If critical angle for a material to air is `30^(@)`, the refractive index of the material will be

Options

A. 1
B. 1.5
C. 2
D. 2.5

Answer

To find the refractive index of a material given the critical angle with respect to air, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Critical Angle**: The critical angle (Ic) is the angle of incidence in the denser medium (the material) at which the angle of refraction in the less dense medium (air) is 90 degrees. Beyond this angle, total internal reflection occurs.

2. **Identify Given Values**:
- Critical angle (Ic) = 30 degrees
- Refractive index of air (n2) = 1 (since air is the less dense medium)

3. **Use Snell's Law**: According to Snell's Law:
\[ n_1 \sin(I_c) = n_2 \sin(R) \]
where:
- \( n_1 \) = refractive index of the material (what we need to find)
- \( n_2 \) = refractive index of air = 1
- \( I_c \) = critical angle = 30 degrees
- \( R \) = angle of refraction = 90 degrees (at critical angle)

4. **Apply Snell's Law**: Substitute the known values into Snell's Law:
\[ n_1 \sin(30^\circ) = 1 \cdot \sin(90^\circ) \]

5. **Calculate Sine Values**:
- We know that \( \sin(90^\circ) = 1 \)
- We also know that \( \sin(30^\circ) = \frac{1}{2} \)

6. **Substitute Sine Values into the Equation**:
\[ n_1 \cdot \frac{1}{2} = 1 \]

7. **Solve for Refractive Index (n1)**:
\[ n_1 = 1 \cdot 2 = 2 \]

8. **Conclusion**: The refractive index of the material is 2.

### Final Answer:
The refractive index of the material is **2**.

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