Problem 26 - University Physics with Modern Physics

Question

(a) At what angle above the horizontal is the sun if sunlight reflected from the surface of a calm lake is completely polarized? (b) What is the plane of the electric-field vector in the reflected light?

Step-by-Step Solution

Answer

(a) The sun is at an angle of 53.1° above the horizontal. (b) The electric-field vector is parallel to the water surface.

Step 1: Understanding Brewster's Angle

In order to determine the angle at which sunlight is completely polarized by reflection from a calm lake, we need to use Brewster's Angle, which is the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. The relationship is given by: \( \tan(\theta_B) = \frac{n_2}{n_1} \), where \( \theta_B \) is Brewster's angle, and \( n_1 \) and \( n_2 \) are the indices of refraction of the two media. For air and water, these are approximately \( n_1 = 1.0 \) and \( n_2 = 1.33 \).

Step 2: Calculate Brewster's Angle

Using the formula, \( \tan(\theta_B) = \frac{1.33}{1.0} \), we find that \( \theta_B = \tan^{-1}(1.33) \). Calculate \( \theta_B \) to find the angle.

Step 3: Compute the Angle with a Calculator

Now compute \( \theta_B = \tan^{-1}(1.33) \) using a calculator. This yields \( \theta_B \approx 53.1^\circ \). This is the angle of incidence for complete polarization.

Step 4: Identify the Plane of Polarization

When light is polarized by reflection, the electric field vector of the polarized light is parallel to the reflecting surface. Thus, for the lake, the plane of the electric-field vector in the reflected light is parallel to the water's surface.

Key Concepts

Polarization

Polarization is a fascinating phenomenon of light. It occurs when the vibrations of light waves are aligned in a specific direction. Typically, light waves vibrate in all directions perpendicular to the direction of their travel. However, under certain conditions, these waves can become restricted to a single direction, which is known as polarization.
This can be achieved through various methods, such as reflection, refraction, or using polarizing filters. When light reflects off surfaces like water or glass, it can become polarized. In our exercise, sunlight reflecting off a calm lake becomes completely polarized at a specific angle, known as Brewster's Angle.
In polarized light, the electric field oscillates in one direction, making it useful in many applications. This includes reducing glare in sunglasses, enhancing contrast in photography, and various scientific instruments.

Angle of Incidence

The angle of incidence is a crucial concept in understanding how light interacts with surfaces. This is the angle at which an incoming light ray hits a surface, measured between the ray itself and a line perpendicular to the surface, known as the normal.
Brewster's Angle is a special case of the angle of incidence. At Brewster's Angle, the reflected light becomes completely polarized. For different materials and their respective indices of refraction, this angle changes. It's calculated using the equation: \( \tan(\theta_B) = \frac{n_2}{n_1} \)where \( \theta_B \) stands for Brewster's Angle.
In the context of the exercise, sunlight reflects from the water surface at Brewster's Angle, which we calculated to be approximately \( 53.1^\circ \).
This angle is essential in various applications, such as designing anti-reflective coatings and understanding light behavior in nature.

Indices of Refraction

Indices of refraction, denoted as \(n\), are fundamental in understanding how light travels through different media. Each medium has its own index of refraction, which defines how much the speed of light is reduced inside the material compared to the speed of light in a vacuum.
The ratio of indices of refraction between two media influences the bending of the light, also known as refraction. In the exercise, indices of refraction for air \( n_1 = 1.0 \) and water \( n_2 = 1.33 \) are used to find Brewster's Angle.
The formula \( \tan(\theta_B) = \frac{n_2}{n_1} \) is used to calculate the angle of light needed for complete polarization upon reflection. The difference in indices explains why light bends as it moves from air into water. Knowing these values helps in designing lenses and optical devices and understanding basic principles of optics.