A ray of monochromatic light is incident on one refracting face of a prism of angle `75^(@)`. It passes thorugh the prism and is incident on the other face at the critical angle. If the refractive index of the material of the prism is `sqrt(2)`, the angle of incidence on the first face of the prism is

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. None of these

Answer

To solve the problem, we need to find the angle of incidence on the first face of the prism when a ray of monochromatic light passes through a prism with an angle of 75 degrees and a refractive index of √2, and it strikes the second face at the critical angle.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Angle of the prism (A) = 75 degrees
- Refractive index of the prism (μ) = √2
- The angle of incidence on the second face is the critical angle (Ic).

2. **Calculate the Critical Angle (Ic):**
- The critical angle can be calculated using the formula:
\[ \sin(I_c) = \frac{1}{\mu} \]
- Substituting the value of μ:
\[ \sin(I_c) = \frac{1}{\sqrt{2}} \]
- Therefore, the critical angle (Ic) is:
\[ I_c = 45^\circ \]

3. **Relate the Angles in the Prism:**
- According to the geometry of the prism, the angle of incidence (I) at the first face and the angle of refraction (R1) at the first face can be related to the angle of the prism:
\[ A = R_1 + I_c \]
- Substituting the known values:
\[ 75^\circ = R_1 + 45^\circ \]
- Solving for R1:
\[ R_1 = 75^\circ - 45^\circ = 30^\circ \]

4. **Apply Snell's Law at the First Face:**
- Using Snell's law at the first face of the prism:
\[ \mu_1 \sin(I) = \mu_2 \sin(R_1) \]
- Where:
- μ1 (refractive index of air) = 1
- μ2 (refractive index of the prism) = √2
- R1 = 30 degrees
- Substituting the values:
\[ 1 \cdot \sin(I) = \sqrt{2} \cdot \sin(30^\circ) \]
- Since \(\sin(30^\circ) = \frac{1}{2}\):
\[ \sin(I) = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2} \]

5. **Calculate the Angle of Incidence (I):**
- Therefore, we have:
\[ \sin(I) = \frac{\sqrt{2}}{2} \]
- This implies:
\[ I = 45^\circ \]

### Final Answer:
The angle of incidence on the first face of the prism is **45 degrees**.