The refractive index of the material of a prism is `sqrt(2)` and the angle of the prism is `30^(@)`. One of the two refracting surfaces of the prism is made a mirror inwards, by silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if its angle of incidence on the prism is

Options

A. `30^(@)`
B. `45^(@)`C
C. `60^(@)`C
D. zero

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the Setup
We have a prism with a refractive index \( n = \sqrt{2} \) and an angle of the prism \( A = 30^\circ \). One of the refracting surfaces (let's say surface AC) is silvered, making it act like a mirror.

### Step 2: Identify the Conditions for Retracing the Path
For the light beam to retrace its path after reflecting off the silvered surface, it must strike the mirror perpendicularly. This means that the angle of incidence on the mirror must be \( 90^\circ \).

### Step 3: Analyze the Geometry
Let’s denote:
- The point where the light ray enters the prism as point D.
- The point where the light ray hits the silvered surface as point O.
- The angle of incidence at point O as \( i \).
- The angle of refraction at point O as \( r \).

Since the prism angle \( A = 30^\circ \), we can deduce the angles in the triangle formed by the points A, D, and O.

### Step 4: Calculate the Angles
In triangle ADO:
- The angle at point A is \( 30^\circ \).
- The angle at point D (the angle of incidence) is \( 90^\circ \) (since the ray is perpendicular to the mirror).
- Therefore, the angle at point O can be calculated as:
\[ \text{Angle O} = 180^\circ - 90^\circ - 30^\circ = 60^\circ \]

### Step 5: Relate the Angles Using Snell's Law
Using Snell's Law at the interface between air and the prism:
\[ n_1 \sin(i) = n_2 \sin(r) \]
Where:
- \( n_1 = 1 \) (refractive index of air)
- \( n_2 = \sqrt{2} \) (refractive index of the prism)
- \( r = 30^\circ \) (the angle of refraction we calculated)

Substituting the values:
\[ 1 \cdot \sin(i) = \sqrt{2} \cdot \sin(30^\circ) \]
Since \( \sin(30^\circ) = \frac{1}{2} \):
\[ \sin(i) = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2} \]

### Step 6: Solve for the Angle of Incidence
To find \( i \):
\[ \sin(i) = \frac{\sqrt{2}}{2} \]
This implies:
\[ i = 45^\circ \]

### Final Answer
The angle of incidence on the prism must be \( 45^\circ \).

---