A ray of light is incident at an angle of `60^@` on the face of a prism having refracting angle `30^@.` The ray emerging out of the prism makes an angle `30^@` with the incident ray. Show that the emergent ray is perpendicular to the face through which it emerges and calculate the refractive index of the material of prism.

Explanation

To solve the problem step by step, we will analyze the situation involving the prism, the angles of incidence, refraction, and the refractive index.

### Step 1: Understand the Geometry of the Prism
- A ray of light is incident on the prism at an angle of \( i_1 = 60^\circ \).
- The refracting angle of the prism is \( A = 30^\circ \).
- The angle of deviation \( D \) is given as \( 30^\circ \).

### Step 2: Use the Angle of Deviation Formula
The angle of deviation \( D \) can be expressed as:
\[ D = i_1 + i_2 - A \]
Where:
- \( i_1 \) is the angle of incidence,
- \( i_2 \) is the angle of refraction at the second face of the prism,
- \( A \) is the angle of the prism.

Substituting the known values:
\[ 30^\circ = 60^\circ + i_2 - 30^\circ \]

### Step 3: Solve for \( i_2 \)
Rearranging the equation:
\[ i_2 = 30^\circ - 60^\circ + 30^\circ \]
\[ i_2 = 0^\circ \]

### Step 4: Analyze the Emergent Ray
Since \( i_2 = 0^\circ \), this means that the ray emerges perpendicular to the surface of the prism. Thus, the emergent ray is indeed perpendicular to the face through which it emerges.

### Step 5: Calculate the Refractive Index
We can use Snell's Law at the first interface of the prism:
\[ \mu_1 \sin(i_1) = \mu_2 \sin(r_1) \]
Where:
- \( \mu_1 \) is the refractive index of air (approximately 1),
- \( r_1 \) is the angle of refraction inside the prism.

Since \( i_2 = 0^\circ \), the angle of refraction at the second face \( r_2 \) is also \( 0^\circ \). Therefore, at the first face, we can find \( r_1 \):
\[ r_1 = A - i_2 = 30^\circ - 0^\circ = 30^\circ \]

Now, applying Snell's Law:
\[ 1 \cdot \sin(60^\circ) = \mu_2 \cdot \sin(30^\circ) \]

### Step 6: Solve for \( \mu_2 \)
Substituting the values:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \]
\[ \frac{\sqrt{3}}{2} = \mu_2 \cdot \frac{1}{2} \]
\[ \mu_2 = \frac{\sqrt{3}}{1} = \sqrt{3} \]

### Final Answer
- The emergent ray is perpendicular to the face through which it emerges.
- The refractive index of the material of the prism is \( \mu_2 = \sqrt{3} \).