Problem 36 - Physics

Question

When red light in a vacuum is incident at the Brewster angle on a certain type of glass, the angle of refraction is \(29.9^{\circ} .\) What are (a) the Brewster angle and (b) the index of refraction of the glass?

Step-by-Step Solution

Answer

Brewster's angle is approximately 56.3° and the index of refraction of the glass is about 1.49.

Step 1: Understanding Brewster's Angle

Brewster's angle is the angle of incidence at which light is perfectly polarized upon reflection. It occurs when the reflected and refracted rays are at a 90-degree angle to each other. The formula to find Brewster's angle is \( \tan(\theta_B) = n \), where \( \theta_B \) is Brewster's angle and \( n \) is the index of refraction of the glass.

Step 2: Using Snell's Law

According to Snell's Law, \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \), where \( n_1 \) and \( n_2 \) are the refractive indices of the two media, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction respectively. Here, \( n_1 = 1 \) for air, \( \theta_2 = 29.9^{\circ} \), and \( \theta_1 = \theta_B \).

Step 3: Solving for Brewster's Angle

Since we're looking for \( \theta_B \), we use \( \tan(\theta_B) = n \). Also from Snell's Law previously discussed is that \( \tan(\theta_B) = \frac{\sin(29.9^{\circ})}{\cos(29.9^{\circ})} \). Solving it gives us \( \theta_B \approx 56.3^{\circ} \).

Step 4: Finding the Index of Refraction

From \( \tan(\theta_B) = n \), we substitute the Brewster's angle we calculated. \( n = \tan(56.3^{\circ}) \). Calculating gives \( n \approx 1.49 \).

Key Concepts

Index of Refraction

The index of refraction, sometimes called the refractive index, is a measure of how much light slows down when entering a medium from a vacuum (or air). This indicates how much a medium bends the light compared to when it's traveling through a vacuum.

• It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium: \( n = \frac{c}{v} \).
• Here, \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \text{ m/s} \), and \( v \) is the speed of light in the medium.
• For example, the index of refraction for ordinary glass is often around 1.5, meaning light travels 1.5 times slower in glass than in a vacuum.

Different materials have different indices of refraction, influencing how light behaves at the interface between two media. Understanding this concept helps in designing lenses and other optical devices. In the previous exercise, the calculated index of refraction for the glass is \( n \approx 1.49 \), indicating how glass slows down and bends the light entering it.

Snell's Law

Snell's Law is a key concept in the study of optics. It governs how light is refracted, or bends, as it passes from one medium to another. The law is mathematically expressed as:
\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]

• Here, \( n_1 \) and \( n_2 \) are the indices of refraction for the first and second medium, respectively.
• \( \theta_1 \) is the angle of incidence, the angle between the incoming light ray and the normal (an imaginary line perpendicular to the surface of the medium).
• \( \theta_2 \) is the angle of refraction, or the angle between the refracted light ray and the normal.

When light travels from a medium with a lower index of refraction to a higher one, it slows down and bends towards the normal. Conversely, if it moves to a medium with a lower index, it speeds up and bends away from the normal. In the original exercise, Snell's Law helped determine that the angle of refraction was \( 29.9^{\circ} \) in the glass.

Polarization of Light

Polarization refers to the direction in which the electric field of a light wave oscillates. In unpolarized light, these oscillations occur in multiple directions perpendicular to the direction of propagation. Polarization, however, constrains these oscillations to a single plane.
Brewster's angle, mentioned in the exercise, is closely tied to polarization. At this angle, the reflected light becomes completely polarized perpendicular to the plane of incidence. This happens when the angle of reflection and the angle of refraction are such that they add up to \( 90^{\circ} \).

• Brewster's angle can be calculated using the formula \( \tan(\theta_B) = n \), where \( n \) is the index of refraction of the medium.
• At Brewster's angle, the reflected and refracted rays are perpendicular, leading to minimal reflection and maximum refraction.
• This property is used in reducing glare in photography and making polarized sunglasses that block certain reflections.

In the given scenario, the incident light at Brewster's angle was reflected perfectly polarized, as calculated in the original solution. Polarization is crucial in many optical applications, ensuring clearer images by managing light reflections.