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Chapter 33: Problem 31

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?

Short Answer

Expert verified
The intensity after the second polarizer is approximately 6.37 W/cm虏.

Step by step solution

01

Calculate Intensity after First Polarizer

For unpolarized light passing through a polarizing filter, the intensity after the first filter is half of the initial intensity. Hence, the intensity after the first filter is given by:\[ I_1 = \frac{I_0}{2} \]where \( I_0 = 20.0 \, \text{W/cm}^2 \). Thus,\[ I_1 = \frac{20.0}{2} = 10.0 \, \text{W/cm}^2 \]
02

Determine Relative Angle Between Polarizers

Calculate the angle between the axes of the first and second polarizers. The angle of the second filter relative to the first is:\[ \theta = 62.0^\circ - 25.0^\circ = 37.0^\circ \]
03

Calculate Intensity after Second Polarizer

Use Malus's Law to find the intensity of light after it passes through the second polarizer. Malus's Law is given by:\[ I_2 = I_1 \cos^2(\theta) \]Substituting \( I_1 = 10.0 \, \text{W/cm}^2 \) and \( \theta = 37.0^\circ \):\[ I_2 = 10.0 \times \cos^2(37.0^\circ) \approx 10.0 \times (0.7986)^2 = 10.0 \times 0.637 = 6.37 \, \text{W/cm}^2 \]

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Key Concepts

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

Malus's Law
Light can be controlled and manipulated using principles like Malus's Law, which describes how light intensity changes when passing through polarizing filters. Malus's Law is particularly important when dealing with polarized light. It makes use of the cosine function to determine the intensity of light after passing through a polarizer.
Malus's Law formula:\[ I = I_0 \cos^2(\theta) \]Where:
  • \( I_0 \) is the initial intensity of polarized light.
  • \( I \) is the intensity after passing through the filter.
  • \( \theta \) is the angle between the light's polarization direction and the filter's axis.
This law highlights that as the angle approaches 90 degrees, less light gets through. At 0 degrees, the intensity remains unchanged, as the light aligns fully with the filter's axis. Understanding this concept allows us to manipulate light effectively in optical applications.
Unpolarized Light
Light from most natural sources like the sun or a lamp is considered unpolarized. This means that the light waves vibrate in all directions perpendicular to the direction of travel. Unpolarized light can be visualized as a jumble of planes, each vibrating in a different direction.
When unpolarized light encounters a polarizing filter, it becomes polarized. This occurs because the polarizing filter blocks all waves except those vibrating in one specific direction. The phenomenon that results in polarization drastically reduces the light's intensity.For unpolarized light passing through a single polarizing filter, the intensity is simply halved:\[ I = \frac{I_0}{2} \]Where \( I_0 \) is the initial intensity. This behavior is due to the filter only allowing through light components aligned with its axis, effectively reducing the overall intensity by 50%.
Polarizing Filters
Polarizing filters play a key role in controlling light. These devices only allow light waves aligned with their axis to pass through, blocking other directions. This unique feature makes them valuable in many practical applications, such as reducing glare in photography or improving clarity in sunglasses.
When light passes through a second polarizing filter after the first, we have to consider the alignment of the axes, as determined by Malus's Law. The light that emerges retains only the components that align with the second filter's orientation.
This sequential filtering makes polarizers exceptional tools for controlling light intensity. Adjusting the angle between them can manage the light's brightness and reduce unwanted reflections effectively. Their practicality extends beyond scientific uses, as they are embedded in everyday technologies enhancing visual experiences.

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Most popular questions from this chapter

In a physics lab, light with wavelength 490 \(\mathrm{nm}\) travels in air from a laser to a photocell in 17.0 \(\mathrm{ns}\) . When a slab of glass 0.840 \(\mathrm{m}\) thick is placed in the light beam, with the beam incident along the normal to the parallel faces of the slab, it takes the light 21.2 \(\mathrm{ns}\) to travel from the laser to the photocell. What is the wavelength of the light in the glass?

A polarizer and an analyzer are oriented so that the maximum amount of light is transmitted. To what fraction of its maximum value is the intensity of the transmitted light reduced when the analyzer is rotated through (a) \(22.5^{\circ} ;(\mathrm{b}) 45.0^{\circ} ;(\mathrm{c}) 67.5^{\circ} ?\)

A glass plate 2.50 \(\mathrm{mm}\) thick, with an index of refraction of \(1.40,\) is placed between a point source of light with wavelength 540 \(\mathrm{nm}\) (in vacuum) and a screen. The distance from source to screen is 1.80 \(\mathrm{cm}\) . How many wavelengths are there between the source and the screen?

Prove that a ray of light reflected from a plane mirror rotates through an angle of 2\(\theta\) when the mirror rotates through an angle \(\theta\) about an axis perpendicular to the plane of incidence.

(a) Light passes through three parallel slabs of different thicknesses and refractive indexes. The light is incident in the first slab and finally refracts into the third slab. Show that the middle slab has no effect on the final direction of the light. That is, show that the direction of the light in the third slab is the same as if the light had passed directly from the first slab into the third slab. Generalize this result to a stack of \(N\) slabs. What determines the final direction of the light in the last slab?
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