/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 For a particular transparent med... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 38: Problem 46

For a particular transparent medium surrounded by air, show that the critical angle for total internal reflection and the polarizing angle are related by \(\cot \theta_{p}=\sin \theta_{c}\)

Short Answer

Expert verified
Given relation \(\cot \theta_{p} = \sin \theta_{c} \) is verified using Brewster's Law and Snell's Law. This indicates the key connection between the polarizing angle and the critical angle for total internal reflection.

Step by step solution

01

Recall and Apply Brewster鈥檚 Law

Using Brewster's Law, we know that the polarizing angle \(\theta_{p}\) satisfies the equation \(\tan \theta_{p} = n \), where n is the refractive index of the denser medium with respect to the other. Hence, \(\cot \theta_{p} = 1/n \) .
02

Recall and Apply the concept of Critical Angle

The critical angle for total internal reflection, \(\theta_{c}\), is obtained using Snell's Law \(\sin \theta_{c} = 1/n \). Here n is the refractive index of the denser medium relative to air.
03

Compare the results of the above two steps

From Step 1, we have \(\cot \theta_{p} = 1/n \) and from Step 2, we have \(\sin \theta_{c} = 1/n \). Hence, their right hand sides are equal resulting in \(\cot \theta_{p} = \sin \theta_{c} \). This confirms the given relation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Brewster's Law
Brewster's Law is an essential principle in optics that deals with the reflection and polarization of light. It states that when light enters a medium from air at an angle known as the polarizing angle or Brewster's angle, \( \theta_p \), the reflected light will be perfectly polarized if it is reflected at a right angle to the refracted light.

This law is mathematically expressed as \( \tan \theta_p = n \), where \( n \) is the refractive index of the medium. The refractive index is a measure of how much the speed of light decreases inside the medium compared to the speed of light in a vacuum.

Critical Angle
The critical angle is a crucial concept in the phenomenon of total internal reflection. It is defined as the minimum angle of incidence within a denser medium at which light is completely reflected inside the medium, with no transmission into the less dense medium such as air or vacuum.

When light travels from a medium with a higher refractive index to a medium with a lower refractive index, there's an angle larger than which all the light is reflected internally. This specific angle is called the critical angle, \( \theta_c \). It can be determined using Snell's Law.
Snell's Law
Snell's Law, also known as the law of refraction, is a formula that describes how light bends when it passes between different materials. The bending of light, or refraction, is due to the change in speed as light enters from one medium with a certain refractive index into another medium with a different refractive index.

The law is expressed as \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), where \( n_1 \) and \( n_2 \) are the refractive indices of the first and second medium, respectively, while \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction.
Polarizing Angle
The polarizing angle, often referred to as Brewster's angle, is an important angle in optics where light is preferentially reflected or transmitted at specific polarizations. It is the angle of incidence at which no reflection occurs for one polarization orientation.

When light strikes a surface at this angle, the light that is reflected is completely polarized. This phenomenon is used in various applications, such as reducing glare in photography and enhancing contrast in LCD displays. The value of this angle is given by the inverse tangent of the refractive index of the medium, as Brewster's Law illustrates.
Refractive Index
The refractive index of a medium is a dimensionless number that indicates how light or any other wave propagates through that medium compared to its speed in a vacuum. It's defined by the ratio of the speed of light in a vacuum to the speed of light in the medium.

The refractive index determines how much light is bent, or refracted, when entering a medium. This value is essential when calculating the critical angle and polarizing angle, as seen in the solutions to the given exercise. Different materials have different refractive indices, which is why light bends differently through glass, water, air, and so on.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A helium-neon laser emits light that has a wavelength of \(632.8 \mathrm{nm} .\) The circular aperture through which the beam emerges has a diameter of \(0.500 \mathrm{cm} .\) Estimate the diameter of the beam \(10.0 \mathrm{km}\) from the laser.

Two wavelengths \(\lambda\) and \(\lambda+\Delta \lambda\) (with \(\Delta \lambda<<\lambda\) ) are incident on a diffraction grating. Show that the angular separation between the spectral lines in the \(m\) th-order spectrum is $$ \Delta \theta=\frac{\Delta \lambda}{\sqrt{(d / m)^{2}-\lambda^{2}}} $$ where \(d\) is the slit spacing and \(m\) is the order number.

A pinhole camera has a small circular aperture of diameter \(D\) Light from distant objects passes through the aperture into an otherwise dark box, falling on a screen located a distance L away. If \(D\) is too large, the display on the screen will be fuzzy, because a bright point in the field of view will send light onto a circle of diameter slightly larger than \(D\). On the other hand, if \(D\) is too small, diffraction will blur the display on the screen. The screen shows a reasonably sharp image if the diameter of the central disk of the diffraction pattern, specified by Equation \(38.9,\) is equal to \(D\) at the screen. (a) Show that for monochromatic light with plane wave fronts and \(L \gg D\), the condition for a sharp view is fulfilled if \(D^{2}=2.44 \lambda L .\) (b) Find the optimum pinhole diameter for 500 -nm light projected onto a screen \(15.0 \mathrm{cm}\) away.

A spy satellite can consist essentially of a large-diameter concave mirror forming an image on a digital-camera detector and sending the picture to a ground receiver by radio waves. In effect, it is an astronomical telescope in orbit, looking down instead of up. Can a spy satellite read a license plate? Can it read the date on a dime? Argue for your answers by making an order-of- magnitude calculation, specifying the data you estimate.

Narrow, parallel, glowing gas-filled tubes in a variety of colors form block letters to spell out the name of a night club. Adjacent tubes are all \(2.80 \mathrm{cm}\) apart. The tubes forming one letter are filled with neon and radiate predominantly red light with a wavelength of \(640 \mathrm{nm}\). For another letter, the tubes emit predominantly violet light at \(440 \mathrm{nm}\). The pupil of a dark-adapted viewer's eye is \(5.20 \mathrm{mm}\) in diameter. If she is in a certain range of distances away, the viewer can resolve the separate tubes of one color but not the other. Which color is easier to resolve? The viewer's distance must be in what range for her to resolve the tubes of only one color?
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Magnetism

Read Explanation

Thermodynamics

Read Explanation

Fundamentals of Physics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Torque and Rotational Motion

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.