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Chapter 33: Problem 47

Light that is traveling in water (with an index of refraction of 1.33) is incident on a plate of glass (with index of refraction 1.71). At what angle of incidence does the reflected light end up fully polarized?

Short Answer

Expert verified
The angle of incidence for fully polarized reflected light is approximately 52.78 degrees.

Step by step solution

01

Understanding Brewster's Angle

When the light is fully polarized upon reflection, it means it reflects at Brewster's angle. Brewster's angle occurs when the reflected and refracted rays are at 90 degrees to each other.
02

Using Brewster's Law

Brewster's Law is given by the formula: \(n_1 \tan(\theta_B) = n_2\), where \(\theta_B\) is Brewster's angle, \(n_1\) is the index of refraction of the initial medium (water, 1.33), and \(n_2\) is the index of refraction of the second medium (glass, 1.71).
03

Solving for Brewster's Angle

Rearrange Brewster's law to find \(\theta_B\): \(\tan(\theta_B) = \frac{n_2}{n_1}\). So, \(\tan(\theta_B) = \frac{1.71}{1.33}\).
04

Calculating Brewster's Angle

Find \(\theta_B\) by taking the arctangent of \(\frac{1.71}{1.33}\). Thus, \(\theta_B = \tan^{-1}\left(\frac{1.71}{1.33}\right)\).
05

Result

Calculate \(\theta_B\), which is approximately \(52.78\) degrees. This is the angle of incidence at which the reflected light becomes fully polarized.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polarization of Light
Polarization of light is a fascinating phenomenon where light waves vibrate in particular directions. Usually, light waves vibrate in multiple directions as they travel. However, when light becomes polarized, these vibrations get restricted to a single plane. This can happen by reflection, refraction, or by passing through a polarizing filter.

Understanding polarization is key to grasping concepts related to Brewster's angle. When light reflects off surfaces like water, glass, or metallic surfaces at specific angles, the reflected light becomes polarized. This means the light waves are oscillating parallel to the surface, giving us useful information about the angle and material properties.

Polarized light has real-world applications in reducing glare. This is why polarized sunglasses are so popular鈥攖hey filter out glare by blocking horizontal waves, allowing only vertically polarized light to pass through.
  • Polarized light vibrates in one direction.
  • Applications include sunglasses and photography.
  • Reflection and refraction can both cause polarization.
Index of Refraction
The index of refraction, or refractive index, is a measure of how much a medium slows down light compared to air. Each substance has its own specific refractive index, which determines how light propagates through it.

In our scenario, light travels from water (with an index of refraction of 1.33) to glass (with an index of 1.71). The higher the refractive index, the slower light travels through that medium. This concept is vital when calculating Brewster's angle as it dictates how light bends and reflects.

The refractive index plays a crucial role in optics, affecting not just speed, but also the bending of light, as described by Snell's Law. It affects lenses in glasses, cameras, and even in oceanography studies.
  • Higher index means slower light in that medium.
  • Determines bending of light at interfaces.
  • Critical for calculating polarization angles.
Brewster's Law
Brewster's Law offers insight into optics by explaining the condition under which light becomes polarized upon reflection. Named after Sir David Brewster, this law states that light is perfectly polarized at an angle where reflected and refracted rays are perpendicular.

The formula for Brewster's Law is given by \( n_1 \tan(\theta_B) = n_2 \), where \( \theta_B \) is Brewster's angle, \( n_1 \) is the refractive index of the first medium, and \( n_2 \) is the refractive index of the second medium. Calculating Brewster's angle helps determine the precise angle of incidence for achieving maximum polarization.

In the provided exercise, the reflected light becomes fully polarized at approximately \(52.78\) degrees when traveling from water to glass. This angle occurs because of the refractive indices of both materials, allowing us to understand how light behaves at these boundaries.
  • Applies when reflected light is fully polarized.
  • Formula: \( n_1 \tan(\theta_B) = n_2 \).
  • Critical in optical applications and glare reduction.

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Most popular questions from this chapter

It has been proposed that a spaceship might be propelled in the solar system by radiation pressure, using a large sail made of foil. How large must the surface area of the sail be if the radiation force is to be equal in magnitude to the Sun's gravitational attraction? Assume that the mass of the ship \(+\) sail is \(1800 \mathrm{~kg}\), that the sail is perfectly reflecting, and that the sail is oriented perpendicular to the Sun's rays. See Appendix C for needed data. (With a larger sail, the ship is continuously driven away from the Sun.)

A small spaceship with a mass of only \(1.5 \times 10^{3} \mathrm{~kg}\) (including an astronaut) is drifting in outer space with negligible gravitational forces acting on it. If the astronaut turns on a \(25 \mathrm{~kW}\) laser beam, what speed will the ship attain in \(45.0\) day because of the momentum carried away by the beam?

A plane electromagnetic wave traveling in the positive direction of an \(x\) axis in vacuum has components \(E_{x}=E_{y}=0\) and \(E_{z}=(4.0 \mathrm{~V} / \mathrm{m}) \cos \left[\left(\pi \times 10^{15} \mathrm{~s}^{-1}\right)(t-x / c)\right] .\) (a) What is the amplitude of the magnetic field component? (b) Parallel to which axis does the magnetic field oscillate? (c) When the electric field component is in the positive direction of the \(z\) axis at a certain point \(P\), what is the direction of the magnetic field component there? (d) In what direction is the wave moving?

Light in vacuum is incident on the surface of a glass slab. In the vacuum the beam makes an angle of \(32.0^{\circ}\) with the normal to the surface, while in the glass it makes an angle of \(16.0^{\circ}\) with the normal. What is the index of refraction of the glass?

Prove, for a plane electromagnetic wave that is normally incident on a flat surface, that the radiation pressure on the surface is equal to the energy density in the incident beam. (This relation between pressure and energy density holds no matter what fraction of the incident energy is reflected.)
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