Problem 43 - College Physics

Question

\(\bullet$$\bullet\) A parallel-sided plate of glass having a refractive index of 1.60 is in contact with the surface of water in a tank. A ray coming from above makes an angle of incidence of \(32.0^{\circ}\) with the top surface of the glass. What angle does this ray make with the normal in the water?

Step-by-Step Solution

Answer

The angle in the water is approximately \( 39.6^\circ \).

Step 1: Identify the Known Quantities

We are given the refractive index of the glass as \( n_1 = 1.60 \) and for water, by default, it's \( n_2 = 1.33 \). The angle of incidence \( \theta_1 \) is given as \( 32.0^\circ \). We need to find the angle of refraction in the water \( \theta_2 \).

Step 2: Apply Snell's Law at the Glass-Water Interface

Snell's Law is given by \[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \] We need to rearrange this equation to solve for \( \theta_2 \).\[ \sin(\theta_2) = \frac{n_1}{n_2} \sin(\theta_1) \]

Step 3: Substitute the Known Values

Substitute the values into the equation:\[ \sin(\theta_2) = \frac{1.60}{1.33} \sin(32.0^\circ) \] Calculate \( \sin(32.0^\circ) \) and compute the right-hand side.

Step 4: Calculate the Angle of Refraction (\theta_2)

Use a calculator to find \( \sin(32.0^\circ) \approx 0.5299 \).\[ \sin(\theta_2) = \frac{1.60}{1.33} \times 0.5299 \approx 0.637 \]Now, find \( \theta_2 \) using the inverse sine function: \[ \theta_2 = \sin^{-1}(0.637) \approx 39.6^\circ \]

Step 5: Conclusion

The ray makes an angle of approximately \( 39.6^\circ \) with the normal in the water.

Key Concepts

Refractive Index

The refractive index is a fundamental concept in optical physics. It quantifies how much light, or any radiation, bends or refracts when entering a different medium. This value is a ratio that compares the speed of light in a vacuum to the speed of light in the medium being considered.
\[ n = \frac{c}{v} \]
Here, \( n \) is the refractive index, \( c \) represents the speed of light in a vacuum, and \( v \) is the speed of light in the medium. A higher refractive index indicates that light travels slower in that material.
This concept is crucial, as it helps determine how light will behave at the interface of two mediums, such as glass contacting water, as in our exercise. For instance, the glass has a refractive index of 1.60, meaning it significantly slows down light compared to the water, which has a refractive index of 1.33.

Angle of Incidence

The angle of incidence is the angle between the incoming ray and the normal—a line perpendicular to the surface—at the point where the ray strikes an optical surface. Understanding this angle is important because it directly influences the path and angle at which the light will bend.
In our example, a ray of light hits the upper surface of the glass at an angle of incidence of \( 32.0^{\circ} \). This angle determines how much the light will bend once it enters the glass medium. It's measured from the normal, not the surface, which is a common misconception.

Angle of Refraction

The angle of refraction is the angle between the refracted ray and the normal after the light has passed into a new medium. The behavior of the angle of refraction is dictated by Snell's Law, which helps us calculate how the light bends.
The formula for Snell's Law is:
\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]
This determines the relationship between the angles and refractive indices of the two media. In our problem, the light ray refracts as it crosses from glass (with \( n_1 = 1.60 \)) to water (with \( n_2 = 1.33 \)), creating an angle of refraction of approximately \( 39.6^{\circ} \).
Calculating this requires simple trigonometric functions and a good understanding of Snell’s Law.

Optical Physics

Optical physics focuses on the study of light and its interactions with matter. It encompasses the study of phenomena such as reflection, refraction, diffraction, and the behavior of light in different media. Understanding the fundamental principles of optical physics allows us to predict and manipulate light paths, which is essential in many technologies and natural phenomena.
Snell’s Law is a key principle in optical physics, explaining refraction, or the bending of light as it moves through different media. By using known refractive indices of materials, one can predict the path of light, which is how we solved the problem of determining the angle that light makes with the normal in water. Grasping these concepts is essential for further exploration into advanced areas like optics in engineering and technology.