A glass prism has a refractive angle of `90^(@)` and a refractive index of 1.5. A ray is incident at an angle of `30^(@)`. The ray emerges from an adjacent face at an angle of

Options

A. `60^(@)`
B. `30^(@)`
C. `45^(@)`
D. the ray does not emerge

Explanation

To solve the problem step by step, we will follow the principles of refraction and apply Snell's law.

### Step 1: Identify Given Values
- Refractive angle of the prism (A) = 90°
- Refractive index of glass (μ) = 1.5
- Angle of incidence (i) = 30°

### Step 2: Calculate the Critical Angle
The critical angle (θc) can be calculated using the formula:
\[ \theta_c = \sin^{-1}\left(\frac{1}{\mu}\right) \]
Substituting the value of μ:
\[ \theta_c = \sin^{-1}\left(\frac{1}{1.5}\right) = \sin^{-1}\left(\frac{2}{3}\right) \approx 41.8° \]

### Step 3: Apply Snell's Law at the First Surface
Using Snell's law at the first surface (air to prism):
\[ \mu_{\text{air}} \cdot \sin(i) = \mu_{\text{glass}} \cdot \sin(r_1) \]
Where:
- μ_air = 1 (for air)
- i = 30°
- μ_glass = 1.5
- r_1 = angle of refraction in the prism

Substituting the values:
\[ 1 \cdot \sin(30°) = 1.5 \cdot \sin(r_1) \]
Since \(\sin(30°) = \frac{1}{2}\):
\[ \frac{1}{2} = 1.5 \cdot \sin(r_1) \]
\[ \sin(r_1) = \frac{1}{2 \cdot 1.5} = \frac{1}{3} \]
Now, calculate \(r_1\):
\[ r_1 = \sin^{-1}\left(\frac{1}{3}\right) \approx 19.5° \]

### Step 4: Calculate the Angle of Refraction at the Second Surface
The angle of the prism (A) is given by:
\[ A = r_1 + r_2 \]
Where \(r_2\) is the angle of refraction at the second surface. Given that \(A = 90°\):
\[ 90° = 19.5° + r_2 \]
Solving for \(r_2\):
\[ r_2 = 90° - 19.5° = 70.5° \]

### Step 5: Check if the Ray Emerges
Now, we need to check if \(r_2\) is greater than the critical angle:
- Critical angle \(θ_c \approx 41.8°\)
- \(r_2 = 70.5°\)

Since \(r_2\) (70.5°) is greater than the critical angle (41.8°), the ray will not emerge from the second surface.

### Conclusion
The ray does not emerge from the adjacent face of the prism.

### Final Answer
The ray emerges from the adjacent face at an angle of **not applicable** (it does not emerge).

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