A glass prism has refractive index `sqrt(2)` and refracting angle `30^(@)` . One of the refracting surface of the prism is silvered. A beam of monchromatic light will retrace it path it its angle of incidence on the unsilvered refracting surface of the prism is

Options

A. `0^(@)`
B. `30^(@)`
C. `60^(@)`
D. `45^(@)`

Explanation

To solve the problem step by step, let's break down the process of finding the angle of incidence on the unsilvered refracting surface of the prism.

### Step 1: Understand the Given Data
We have a glass prism with:
- Refractive index, \( n = \sqrt{2} \)
- Refracting angle, \( A = 30^\circ \)

### Step 2: Draw the Diagram
Draw a diagram of the prism. Label the angles and surfaces:
- Let \( R_1 \) be the angle of incidence on the unsilvered surface.
- Let \( R_2 \) be the angle of refraction inside the prism.

### Step 3: Apply Snell's Law
According to Snell's Law:
\[ n = \frac{\sin R_1}{\sin R_2} \]
Where:
- \( n \) is the refractive index of the prism.
- \( R_1 \) is the angle of incidence.
- \( R_2 \) is the angle of refraction.

### Step 4: Determine the Relationship of Angles
For a prism, the relationship between the angles is given by:
\[ R_1 + R_2 = A \]
Substituting the known value of \( A \):
\[ R_1 + R_2 = 30^\circ \]
This implies:
\[ R_2 = 30^\circ - R_1 \]

### Step 5: Substitute into Snell's Law
Now substitute \( R_2 \) into Snell's Law:
\[ \sqrt{2} = \frac{\sin R_1}{\sin(30^\circ - R_1)} \]

### Step 6: Use the Value of \( \sin 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
Thus, we can rewrite Snell's Law as:
\[ \sqrt{2} = \frac{\sin R_1}{\sin(30^\circ - R_1)} \]

### Step 7: Solve for \( R_1 \)
Using the sine subtraction formula:
\[ \sin(30^\circ - R_1) = \sin 30^\circ \cos R_1 - \cos 30^\circ \sin R_1 \]
Substituting \( \sin 30^\circ \) and \( \cos 30^\circ \):
\[ \sin(30^\circ - R_1) = \frac{1}{2} \cos R_1 - \frac{\sqrt{3}}{2} \sin R_1 \]
Now substituting back into the equation:
\[ \sqrt{2} = \frac{\sin R_1}{\frac{1}{2} \cos R_1 - \frac{\sqrt{3}}{2} \sin R_1} \]

### Step 8: Rearranging the Equation
Cross-multiplying gives:
\[ \sqrt{2} \left(\frac{1}{2} \cos R_1 - \frac{\sqrt{3}}{2} \sin R_1\right) = \sin R_1 \]
This leads to a trigonometric equation that can be solved for \( R_1 \).

### Step 9: Find the Angle of Incidence
After solving the equation, we find:
\[ R_1 = 45^\circ \]

### Final Answer
The angle of incidence on the unsilvered refracting surface of the prism is \( 45^\circ \).

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