Problem 74 - College Physics

Question

\(\bullet$$\bullet\) A thick layer of oil is floating on the surface of water in a tank. A beam of light traveling in the oil is incident on the water interface at an angle of \(30.0^{\circ}\) from the normal. The refracted beam travels in the water at an angle of \(45.0^{\circ}\) from the normal. What is the refractive index of the oil?

Step-by-Step Solution

Answer

The refractive index of the oil is approximately 1.88.

Step 1: Understand Snell's Law

Snell's Law is given by the equation \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), where \( n_1 \) and \( n_2 \) are the refractive indices of the media, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively. In this scenario, \( n_1 \) is the refractive index of oil, \( n_2 \) is the refractive index of water, \( \theta_1 = 30.0^{\circ} \) (angle in oil), and \( \theta_2 = 45.0^{\circ} \) (angle in water).

Step 2: Refractive Index of Water

It's a well-known fact that the refractive index of water is approximately \( n_2 = 1.33 \). We will use this value in our calculations.

Step 3: Rearrange Snell's Law for Oil's Refractive Index

We want to find the refractive index of oil \( n_1 \). Arrange Snell's Law: \( n_1 = \frac{n_2 \sin \theta_2}{\sin \theta_1} \). Substitute \( n_2 = 1.33 \), \( \theta_2 = 45.0^{\circ} \), and \( \theta_1 = 30.0^{\circ} \).

Step 4: Calculate Sin Values

Calculate \( \sin 30.0^{\circ} \) and \( \sin 45.0^{\circ} \). We have \( \sin 30.0^{\circ} = 0.5 \) and \( \sin 45.0^{\circ} = \frac{\sqrt{2}}{2} \).

Step 5: Plug in Values

Substitute everything into the equation: \( n_1 = \frac{1.33 \times \frac{\sqrt{2}}{2}}{0.5} \). Simplify this expression to get \( n_1 = 1.33 \times \sqrt{2} \).

Step 6: Final Calculation

Calculate \( 1.33 \times \sqrt{2} \approx 1.33 \times 1.414 = 1.88 \). Thus, the refractive index of the oil is approximately \( 1.88 \).

Key Concepts

Refractive Index

The refractive index is an important property of materials in optics. It describes how light propagates through a medium. When light travels from one material to another, its speed changes, which bends the light path, a phenomenon known as refraction.

The refractive index, often denoted as \( n \), is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. Mathematically, this is given by:

• \( n = \frac{c}{v} \)

where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium. The higher the refractive index, the more the light is bent when entering the material.

For practical uses, refractive indices are typically measured for visible light. Each material has a characteristic refractive index, which is crucial for designing optical devices, such as lenses and prisms.

Angles of Incidence and Refraction

The angles of incidence and refraction are key components in understanding light's behavior at the boundary between two media. When a light ray hits this boundary, the angle between the ray and an imaginary line perpendicular to the surface, called the "normal," is the angle of incidence.

Similarly, the angle between the refracted ray (the bent path) and the normal is called the angle of refraction. These angles are essential in applying Snell's Law, an equation that describes how light will refract based on the different refractive indices of the two media. Snell's Law is expressed as:

• \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \)

Here, \( n_1 \) and \( n_2 \) are the refractive indices of the initial and second media, respectively; \( \theta_1 \) is the angle of incidence, and \( \theta_2 \) is the angle of refraction.

Understanding these angles helps predict how and where light will travel when transitioning between substances such as air, water, or oil.

Optics

Optics is the branch of physics that deals with light and its interactions with matter. It encompasses a range of phenomena, including reflection, refraction, diffraction, and interference, all of which determine how light behaves and travels.

One key aspect of optics is understanding how light bends when it moves between different media, explained by Snell's Law. In designing optical equipment like glasses, cameras, and microscopes, the principles of optics ensure light is manipulated in the desired way to achieve clear images or specific effects.

Optical systems often use lenses and mirrors to direct or modify light. These elements rely heavily on refractive indices and proper calculation of angles of incidence and reflection/refraction to fulfill their functions. This makes optics crucial in various technological and scientific applications, ranging from everyday eyeglasses to advanced telescopic lenses for astronomy.