Problem 31 - Physics

Question

Light is reflected from a glass coffee table. When the angle of incidence is \(56.7^{\circ},\) the reflected light is completely polarized parallel to the surface of the glass. What is the index of refraction of the glass?

Step-by-Step Solution

Answer

The refractive index of the glass is approximately 1.50.

Step 1: Understanding Brewster's Angle

When light is completely polarized upon reflection, the angle of incidence is known as Brewster's angle (\( \theta_B \)). According to Brewster's law, the tangent of Brewster's angle equals the refractive index (n) of the material if the refractive medium is air.

Step 2: Brewster's Law

Brewster's law states that \( \tan(\theta_B) = n \). Given that \( \theta_B = 56.7^{\circ} \), applying Brewster's law will allow us to find the refractive index of the glass.

Step 3: Calculate the Refractive Index

Use the formula \( n = \tan(\theta_B) \). Substitute \( \theta_B = 56.7^{\circ} \) into the equation: \( n = \tan(56.7^{\circ}) \). Calculate to find that \( n \approx 1.50 \).

Key Concepts

Angle of Incidence

The angle of incidence is a fundamental concept in the study of light and optics. It refers to the angle between the incoming light ray and the normal line, an imaginary line perpendicular to the surface at the point of incidence. Understanding this angle is crucial because it affects how light behaves upon hitting a surface. This behavior can include reflection, refraction, or absorption depending on the material and the angle. For instance, when light hits a glass surface at an angle, part of it is reflected while the rest enters the glass, bending slightly—a phenomenon known as refraction. This angle is especially important in determining conditions like polarization of light, which occurs particularly at Brewster's angle.

Polarization of Light

Polarization of light refers to the alignment of light waves in particular directions. In most everyday contexts, light waves vibrate in multiple directions. However, when light is polarized, its waves align along a specific direction. This can occur naturally or through specific processes like reflection. When light reflects off surfaces such as water or glass at certain angles, it can become polarized. This is because the reflected light waves vibrate primarily in a plane parallel to the reflecting surface. Understanding polarization is crucial in various applications, ranging from reducing glare with polarized sunglasses to enhancing image clarity in photography.

Index of Refraction

The index of refraction, often symbolized as "n," is a measure of how much a material can bend light. It is a dimensionless number that indicates how fast light travels in a medium compared to a vacuum. The higher the index of refraction, the slower light moves through the material. For example, the index of refraction for air is approximately 1.00, meaning light travels slightly slower in air compared to a vacuum. In the case of glass, the index is typically higher, indicating that light slows down as it enters the glass. This property is essential in calculating angles of refraction using Snell's Law and understanding light behavior at interfaces.

Brewster's Law

Brewster's Law gives a precise relationship between the index of refraction and the angle at which light becomes perfectly polarized upon reflection. Named after Sir David Brewster, this law states that the tangent of Brewster's angle (the angle of incidence at which reflected light is totally polarized) equals the refractive index of the material. In mathematical terms, it is represented as \( \tan(\theta_B) = n \). Thus, when polarized light reflects off a surface, one can use the angle of incidence to find the material's index of refraction. In the provided example, Brewster's angle was given as \(56.7^{\circ}\). By applying Brewster's Law, the calculation \( n = \tan(56.7^{\circ}) \) reveals that the index of refraction for glass is approximately 1.50. Such insights are pivotal in designing optical devices and improving photography techniques.