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Chapter 33: Problem 33

A beam of polarized light passes through a polarizing filter. When the angle between the polarizing axis of the filter and the direction of polarization of the light is \(\theta\) , the intensity of the emerging beam is \(I\) . If you now want the intensity to be \(I / 2,\) what should be the angle (in terms of \(\theta\) ) between the polarizing angle of the filter and the original direction of polarization of the light?

Short Answer

Expert verified
The angle should be \(\pm 45^\circ\) (or \(\pm \frac{\pi}{4}\) radians).

Step by step solution

01

Understanding the Initial Problem

When polarized light passes through a polarizing filter, the intensity of the light that emerges is determined by Malus's Law. According to this law, the transmitted intensity is given by the formula:\[ I = I_0 \cos^2(\theta) \]where \(I_0\) is the initial intensity of the light, and \(\theta\) is the angle between the light's initial polarization direction and the filter's axis.
02

Set Up the Target Intensity

We want to find the angle \(\theta'\) that makes the emerging intensity \(I' = \frac{I}{2}\). We use Malus's Law again for this desired condition:\[ \frac{I_0}{2} = I_0 \cos^2(\theta') \] We can cancel \(I_0\) from both sides since it is non-zero, leading to:\[ \frac{1}{2} = \cos^2(\theta') \]
03

Solve for the New Angle

To find \(\theta'\), we solve the equation:\[ \cos^2(\theta') = \frac{1}{2} \]Taking the square root of both sides gives us:\[ \cos(\theta') = \pm \frac{1}{\sqrt{2}} \]Thus, \(\theta'\) can be either:\[ \theta' = \frac{\pi}{4} + k\pi \text{ or } \theta' = -\frac{\pi}{4} + k\pi \]where \(k\) is any integer, since cosine is periodic with period \(\pi\). However, typically we refer to the principal value for \(\theta'\), which is \(\theta' = \frac{\pi}{4}\) (45 degrees).

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Key Concepts

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

Polarized Light
Light can be made up of waves vibrating in many directions. When these waves are restricted to a single plane, we have polarized light. This type of light travels with waves oscillating in one specific direction, making it different from typical unpolarized light, which vibrates in multiple directions.
Polarization is a common phenomenon in nature. For example, light reflecting off surfaces like water or glass often becomes polarized. Sunglasses use polarizing filters to block this specific plane of light, reducing glare. Understanding polarized light helps us grasp a wide range of optics applications.
In physics, polarizing light helps us explore wave behavior more deeply. It allows the study of phenomena like birefringence or the interaction of light with different materials.
Intensity
Intensity refers to the brightness or power of light traveling through a given area. When dealing with polarized light, intensity helps us understand how much light makes it through the polarizing filter.
The intensity of polarized light passing through a filter is not constant. It depends on the angle between the light's polarization direction and the filter's axis. The formula to express this intensity is defined by Malus's Law:
  • \( I = I_0 \cos^2(\theta) \)
where \( I \) is the transmitted intensity, \( I_0 \) is the initial intensity, and \( \theta \) is the angle in question.
Intensity is crucial for applications like photography and vision sciences, as it affects how images are captured and perceived.
Polarizing Filter
A polarizing filter is a device used to block certain orientations of light waves. When unpolarized or partially polarized light passes through a polarizing filter, only the wave components aligned with the filter's axis are transmitted.
Polarizing filters are used in many areas, including:
  • Cameras and sunglasses to reduce glare
  • Optical instruments to control light
  • Scientific experiments to study light properties
The effectiveness of a polarizing filter in blocking non-aligned light is observed in the intensity changes as indicated by Malus's Law. Filters help us analyze and measure polarization in experimental and practical settings.
Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between angles and sides in triangles. It becomes essential when working with polarized light and other physics concepts involving angles.
In our context, trigonometric functions, like the cosine function, play a key role. They help in calculating changes in light intensity as light passes through a polarizing filter:
  • \( \cos(\theta) \) and \( \cos^2(\theta) \) are directly related to light's intensity.
By using trigonometry:
  • We quantify the angle-dependency aspects of light behavior.
  • We reach solutions that explain how polarizing angles affect light.
Understanding trigonometric relationships allows for precise manipulation and prediction of light behavior in complex optical systems.
Angle of Polarization
The angle of polarization is the angle between the direction of polarized light and the axis of a polarizing filter. This angle significantly influences the amount of light that can pass through the filter.
In practice, knowing this angle helps:
  • Predict the behavior of light in experimental setups.
  • Adjust equipment to optimize light intensity for specific applications.
For example, the exercise showed how different angles alter the intensity based on Malus's Law:
  • When the angle is \( \theta' \), and we want half intensity \( (I/2) \), we solve for this angle using trigonometric equations.
By manipulating the angle of polarization, scientists can finely control light's interaction with materials, leading to innovative breakthroughs in technology and science.

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Most popular questions from this chapter

Heart Sonogram. Physicians use high-frequency \((f=1-5 \mathrm{MHz})\) sound waves, called ultrasound, to image internal organs. The speed of these ultrasound waves is 1480 \(\mathrm{m} / \mathrm{s}\) in muscle and 344 \(\mathrm{m} / \mathrm{s}\) in air. We define the index of refraction of a material for sound waves to be the ratio of the speed of sound in air to the speed of sound in the material. Snell's law then applies to the refraction of sound waves. (a) At what angle from the normal does an ultrasound beam enter the heart if it leaves the lungs at an angle of \(9.73^{\circ}\) from the normal to the heart wall? (Assume that the speed of sound in the lungs is 344 \(\mathrm{m} / \mathrm{s} .\) ) (b) What is the critical angle for sound waves in air incident on muscle?

Three Polarizing Filters. Three polarizing filters are stacked with the polarizing axes of the second and third at \(45.0^{\circ}\) and \(90.0^{\circ},\) respectively, with that of the first. (a) If unpolarized light of intensity \(I_{0}\) is incident on the stack, find the intensity and state of polarization of light emerging from each fitter. (b) If the second filter is removed, what is the intensity of the light emerging from each remaining filter?

A certain birefringent material has indexes of refraction \(n_{1}\) and \(n_{2}\) for the two per- pendicular components of linearly polarized light passing through it. The corresponding wavelengths are \(\lambda_{1}=\lambda_{0} / n_{1}\) and \(\lambda_{0} / n_{2},\) where \(\lambda_{0}\) is the wavelength in vacuum. (a) If the crystal is to function as a quarter-wave plate, the number of wavelengths of each component within the material must differ by \(\frac{1}{4}\) . Show that the minimum thickness for a quarter-wave plate is $$d=\frac{\lambda_{0}}{4\left(n_{1}-n_{2}\right)}$$ (b) Find the minimum thickness of a quarter-wave plate made of siderite \(\left(\mathrm{FeO} \cdot \mathrm{CO}_{2}\right)\) if the indexes of refraction are \(n_{1}=1.875\) and \(n_{2}=1.635\) and the wavelength in vacuum is \(\lambda_{0}=589 \mathrm{nm} .\)

A parallel beam of light in air makes an angle of \(47.5^{\circ}\) with the surface of a glass plate having a refractive index of 1.66 . (a) What is the angle between the reflected part of the beam and the surface of the glass? (b) What is the angle between the refracted beam and the surface of the glass?

A beam of light is traveling inside a solid glass cube having index of refraction \(1.53 .\) It strikes the surface of the cube from the inside. (a) If the cube is in air, at what minimum angle with the normal inside the glass will this light not enter the air at this surface? (b) What would be the minimum angle in part (a) if the cube were immersed in water?
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