The index of refraction of benzene is `1.8` . What is the critical angle for a light ray traveling in benzene toward a flat layer of air above the benzene ?

Answer

To find the critical angle for a light ray traveling from benzene to air, we can use Snell's Law and the concept of critical angle in optics. Here’s a step-by-step solution:

### Step 1: Understand the Concept of Critical Angle
The critical angle is the angle of incidence above which total internal reflection occurs when light travels from a denser medium to a rarer medium.

### Step 2: Identify the Refractive Indices
- The refractive index of benzene (denser medium) is given as \( n_2 = 1.8 \).
- The refractive index of air (rarer medium) is approximately \( n_1 = 1.0 \).

### Step 3: Use the Formula for Critical Angle
The formula for the critical angle \( \theta_c \) is given by:
\[ \sin \theta_c = \frac{n_1}{n_2} \]
where \( n_1 \) is the refractive index of the rarer medium (air) and \( n_2 \) is the refractive index of the denser medium (benzene).

### Step 4: Substitute the Values
Substituting the values into the formula:
\[ \sin \theta_c = \frac{1.0}{1.8} \]

### Step 5: Calculate the Sine of the Critical Angle
Calculating the right-hand side:
\[ \sin \theta_c = 0.5556 \quad (\text{approximately}) \]

### Step 6: Find the Critical Angle
To find \( \theta_c \), take the inverse sine:
\[ \theta_c = \sin^{-1}(0.5556) \]

Using a calculator, we find:
\[ \theta_c \approx 34.0^\circ \]

### Final Answer
The critical angle for a light ray traveling from benzene to air is approximately \( 34^\circ \).

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