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Chapter 23: Problem 87

\(\bullet$$\bullet\) (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. (b) Generalize this result to a stack of \(N\) slabs. What determines the final direction of the light in the last slab?

Short Answer

Expert verified
The final direction of the light depends only on the first and last slabs' refractive indices.

Step by step solution

01

Understand the Problem

We need to show that when light passes through the middle slab, it does not affect the final direction of the light in the third slab, and generalize this to N slabs. We will use Snell's Law to analyze how light refracts through each slab.
02

Apply Snell's Law to the First Slab

For the light entering the first slab from air (or a medium with refractive index \(n_0\)), use Snell's Law: \(n_0 \sin \theta_0 = n_1 \sin \theta_1\), where \(\theta_0\) is the angle of incidence and \(\theta_1\) is the angle of refraction in the first slab, and \(n_1\) is the refractive index of the first slab.
03

Light Travels Through the Middle Slab

As the light exits the first slab and enters the middle slab with refractive index \(n_2\), it bends again according to Snell's Law: \(n_1 \sin \theta_1 = n_2 \sin \theta_2\). This shows that the angle within the middle slab is \(\theta_2\).
04

Show that Final Angle Depends Only on the First and Last Slab

When light exits the middle slab and enters the last slab with refractive index \(n_3\), it refracts once more: \(n_2 \sin \theta_2 = n_3 \sin \theta_3\). Combining these equations, we return to \(n_0 \sin \theta_0 = n_3 \sin \theta_3\), showing that \(\theta_3\) depends directly on \(\theta_0\) and does not involve \(n_2\).
05

Generalize to N Slabs

For a stack of N slabs, each slab's refraction cancels out, resulting in \( n_0 \sin \theta_0 = n_N \sin \theta_N\) for the first and last slab's interface. Each intermediate slab refracts light back to the same angle as given to it, resulting in the final direction being determined solely by the refractive indices of the initial and final slabs.

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Key Concepts

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

Refraction
Refraction is the bending of light as it passes from one medium to another with a different refractive index. This phenomenon occurs because the speed of light changes with the medium.
When light travels from a medium with a lower refractive index to one with a higher refractive index, it bends towards the normal (the perpendicular line to the surface at the point of incidence). Conversely, if it travels from a higher to a lower index, it bends away from the normal.
Refraction is significant in daily life, allowing lenses to focus light and enabling devices like glasses and cameras to function effectively.
Refractive Index
The refractive index is a measure of how much a material can bend light. It is represented by the symbol \(n\) and is defined as the ratio of the speed of light in a vacuum to the speed of light in the material.
  • A higher refractive index means light travels slower in that material.
  • Different materials have distinct refractive indices, influencing how much they can bend light.
The refractive index is crucial in designing optical equipment, affecting lenses, prisms, and fiber optics.
Light Propagation
Light propagation refers to the way light travels through different materials. It generally moves in straight lines in a uniform medium, but when encountering different media, its path changes due to refraction.
In complex systems like multi-layered materials, light can bend multiple times. Despite these changes, Snell's Law ensures that in certain conditions, such as similar parallel slabs, the overall path remains consistent from the initial to the final medium.
Understanding light propagation is essential for designing and understanding optical devices, and helps predict how light will behave in various environments.
Angle of Incidence
The angle of incidence is the angle formed between the incoming light ray and the normal at the point where the light hits the surface. This angle directly influences how much the light will bend or refract.
According to Snell's Law, the angle of incidence helps determine the angle of refraction through the formula \(n_1 \sin \theta_1 = n_2 \sin \theta_2\).
The concept of the angle of incidence is pivotal in both theoretical optics and practical applications, such as optical engineering and even in everyday experiences like seeing a rainbow.
Optics
Optics is the branch of physics dedicated to studying light and its interactions with different materials. This field encompasses various phenomena such as reflection, refraction, and diffraction.
Tools used in optics include lenses and mirrors, which are designed based on principles like the refractive index and angles of incidence and refraction.
  • Optics enables advancements in technology, contributing to the development of devices like cameras, microscopes, and telescopes.
  • It also plays a critical role in understanding natural spectacles and improving visual experiences through corrective lenses.
Optics integrates principles from mathematics and physics to innovate solutions across multiple industries.

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Most popular questions from this chapter

\(\bullet$$\bullet\) 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 instead 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?

\(\bullet\) The refractive index of a certain glass is \(1.66 .\) For what angle of incidence is light that is reflected from the surface of this glass completely polarized if the glass is immersed in (a) air or (b) water?

\(\bullet$$\bullet\) In a physics lab, light with wavelength 490 nm travels in air from a laser to a photocell in 17.0 ns. When a slab of glass 0.840 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 ns to travel from the laser to the photocell. What is the wavelength of the light in the glass?

\(\bullet$$\bullet\) Plane-polarized light passes through two polarizers whose axes are oriented at \(35.0^{\circ}\) to each other. If the intensity of the original beam is reduced to \(15.0 \%,\) what was the polarization direction of the original beam, relative to the first polarizer?

\(\bullet\) Light inside the eye. The vitreous humor, a transparent, gelatinous fluid that fills most of the eyeball, has an index of refraction of 1.34 . Visible light ranges in wavelength from 400 \(\mathrm{nm}\) (violet) to \(700 \mathrm{nm}(\mathrm{red}),\) as measured in air. This light travels through the vitreous humor and strikes the rods and cones at the surface of the retina. What are the ranges of (a) the wavelength, (b) the frequency, and (c) the speed of the light just as it approaches the retina within the vitreous humor?
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