https://file.aiquickdraw.com/mathbot/file/1731024500427iis4zrt4.jpg

To solve this problem, let's analyze it step-by-step.

### Given Data:
- **Refractive indices**:
  - \( n_1 = 1.60 \)
  - \( n_2 = 1.40 \)
  - \( n_3 = 1.16 \)

### Questions:
1. **(a)** Find the angle \(\theta\) at which the light is incident on the interface between materials 2 and 3 such that it meets the critical angle.
2. **(b)** Determine whether there is refraction into material 3 if \(\theta\) is increased.

### Step 1: Determine the Critical Angle for the Interface Between Materials 2 and 3
The critical angle \( \theta_c \) for light traveling from a material with refractive index \( n_2 \) to one with refractive index \( n_3 \) can be found using Snell's law:

\[
\sin \theta_c = \frac{n_3}{n_2}
\]

Plugging in the values:
\[
\sin \theta_c = \frac{1.16}{1.40}
\]
\[
\sin \theta_c = 0.8286
\]

To find \(\theta_c\):
\[
\theta_c = \arcsin(0.8286)
\]
\[
\theta_c \approx 56.4^\circ
\]

Thus, for **(a)**, \(\theta \approx 56.4^\circ\) to achieve the critical angle at the interface between materials 2 and 3.

### Step 2: Analyzing (b) If \(\theta\) is Increased
If \(\theta\) is increased beyond the critical angle, total internal reflection will occur at the interface between materials 2 and 3. This means that no light will refract into material 3; instead, all of it will be reflected back into material 2.

### Summary of Answers:
- **(a)** \(\theta \approx 56.4^\circ\)
- **(b)** If \(\theta\) is increased, there will be no refraction into material 3.

Would you like further details on these calculations or have any questions?

Here are some related questions you might find interesting:
1. How does the critical angle change if \( n_3 \) is decreased?
2. What would happen if \( n_1 > n_2 \)?
3. How would you calculate the angle of refraction if total internal reflection did not occur?
4. What are the conditions for total internal reflection?
5. How does light behave at the critical angle?

**Tip**: Remember, total internal reflection only occurs when light travels from a denser medium to a less dense medium and meets or exceeds the critical angle.

In the figure, light initially in material 1 refracts into material 2, crosses that material, and is then incident at the critical angle on the interface between materials 2 and 3. The indices of refraction are n1 = 1.60, n2 = 1.40, n3 = 1.16. (a) What is angle θ? (b) If θ is increased, is there refraction of light into material 3?

Explore how to determine the critical angle for light passing between materials with different refractive indices, using Snell's Law. Given indices n1 = 1.60, n2 = 1.40, and n3 = 1.16, learn to calculate the angle θ for total internal reflection, and understand conditions where light will refract.

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