- Since \( 45° > 41.81° \), TIR will occur.

6. **Total Internal Reflection:**
- The ray will undergo total internal reflection at the 45° surface.
- The angle of reflection will also be 45°.

7. **Path of the Ray:**
- After reflecting off the 45° surface, the ray will exit the prism through the hypotenuse.
- The angle of incidence at the hypotenuse will again be 45°, which is greater than the critical angle, leading to TIR.

8. **Final Path:**
- The ray will reflect off the 45° surface and exit the prism at an angle of 45° with respect to the normal at the hypotenuse.

### Conclusion:
The path of the ray of light incident normally on the hypotenuse of the prism will undergo total internal reflection at the 45° surfaces and exit at an angle of 45°.