A ray of light is incident on a medium at an angle i. It is found that the reflected ray is at right angles to the refracted ray . The refractive index of the medium is given by

Options

A. `sini`
B. `cosi`
C. `tan i`
D. cosec `i`

Explanation

To solve the problem step by step, we will analyze the situation involving the incident ray, reflected ray, and refracted ray, and apply Snell's law.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- A ray of light is incident on a medium at an angle \( i \).
- The reflected ray is at right angles (90 degrees) to the refracted ray.

2. **Identifying Angles**:
- Let \( r \) be the angle of reflection. According to the law of reflection, the angle of incidence \( i \) is equal to the angle of reflection \( r \). Therefore, \( r = i \).

3. **Setting Up the Angle Relationship**:
- Since the reflected ray and the refracted ray are at right angles to each other, we can express this relationship mathematically.
- If we consider the angles around the point where the rays meet, we can write:
\[ i + 90^\circ + r = 180^\circ \]
- This simplifies to:
\[ i + r = 90^\circ \]

4. **Expressing the Angle of Reflection**:
- From the equation \( i + r = 90^\circ \), we can express \( r \) in terms of \( i \):
\[ r = 90^\circ - i \]

5. **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 (which is approximately 1), and \( n_2 \) is the refractive index of the medium we are trying to find.
- Substituting \( n_1 = 1 \) and \( r = 90^\circ - i \):
\[ \sin(i) = n_2 \sin(90^\circ - i) \]

6. **Using the Trigonometric Identity**:
- We know from trigonometric identities that:
\[ \sin(90^\circ - i) = \cos(i) \]
- Therefore, we can rewrite the equation as:
\[ \sin(i) = n_2 \cos(i) \]

7. **Solving for the Refractive Index**:
- Rearranging the equation gives us:
\[ n_2 = \frac{\sin(i)}{\cos(i)} \]
- This simplifies to:
\[ n_2 = \tan(i) \]

8. **Final Result**:
- Thus, the refractive index of the medium is:
\[ \mu = \tan(i) \]

### Conclusion:
The refractive index of the medium is given by \( \mu = \tan(i) \).

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