Problem 62 - College Physics

Question

\(\bullet\) The polarizing angle for light in air incident on a glass plate is \(57.6^{\circ} .\) What is the index of refraction of the glass?

Step-by-Step Solution

Answer

The index of refraction of the glass is approximately 1.55.

Step 1: Understanding the polarizing angle

The polarizing angle, also known as Brewster's angle, is the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. This occurs when the reflected and refracted rays are at 90° to each other.

Step 2: Applying Brewster's Law

Brewster's Law states that \( \tan(\theta_p) = n \), where \( \theta_p \) is the polarizing angle and \( n \) is the index of refraction of the glass. Given that \( \theta_p = 57.6^{\circ} \), we can find \( n \).

Step 3: Solving for the index of refraction

Using the equation \( n = \tan(57.6^{\circ}) \), we calculate the tangent of 57.6 degrees to find \( n \).

Step 4: Calculating the tangent

\( \tan(57.6^{\circ}) \) can be calculated using a scientific calculator. Performing this calculation gives \( \tan(57.6^{\circ}) \approx 1.55 \).

Step 5: Conclusion

Thus, the index of refraction of the glass is approximately 1.55.

Key Concepts

Index of Refraction

The index of refraction is a crucial concept in optics that describes how light propagates through a medium. It is a measure that indicates how much the speed of light is reduced inside the medium, compared to its speed in a vacuum. The index of refraction, usually denoted as \( n \), expresses the ratio of the speed of light in a vacuum \( c \) to the speed of light in the medium \( v \). The formula is given by \( n = \frac{c}{v} \). This value is always greater than or equal to 1. This fundamental property affects how much light bends, or refracts, when entering from one medium into another. For example, when light enters a glass medium from air, it slows down and changes direction due to the glass's index of refraction. Knowing this helps in understanding phenomena like underwater vision, the efficiency of lenses, and of course, the principle behind Brewster's angle.

Polarization

Polarization refers to the orientation of the oscillations of light waves in relation to their direction of travel. Light waves can vibrate in multiple planes; these are unpolarized light waves, like sunlight and most artificial light sources. However, polarization refers to light waves that are confined to a single plane.
Polarization occurs naturally and can be achieved artificially through various methods such as reflection, refraction, and using polarizing filters. For instance, sunglasses use polarization to reduce glare by blocking certain orientations of light waves. In the context of Brewster’s angle, light is polarized upon reflection at a specific angle, where the reflected and refracted rays are perpendicular to one another. This understanding aids in designing optical devices and improving visual technologies.

Optics

Optics is the branch of physics that delves into the behavior and properties of light. It encompasses the study of how light interacts with different materials, the development of tools like lenses, and the exploration of phenomena like diffraction and refraction.
In practical terms, optics is foundational in technologies like eyeglasses, cameras, telescopes, and microscopes. Scientists and engineers use optical principles to manipulate light, solving problems and creating innovations in fields ranging from medicine to telecommunications. Understanding optics is essential for deciphering how light can be controlled, enhancing various scientific and everyday applications. For instance, figuring out how light bends and spreads through lens systems is crucial for correcting vision and capturing high-quality images.

Brewster's Law

Brewster's Law provides a fascinating insight into the behavior of light at non-metallic surfaces. Named after Sir David Brewster, this law states that when unpolarized light hits a surface at a specific angle, called Brewster's angle, the reflected light is perfectly polarized perpendicular to the incident plane.
Mathematically, Brewster’s Law is expressed as \( \tan(\theta_p) = n \), where \( \theta_p \) is Brewster’s angle and \( n \) is the index of refraction of the medium. At this exact angle, the reflected and refracted light are at 90 degrees to each other. This phenomenon is extremely useful in designing polarizing filters and enhancing photographic techniques, as well as in industrial applications where controlling reflected light is crucial. In our specific problem, using \( \tan(57.6^{\circ}) \), we calculated the index of refraction for glass, finding it to be approximately 1.55, thereby illustrating Brewster's Law in action.