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.

Explanation

To solve the problem step by step, we will use the concepts of refraction through a prism.

### Step 1: Identify the given values
- Angle of incidence (I₁) = 60°
- Refracting angle of the prism (A) = 30°
- Angle of deviation (Δ) = 30° (the angle between the incident ray and the emergent ray)

### Step 2: Use the formula for angle of deviation in a prism
The formula for the angle of deviation (Δ) in a prism is given by:
\[ \Delta = I₁ + I₂ - A \]
Where:
- Δ is the angle of deviation
- I₁ is the angle of incidence
- I₂ is the angle of refraction at the second face of the prism
- A is the angle of the prism

### Step 3: Substitute the known values into the formula
We know:
- Δ = 30°
- I₁ = 60°
- A = 30°

Substituting these values into the formula:
\[ 30° = 60° + I₂ - 30° \]

### Step 4: Simplify the equation to find I₂
Rearranging the equation gives:
\[ I₂ = 30° - 60° + 30° \]
\[ I₂ = 0° \]

### Step 5: Interpret the result
An angle of refraction (I₂) of 0° means that the emergent ray is perpendicular to the face of the prism through which it emerges. This is because if I₂ is 0°, it indicates that the ray is emerging straight out of the prism without bending.

### Conclusion
Thus, we have shown that the emergent ray is perpendicular to the face through which it emerges.

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