The critical angle for diamond (refractive index = 2) is

Options

A. About `20^(@)`
B. `60^(@)`
C. `45^(@)`
D. `30^(@)`

Answer

To find the critical angle for diamond with a refractive index (μ) of 2, we can use Snell's law. The critical angle is defined as the angle of incidence in the denser medium (diamond) for which the angle of refraction in the rarer medium (air) is 90 degrees.

### Step-by-Step Solution:

1. **Understand the Concept of Critical Angle**:
The critical angle (ic) is the angle of incidence at which light is refracted at an angle of 90 degrees in the rarer medium.

2. **Identify the Refractive Indices**:
- For diamond (denser medium), the refractive index (μ1) = 2.
- For air (rarer medium), the refractive index (μ2) = 1.

3. **Apply Snell's Law**:
According to Snell's law:
\[ μ_1 \sin(i_c) = μ_2 \sin(r) \]
where:
- \(i_c\) = critical angle,
- \(r\) = angle of refraction (which is 90 degrees at the critical angle).

4. **Substituting Values**:
Since \(r = 90^\circ\), we know that \(\sin(90^\circ) = 1\). Thus, we can rewrite Snell's law as:
\[ 2 \sin(i_c) = 1 \cdot 1 \]

5. **Solve for the Critical Angle**:
Rearranging the equation gives:
\[ \sin(i_c) = \frac{1}{2} \]
Now, we can find \(i_c\) by taking the inverse sine:
\[ i_c = \sin^{-1}\left(\frac{1}{2}\right) \]

6. **Calculate the Angle**:
The value of \(\sin^{-1}\left(\frac{1}{2}\right)\) is:
\[ i_c = 30^\circ \]

### Final Answer:
The critical angle for diamond is \(30^\circ\).

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