Geometric Optics: Snell's Law

Relevance of the topic

Understanding Geometric Optics, particularly Snell's Law, is the foundation for understanding a set of optical phenomena that occur in our daily lives. From the reflection of lights in mirrors to the deviation of light we experience when submerging an object in a glass of water, these phenomena have their roots in geometric optics.

Snell's Law, also known as the Law of Refraction, describes the mathematical relationship between the angles of incidence and refraction, as well as the variation in the speed of light in different media. It is the main tool for understanding why a pencil appears to break in water (due to the variation in light speed) and how eyeglass lenses correct vision (by maneuvering the direction of light). Thus, mastering this law is a crucial step for understanding things we take for granted in our daily perception.

Contextualization

Snell's Law is part of the broad study of Optics, which is the basis for a variety of areas in Physics, including particle physics, quantum physics, and astrophysics. Moreover, understanding this law is fundamental for understanding related topics in Physics, such as quantum mechanics, which uses concepts of refraction and reflection in the formulation of fundamental principles.

In the high school Physics curriculum, Snell's Law is usually introduced after the study of light wave reflection. This topic serves as a bridge to understanding how light behaves when it passes from one medium to another. By understanding Snell's Law, students will be able to explain a series of phenomena, from why the straw appears broken in water to how prisms work in car optics.

Theoretical Development

Components

• Refraction: Refraction occurs when a light wave passes from one medium to another with a different density. This phenomenon implies a change in the direction of the light wave and, if the speed also varies, a change in its wavelength and frequency.

• Angle of Incidence and Refraction: These are the angles formed by the incident and refracted light waves, respectively, with the normal to the separation surface of the media. The normal is an imaginary line perpendicular to the surface.

• Index of Refraction (n): An intrinsic characteristic of a medium that indicates the degree of refraction that light undergoes. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the considered medium (n=c/v, where c is the speed of light in a vacuum and v is the speed of light in the medium).

Key Terms

• Optical medium: Any substance (solid, liquid, or gaseous) through which light can propagate. Each medium has a characteristic index of refraction.

• Speed of Light: The speed of light in a vacuum is about 300,000 km/s. However, when light travels in other media, such as air, water, or glass, its speed decreases.

• Normal: The normal is an imaginary line drawn perpendicular to the separation surface of the media. It is used to measure the angles of incidence and refraction.

Examples and Cases

• Case of the broken Straw: When we look at a straw that is partially submerged in water, it appears "broken". This occurs because the light coming out of the straw changes direction when passing from air to water. The angle between the original direction of the light and the direction we perceive the straw is the angle of refraction, governed by Snell's Law.

• Prisms: A prism is a solid object, often made of glass, that has two parallel lateral faces of trapezoidal shape and the other faces are rectangular. Prism diverts the direction of light passing through it, just as it happens in refraction. The way light "breaks" when entering and exiting the prism is completely explained by Snell's Law.

• Eyeglasses: Eyeglass lenses correct vision by diverting the direction of light passing through them. The more the light is diverted, the greater the change in its original direction. Again, the physical explanation for this light deviation is Snell's Law.

Detailed Summary

Relevant Points

• Refraction and Index of Refraction (n): Refraction is the change in direction and, in some cases, the speed, of a wave phenomenon, such as light, when it passes from one medium to another with a different density. The index of refraction is a physical property of materials that measures the speed of light in them.

• Angles of Incidence and Refraction: In the refraction of light, there are two important angles: the angle of incidence (the angle formed by the incident light wave with the normal to the separation surface of the media) and the angle of refraction (the angle formed by the refracted light wave with the normal).

• Snell's Law: Also known as the law of refraction, it establishes that the quotient between the sines of the angle of incidence and the angle of refraction is equal to the quotient between the wave speeds in the two media (i.e., the index of refraction of one medium relative to the other). In formula, we have: n1 * sin(θ1) = n2 * sin(θ2), where n1 and n2 are the indices of refraction of media 1 and 2, respectively, and θ1 and θ2 are, respectively, the angles of incidence and refraction.

• Optical media: These are all substances (solid, liquid, or gaseous) that allow the propagation of light.

Conclusions

• Snell's Law is a fundamental tool for understanding the behavior of light in the transition between different optical media.

• The variation of the index of refraction between the media is what causes the refraction of light, and this variation is responsible for the phenomenon of the speed of light being different in different media.

• The study of geometric optics and Snell's Law is essential for explaining various common phenomena, such as the appearance of a broken spoon in a glass of water, the functioning of eyeglasses, and the formation of rainbows.

Exercises

1. Explain the phenomenon of the "breaking" of a spoon when it is partially submerged in a glass of water, using Snell's Law and the concepts of angle of incidence, angle of refraction, and index of refraction.

2. Given a beam of light that passes from a medium with an index of refraction n1 to a medium with an index of refraction n2, with the angle of incidence being θ1 = 30° and the index of refraction of the first medium being n1 = 1.5, determine the angle of refraction in the second medium using Snell's Law.

3. The indices of refraction of water and glass are approximately 1.33 and 1.5, respectively. What is the reason a pencil appears "broken" when it is partially submerged in a glass of water and leaning against the edge of the glass, if the light passes through a piece of glass before reaching the observer?