Question

total internal reflection conceptual question
consider scenarios a to f in which a ray of light traveling in material 1 is incident onto the interface with material 2. (figure 1)
for the scenarios in which total internal reflection is possible, rank the scenarios on the basis of the critical angle, the angle above which total internal reflection occurs. at this angle, the refracted ray is at 90 degrees from the normal.
rank from largest to smallest. to rank items as equivalent, overlap them.
view available hint(s)
figure
1 of 1
n₁
n₂
θ₁
n₁, water = 1.33
n₂, air = 1.00
n₁, diamond = 2.42
n₂, air = 1.00
n₁, diamond = 2.42
n₂, water = 1.33
n₁, benzene = 1.51
n₂, water = 1.33
largest

Explanation:

Step1: Recall Critical Angle Formula

The critical angle \( \theta_c \) for total internal reflection (TIR) is given by \( \sin\theta_c = \frac{n_2}{n_1} \) (where \( n_1 > n_2 \), as TIR occurs when light travels from a denser to a rarer medium). Larger \( \frac{n_2}{n_1} \) means larger \( \theta_c \) (since \( \sin\theta \) is increasing in \( [0, \frac{\pi}{2}] \)).

Step2: Calculate \( \frac{n_2}{n_1} \) for Each Scenario

• Scenario 1 (\( n_1 = \text{water} = 1.33, n_2 = \text{air} = 1.00 \)): \( \frac{n_2}{n_1} = \frac{1.00}{1.33} \approx 0.7519 \)
• Scenario 2 (\( n_1 = \text{diamond} = 2.42, n_2 = \text{air} = 1.00 \)): \( \frac{n_2}{n_1} = \frac{1.00}{2.42} \approx 0.4132 \)
• Scenario 3 (\( n_1 = \text{diamond} = 2.42, n_2 = \text{water} = 1.33 \)): \( \frac{n_2}{n_1} = \frac{1.33}{2.42} \approx 0.5496 \)
• Scenario 4 (\( n_1 = \text{benzene} = 1.51, n_2 = \text{water} = 1.33 \)): \( \frac{n_2}{n_1} = \frac{1.33}{1.51} \approx 0.8808 \)

Step3: Rank by \( \frac{n_2}{n_1} \) (Largest to Smallest)

Larger \( \frac{n_2}{n_1} \) → Larger \( \theta_c \). So:

1. Scenario 4 (\( \approx 0.8808 \))
2. Scenario 1 (\( \approx 0.7519 \))
3. Scenario 3 (\( \approx 0.5496 \))
4. Scenario 2 (\( \approx 0.4132 \))

Answer:

From largest to smallest critical angle:

• Benzene (n₁=1.51) to Water (n₂=1.33)
• Water (n₁=1.33) to Air (n₂=1.00)
• Diamond (n₁=2.42) to Water (n₂=1.33)
• Diamond (n₁=2.42) to Air (n₂=1.00)

(Overlap equivalent items if any, but here all have distinct ranks.)