A ray of monochromatic light is incident on one refracting face of a prism of angle `75^(@)`. It passes through 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. `0^(@)`

Explanation

To solve the problem step by step, we need to determine the angle of incidence on the first face of the prism. Here’s how we can approach it:

### Step 1: Understand the Given Information
- The angle of the prism \( A = 75^\circ \)
- The refractive index of the prism \( \mu = \sqrt{2} \)
- The light ray is incident on the second face of the prism at the critical angle \( C \).

### Step 2: Calculate the Critical Angle
The critical angle \( C \) can be calculated using the formula:
\[ \sin C = \frac{1}{\mu} \]
Substituting the value of \( \mu \):
\[ \sin C = \frac{1}{\sqrt{2}} \]
Thus,
\[ C = \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = 45^\circ \]

### Step 3: Use the Prism Formula
In a prism, the relationship between the angles is given by:
\[ A = R_1 + R_2 \]
where \( R_1 \) is the angle of refraction at the first face and \( R_2 \) is the angle of refraction at the second face. Since the light exits at the critical angle \( C \), we have:
\[ R_2 = C = 45^\circ \]
Thus, we can rearrange the equation to find \( R_1 \):
\[ R_1 = A - C \]
Substituting the known values:
\[ R_1 = 75^\circ - 45^\circ = 30^\circ \]

### Step 4: Apply Snell's Law
Now, we can apply Snell's law at the first face of the prism:
\[ \mu_1 \sin I = \mu_2 \sin R_1 \]
Assuming the light is coming from air (where \( \mu_1 = 1 \)) and entering the prism (where \( \mu_2 = \sqrt{2} \)):
\[ 1 \cdot \sin I = \sqrt{2} \cdot \sin(30^\circ) \]
Since \( \sin(30^\circ) = \frac{1}{2} \), we have:
\[ \sin I = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2} \]

### Step 5: Calculate the Angle of Incidence
Now, we can find the angle \( I \):
\[ I = \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = 45^\circ \]

### Final Answer
The angle of incidence on the first face of the prism is \( 45^\circ \).

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