Angle of a prism is `30^(@)` and its refractive index is `sqrt(2)` and one of the surface is silvered. At what angle of incidence, a ray should be incident on one surface so that after reflection from the silvered surface, it retraces its path ?

Options

A. (a)`30^@`
B. (b)`60^@`
C. (c)`45^@`
D. (d)`sin^(-1)sqrt(1.5)`

Explanation

To solve the problem step by step, we will follow the principles of optics, particularly focusing on the properties of prisms and reflection.

### Step 1: Understand the Geometry of the Prism
We have a prism with an angle \( A = 30^\circ \) and one of its surfaces is silvered. The ray of light will enter the prism, refract, reflect off the silvered surface, and then retrace its path.

**Hint:** Visualize the prism and the path of the light ray as it enters and reflects.

### Step 2: Identify Angles of Refraction
Let \( R_1 \) be the angle of refraction when the light enters the prism, and \( R_2 \) be the angle of refraction when it exits. For the ray to retrace its path after reflecting from the silvered surface, the angle \( R_2 \) must be \( 0^\circ \) (meaning it reflects back along the same path).

**Hint:** Recall that for a ray to retrace its path, the angle of refraction after reflection must be \( 0^\circ \).

### Step 3: Apply the Prism Formula
The relationship between the angles in a prism is given by:
\[ A = R_1 + R_2 \]
Since \( R_2 = 0^\circ \), we have:
\[ 30^\circ = R_1 + 0^\circ \implies R_1 = 30^\circ \]

**Hint:** Use the formula for the angles in a prism to find \( R_1 \).

### Step 4: Use Snell's Law
Now, we apply Snell's Law at the first surface of the prism:
\[ n_1 \sin(i) = n_2 \sin(R_1) \]
where:
- \( n_1 = 1 \) (refractive index of air)
- \( n_2 = \sqrt{2} \) (given refractive index of the prism)
- \( R_1 = 30^\circ \)

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

**Hint:** Remember to substitute the known values into Snell's Law carefully.

### Step 5: Solve for the Angle of Incidence
Now, we find the angle of incidence \( i \):
\[ \sin(i) = \frac{\sqrt{2}}{2} \]
This corresponds to:
\[ i = 45^\circ \]

**Hint:** Use the inverse sine function to find the angle corresponding to the calculated sine value.

### Conclusion
The angle of incidence required for the ray to retrace its path after reflecting from the silvered surface is \( 45^\circ \).

**Final Answer:** The angle of incidence is \( 45^\circ \).