A prism of refractive index `sqrt(3/2)` and refracting angle is `45^(@)` is placed in air. One of the two refracting surface of the prism is silvered and a ray of monochromatic light enters the prism from the other face at an angle `theta`. If the ray retraces its path, then what is the value of `theta` (in degree)?

Answer

To solve the problem, we need to analyze the situation step by step:

### Step 1: Understand the Prism Setup
We have a prism with a refractive index \( n = \sqrt{\frac{3}{2}} \) and a refracting angle \( A = 45^\circ \). One of the refracting surfaces of the prism is silvered, meaning that it reflects light back into the prism.

### Step 2: Identify Angles
When a ray of light enters the prism at an angle \( \theta \), it will refract at the first surface. Let’s denote the angle of incidence at the first surface as \( \theta \) and the angle of refraction as \( r_1 \).

### Step 3: Apply the Prism Formula
For a prism, the relationship between the angle of incidence \( \theta \), angle of refraction \( r_1 \), and the refracting angle \( A \) is given by:
\[ A = r_1 + r_2 \]
Since the other surface is silvered, it reflects the light back, making \( r_2 = 0 \). Therefore, we have:
\[ A = r_1 \]
Given \( A = 45^\circ \), we find:
\[ r_1 = 45^\circ \]

### Step 4: Apply Snell's Law
Using Snell's Law at the first surface, we have:
\[ n_1 \sin(\theta) = n_2 \sin(r_1) \]
Here, \( n_1 = 1 \) (for air) and \( n_2 = \sqrt{\frac{3}{2}} \). Substituting the values, we get:
\[ 1 \cdot \sin(\theta) = \sqrt{\frac{3}{2}} \cdot \sin(45^\circ) \]

### Step 5: Calculate \(\sin(45^\circ)\)
We know that:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \]
Substituting this into the equation gives:
\[ \sin(\theta) = \sqrt{\frac{3}{2}} \cdot \frac{1}{\sqrt{2}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 6: Find the Angle \(\theta\)
To find \( \theta \), we take the inverse sine:
\[ \theta = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
This corresponds to:
\[ \theta = 60^\circ \]

### Conclusion
Thus, the value of \( \theta \) is \( 60^\circ \).