The critical angle for total internal reflection in diamond is `24.5^(@)` The refractive index of the diamond is

Options

A. 2.41
B. 1.41
C. 2.59
D. 1.59

Answer

To find the refractive index of diamond using the critical angle, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Critical Angle**:
The critical angle (α) is the angle of incidence above which total internal reflection occurs. For diamond, this angle is given as 24.5 degrees.

2. **Use Snell's Law**:
Snell's Law states that:
\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]
where \(n_1\) and \(n_2\) are the refractive indices of the two media, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, respectively.

3. **Identify the Mediums**:
In this case, the light is going from diamond (with refractive index \(n_1\)) to air (with refractive index \(n_2 = 1\)). The angle of refraction in air at the critical angle is 90 degrees.

4. **Set Up the Equation**:
At the critical angle, we can set up the equation as follows:
\[ n_{diamond} \sin(24.5^\circ) = 1 \cdot \sin(90^\circ) \]
Since \(\sin(90^\circ) = 1\), we can simplify this to:
\[ n_{diamond} \sin(24.5^\circ) = 1 \]

5. **Solve for the Refractive Index**:
Rearranging the equation gives us:
\[ n_{diamond} = \frac{1}{\sin(24.5^\circ)} \]

6. **Calculate the Value**:
Now, we need to calculate \(\sin(24.5^\circ)\) using a scientific calculator:
\[ \sin(24.5^\circ) \approx 0.4162 \]
Therefore:
\[ n_{diamond} = \frac{1}{0.4162} \approx 2.403 \]

### Final Answer:
The refractive index of diamond is approximately \(2.403\).

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