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Chapter 33: Problem 34

Three polarizing filters are stacked, with the polarizing axis of the second and third filters at \(23.0^{\circ}\) and 62.0 ' respectively, to that of the first. If unpolarized light is incident on the stack, the light has intensity 75.0 \(\mathrm{W} / \mathrm{cm}^{2}\) after it passes through the stack. If the incident intensity is kept constant, what is the intensity of the light after it has passed through the stack if the second polarizer is removed?

Short Answer

Expert verified
The intensity of the light after passing through the first and third polarizers is approximately 7.05 W/cm虏.

Step by step solution

01

Understand Light Intensity Through Polarizers

When unpolarized light passes through a polarizer, its intensity is reduced by half. Therefore, if the incident light through the stack is unpolarized, the intensity after the first polarizer will be \(I_1 = \frac{I_0}{2}\), where \(I_0\) is the initial intensity.
02

Malus's Law Application for the Second Filter

The light after the first polarizer becomes polarized. The intensity before reaching the third polarizer is given by Malus's Law: \(I_2 = I_1 \cos^2 \theta_2\), where \(\theta_2 = 23^{\circ}\) is the angle between the first and second polarizers. After removing the second filter, we need to calculate the intensity through the third filter directly without passing through the second.
03

Use Malus's Law for the Third Filter

Since we are removing the second polarizer, apply Malus's Law directly from the first to the third polarizer: \(I_3 = I_1 \cos^2 \theta_3\), where \(\theta_3 = 62^{\circ}\) represents the angle between the first and third polarizers.
04

Calculate Last Intensity

Given \(I_0 = 75.0\, \mathrm{W/cm^2}\), calculate the final intensity after first and third polarizers: 1. Calculate \(I_1 = \frac{I_0}{2} = \frac{75.0}{2} = 37.5\, \mathrm{W/cm^2}\).2. Then, apply Malus's Law using \(\theta_3\): \(I_3 = 37.5 \cos^2(62^{\circ})\). 3. Solve for \(I_3\): \(I_3 \approx 37.5 \times 0.188 = 7.05\, \mathrm{W/cm^2}\).

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Key Concepts

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

Polarization
Polarization is a fundamental concept in optics. It refers to the orientation of the oscillations in a light wave. Normally, light, such as sunlight, is unpolarized. This means that the light waves vibrate in multiple planes or directions.
Polarization restricts these vibrations to a single direction. This can be achieved using a polarizing filter, which allows only light oscillating in a certain direction to pass through. Therefore, polarized light results in reduced intensity compared to its unpolarized counterpart.
Understanding polarization is crucial in many applications, such as reducing glare in photography, enhancing contrast in LCD screens, and even in sunglasses to minimize reflected light.
Malus's Law
Malus's Law is a vital formula used to calculate the intensity of light passing through polarizing filters. It illustrates how light intensity is affected when it passes through a polarizer and then through additional polarizers that are at an angle to the first.
According to Malus's Law, the intensity of light, after passing through a polarizer, is given by the formula:
  • \( I = I_0 \cos^2 \theta \)
where:
  • \( I \) is the transmitted light intensity
  • \( I_0 \) is the initial light intensity
  • \( \theta \) is the angle between the light's polarization direction and the polarizer's axis
This law helps predict how light will behave with multiple polarizers in a sequence, which is crucial for understanding problems like the one with multiple stacked filters.
Light Intensity
The intensity of light refers to the amount of energy that the light possesses per unit area. It is commonly measured in watts per square centimeter (W/cm虏).
When unpolarized light passes through a polarizer, its intensity is halved. This is a basic rule in optics and is integral to analyzing the behavior of light through polarizing filters.
In systems with multiple polarizers, understanding changes in intensity is crucial. For instance, when light passes through two polarizers at a certain angle, Malus鈥檚 Law is deployed to find the light鈥檚 final intensity.
Polarizers
Polarizers are optical filters that only allow light waves of a specific polarization to pass through. There are different types of polarizers, the most common being linear polarizers.
In an experiment or setup like the exercise described, multiple polarizers can be used to control and modify the light intensity and polarization direction. Each polarizer is set at a specific orientation to achieve the desired effect.
The interaction between light and polarizers helps us understand various phenomena in optics. It also plays a significant role in applications such as photography, optical instruments, and even everyday products like screens and sunglasses.

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Most popular questions from this chapter

A beaker with a mirrored bottom is filled with a liquid whose index of refraction is \(1.63 .\) A light beam strikes the top surface of the liquid at an angle of \(42.5^{\circ}\) from the normal. At what angle from the normal will the beam exit from the liquid after traveling down through the liquid, reflecting from the mirrored bottom, and returning to the surface?

Unpolarized light of intensity 20.0 \(\mathrm{W} / \mathrm{cm}^{2}\) is incident on two polarizing filters. The axis of the first filter is at an angle of \(25.0^{\circ}\) counterclockwise from the vertical (viewed in the direction the light is traveling), and the axis of the second filter is at \(62.0^{\circ}\) counterclockwise from the vertical. What is the intensity of the light after it has passed through the second polarizer?

Light traveling in water strikes a glass plate at an angle of incidence of \(53.0^{\circ}\) : part of the beam is reflected and part is refracted. If the refiected and refracted portions make an angle of \(90.0^{\circ}\) with each other, what is the index of refraction of the glass?

A ray of light in diamond (index of refraction 2.42 ) is incident on an interface with air. What is the largest angle the ray can make with the normal and not be totally reflected back into the diamond?

When the sun is either rising or setting and appears to be just on the horizon, it is in fact below the horizon. The explanation for this seeming paradox is that light from the sun bends slightly when entering the earth's atmosphere, as shown in Fig. 33.53 . Since our perception is based on the idea that light travels in straight lines, we perceive the light to be coming from an apparent position that is an angle \(\delta\) above the sun's true position. (a) Make the simplifying assumptions that the atmosphere has uniform density, and hence uniform index of refraction \(n\) , and extends to a height \(h\) above the earth's surface, at which point it abruptly stops. Show that the angle 8 is given by $$ \delta=\arcsin \left(\frac{n R}{R+h}\right)-\arcsin \left(\frac{R}{R+h}\right) $$ where \(R=6378 \mathrm{km}\) is the radius of the earth. (b) Calculate \(\delta\) using \(n=1.0003\) and \(h=20 \mathrm{km}\) . How does this compare to the angular radius of the sun, which is about one quarter of a degree? (In actuality a light ray from the sun bends gradually, not abruptly, since the density and refractive index of the atmosphere change gradually with altitude.)
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