Problem 119 - Physics

Question

The drawing shows a ray of light traveling through three materials whose surfaces are parallel to each other. The refracted rays (but not the reflected rays) are shown as the light passes through each material. Taking into account the relative sizes of the angles of incidence and refraction, rank the materials according to their indices of refraction, greatest first. Provide reasons for your ranking. A ray of light strikes the \(a-b\) interface at a \(50.0^{\circ}\) angle of incidence. The index of refraction of material \(a\) is \(n_{a}=1.20\). The angles of refraction in materials \(b\) and \(c\) are, respectively, \(45.0^{\circ}\) and \(56.7^{\circ}\). Find the indices of refraction in these two media. Verify that your answers are consistent with your answers to the Concept Qucstion.

Step-by-Step Solution

Answer

Indices are: \( n_b = 1.30 \) and \( n_c = 1.10 \). Ranked: \( b > a > c \).

Step 1: Understand Snell's Law

Snell's Law is used for refraction of light and is given by the formula: \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), where \( n \) is the index of refraction and \( \theta \) is the angle. The subscript 1 represents the initial medium and 2 represents the medium the light enters.

Step 2: Apply Snell's Law between Materials a and b

Given that the angle of incidence in material \( a \) is \( 50.0^{\circ} \) and its index of refraction \( n_a = 1.20 \), and the angle of refraction in material \( b \) is \( 45.0^{\circ} \), we apply Snell's Law:\[1.20 \times \sin(50^{\circ}) = n_b \times \sin(45^{\circ})\]Calculate \( n_b \) by rearranging the equation:\[n_b = \frac{1.20 \times \sin(50^{\circ})}{\sin(45^{\circ})}\]

Step 3: Calculate Index of Refraction for Material b

Substituting the values into the equation:\[n_b = \frac{1.20 \times 0.766}{0.707} = \frac{0.9192}{0.707} \approx 1.30\]

Step 4: Apply Snell's Law between Materials b and c

The angle in material \( b \) is \( 45.0^{\circ} \) and the angle of refraction in \( c \) is \( 56.7^{\circ} \). Applying Snell's Law:\[n_b \times \sin(45^{\circ}) = n_c \times \sin(56.7^{\circ})\]Take \( n_b = 1.30 \) from step 3 to substitute:\[1.30 \times \sin(45^{\circ}) = n_c \times \sin(56.7^{\circ})\] Rearrange to solve for \( n_c \):\[n_c = \frac{1.30 \times \sin(45^{\circ})}{\sin(56.7^{\circ})}\]

Step 5: Calculate Index of Refraction for Material c

Substitute the values into the equation:\[n_c = \frac{1.30 \times 0.707}{0.837} = \frac{0.9191}{0.837} \approx 1.10\]

Step 6: Rank the materials according to indices of refraction

The indices are: \( n_a = 1.20 \), \( n_b = 1.30 \), \( n_c = 1.10 \). In descending order, they are ranked as: \( b > a > c \), so material \( b \) has the highest index of refraction, followed by \( a \), and then \( c \).

Key Concepts

Angle of Incidence

In optics, the angle of incidence is the angle at which a ray of light hits a surface. It is defined as the angle between the incoming light ray and a line perpendicular to the surface, known as the normal line. Understanding this angle is crucial because it affects how light will behave when it enters a new medium. When a light ray strikes at the angle of incidence, it can either be absorbed, reflected, or refracted. In our exercise, the angle of incidence is given as \(50.0^{\circ}\) for the light transitioning from material \(a\) to material \(b\). This initial angle plays a significant role in determining how the light will bend or change direction as it passes into the new medium. In practical applications, keeping track of the angle of incidence helps in predicting the effectiveness of lenses and prisms. It can also be useful in designing glasses and other optical devices that rely on bending and focusing light accurately.

Angle of Refraction

The angle of refraction refers to the angle formed between the refracted ray and the normal line at the point of refraction. This concept comes from the refraction of light, which is the change in direction that a light wave experiences when it passes from one medium to another of different optical density. According to Snell's Law, the angle of refraction is closely related to the index of refraction of the two media involved. In the given exercise, the light refracts at \(45.0^{\circ}\) when moving from material \(a\) to material \(b\), and \(56.7^{\circ}\) when going from material \(b\) to material \(c\). These angles tell us how the light bends and are essential for calculating the indices of refraction of the materials. Understanding the relationship between angles of incidence and refraction can also help in designing optical instruments and applying them in various fields such as microscopy and astronomy, ensuring clarity and precision in viewing distant or minute objects.

Index of Refraction

The index of refraction, denoted by \(n\), is a measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum. Each material has its own index of refraction, influencing how light waves travel through it. In our exercise, we are given that material \(a\) has an index of refraction, \(n_a = 1.20\). From this, we can derive the indices for materials \(b\) and \(c\) using Snell's Law and the data provided. By applying the formula \(n_1 \sin \theta_1 = n_2 \sin \theta_2\), we calculated \(n_b\) as \(1.30\) and \(n_c\) as \(1.10\). These values allow us to rank the three materials based on their optical densities, impacting how much the light bends. The index of refraction is integral in various applications such as designing corrective lenses, creating fiber optics, and even in the study of rainbows and mirages. Knowing an object's refractive index can also aid in identifying substances using refractometers in scientific laboratories.

Refraction of Light

Refraction of light is the bending or change in direction of a light wave as it passes from one medium to another. This phenomenon occurs because light travels at different speeds in different media. The change in speed alters the light wave's direction, an effect beautifully explained by Snell's Law. In our scenario, light begins in one medium, material \(a\), and transitions through two more, materials \(b\) and \(c\). Each transition involves refraction, changing the light's path. The angles of incidence and refraction at each interface provide significant insight into how light behaves. Understanding how to apply Snell's Law helps to determine the indices of refraction for the media involved. Refraction is more than a theoretical concept; it's a cornerstone of practical applications. It's crucial for technologies like lenses in glasses and cameras, crafting fine optical elements in telescopes and microscopes, and forming the basis for phenomenon explanations such as the bending of sunlight in Earth's atmosphere, leading to atmospheric refraction effects like the apparent bending of objects at extreme angles.