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 follow the given information and apply the principles of refraction and geometry of the prism.

### Step 1: Identify the given values
- Angle of incidence (I1) = 60°
- Refracting angle of the prism (A) = 30°
- Angle of deviation (δ) = 30° (since the emergent ray makes an angle of 30° with the incident ray)

### Step 2: Use the formula for angle of deviation
The angle of deviation (δ) for a prism can be expressed as:
\[ \delta = I_1 + I_2 - A \]
Where:
- \(I_2\) is the angle of refraction at the second face of the prism.

### Step 3: Substitute the known values into the deviation formula
Substituting the known values into the equation:
\[ 30° = 60° + I_2 - 30° \]

### Step 4: Solve for \(I_2\)
Rearranging the equation:
\[ 30° = 60° + I_2 - 30° \]
\[ 30° = 30° + I_2 \]
\[ I_2 = 0° \]

### Step 5: Interpret the result for \(I_2\)
Since \(I_2 = 0°\), this means that the ray emerges without bending at the second face of the prism. Therefore, the emergent ray is perpendicular to the face through which it emerges.

### Step 6: Use Snell's Law to find the refractive index
According to Snell's Law:
\[ \mu_1 \sin I_1 = \mu_2 \sin I_2 \]
Where:
- \(\mu_1\) is the refractive index of air (approximately 1),
- \(\mu_2\) is the refractive index of the prism material.

Substituting the known values:
\[ 1 \cdot \sin(60°) = \mu \cdot \sin(30°) \]

### Step 7: Substitute the sine values
Using the values:
- \(\sin(60°) = \frac{\sqrt{3}}{2}\)
- \(\sin(30°) = \frac{1}{2}\)

The equation becomes:
\[ 1 \cdot \frac{\sqrt{3}}{2} = \mu \cdot \frac{1}{2} \]

### Step 8: Solve for \(\mu\)
Rearranging gives:
\[ \mu = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]

### Conclusion
The refractive index of the material of the prism is \(\sqrt{3}\).

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