A ray of light in incident on a glass plate at an angle of `60^@`. What is the refractive index of glass if the reflected and refracted rays are perpendicular to each other?

Options

A. `(1)/(2)`
B. `sqrt((3)/(2))`
C. `(3)/(2)`
D. `sqrt(3)`

Answer

To solve the problem, we need to find the refractive index of glass when a ray of light is incident at an angle of \(60^\circ\) and the reflected and refracted rays are perpendicular to each other.

### Step-by-Step Solution:

1. **Understand the Geometry of the Problem**:
- When light hits a surface, it is both reflected and refracted. The angle of incidence (\(i\)) is given as \(60^\circ\).
- The reflected ray will make the same angle with the normal as the angle of incidence. Therefore, the angle of reflection (\(r\)) is also \(60^\circ\).
- According to the problem, the reflected ray and the refracted ray are perpendicular to each other. This means that the angle between the reflected ray and the refracted ray is \(90^\circ\).

2. **Using the Relationship Between Angles**:
- If the reflected ray and refracted ray are perpendicular, we can express this relationship mathematically:
\[ r + r' = 90^\circ \]
where \(r'\) is the angle of refraction. Since \(r = 60^\circ\), we have:
\[ 60^\circ + r' = 90^\circ \]
Thus,
\[ r' = 90^\circ - 60^\circ = 30^\circ \]

3. **Applying Snell's Law**:
- Snell's Law states that:
\[ n_1 \sin(i) = n_2 \sin(r') \]
- Here, \(n_1\) is the refractive index of air (approximately \(1\)), \(i\) is the angle of incidence (\(60^\circ\)), \(n_2\) is the refractive index of glass (which we need to find), and \(r'\) is the angle of refraction (\(30^\circ\)).
- Substituting the known values into Snell's Law gives:
\[ 1 \cdot \sin(60^\circ) = n_2 \cdot \sin(30^\circ) \]

4. **Calculating the Sine Values**:
- We know:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin(30^\circ) = \frac{1}{2} \]
- Substituting these values into the equation gives:
\[ \frac{\sqrt{3}}{2} = n_2 \cdot \frac{1}{2} \]

5. **Solving for the Refractive Index**:
- Rearranging the equation to solve for \(n_2\):
\[ n_2 = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \]

6. **Final Answer**:
- The refractive index of glass is:
\[ n_2 = \sqrt{3} \approx 1.732 \]