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Chapter 35: Problem 36

For \(589-\) nm light, calculate the critical angle for the following materials surrounded by air: (a) diamond, (b) flint glass, and (c) ice.

Short Answer

Expert verified
The approximate critical angles for \(589-\) nm light passing from diamond, flint glass, and ice to air are \(24.41^\circ\), \(38.66^\circ\), and \(48.76^\circ\), respectively.

Step by step solution

01

Find the refractive indices

We first need to look up the refractive indices for diamond, flint glass, and ice. The refractive index of diamond is \(2.42\), for flint glass it is \(1.62\), and for ice it's \(1.31\).
02

Calculate the critical angle for diamond

The formula for the critical angle \(\theta_c\) is given by \(\theta_c = arcsin(\frac{n_2}{n_1})\). Plugging in the refractive index of diamond (\(n_1 = 2.42\)) into this equation, we get \(\theta_c = arcsin(\frac{1}{2.42})\). Therefore, the critical angle for diamond is approximately \(24.41^\circ\).
03

Calculate the critical angle for flint glass

Using the same equation as above, but with the refractive index of flint glass (\(n_1 = 1.62\)), we find \(\theta_c = arcsin(\frac{1}{1.62})\), which gives a critical angle of approximately \(38.66^\circ\).
04

Calculate the critical angle for ice

Lastly, using the refractive index for ice (\(n_1=1.31\)) in the equation, we calculate \(\theta_c = arcsin(\frac{1}{1.31})\), yielding a critical angle of about \(48.76^\circ\).

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Key Concepts

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

Refractive Index
The refractive index, sometimes called the index of refraction, is a measure of how light propagates through a material. Every material has a unique refractive index, which describes how much the light will bend, or refract, when entering the material. It is denoted by the symbol \( n \).
  • **Formula:** The refractive index is given by \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium.
  • **Values:** For instance, the refractive index for air is approximately \( 1 \), whereas for diamond, it is \( 2.42 \). This means light travels fastest in a vacuum and slower in diamond.

A higher refractive index indicates a stronger "bending" or slowing down of the light path. This property is crucial in calculating the critical angle, which is the angle of incidence above which total internal reflection occurs.
Total Internal Reflection
Total Internal Reflection occurs when light is completely reflected at the boundary of two materials. This happens when light travels from a denser to a less dense medium, like from water to air.
  • **Critical Angle Definition:** The critical angle is the minimum angle of incidence inside a medium at which light is entirely reflected back into that medium. Beyond this angle, all light is reflected back, and none refracts into the next medium.
  • **Calculation:** If \( n_1 \) is the refractive index of the denser medium and \( n_2 \) is the refractive index of the less dense medium (usually air, which has \( n_2 = 1 \)), the critical angle \( \theta_c \) can be calculated using \( \theta_c = \arcsin\left(\frac{n_2}{n_1}\right) \).

This concept is widely used in fiber optics and various optical devices to trap light inside a medium and guide it efficiently.
Optics
Optics is the branch of physics that deals with the study of light and its interactions with different materials. It covers various phenomena, including reflection, refraction, and diffraction.
  • **Reflection:** When light bounces back upon hitting a surface, such as a mirror. Total internal reflection is a special type of reflection.
  • **Refraction:** The bending of light as it passes from one medium to another, which is dependent on the refractive indices of the media involved.
  • **Applications:** Optics is fundamental in designing lenses, microscopes, glasses, cameras, and numerous optical instruments.

Understanding the basics of optics allows us to design improved optical systems. For instance, knowing the critical angle helps in creating lenses or waveguides that ensure efficient light transmission. The study of optics leads to innovations in technology, improving everything from medical imaging to telecommunications.

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Most popular questions from this chapter

Two flat rectangular mirrors, both perpendicular to a horizontal sheet of paper, are set edge to edge with their reflecting surfaces perpendicular to each other. (a) A light ray in the plane of the paper strikes one of the mirrors at an arbitrary angle of incidence \(\theta_{1} .\) Prove that the final direction of the ray, after reflection from both mirrors, is opposite to its initial direction. In a clothing store, such a pair of mirrors shows you an image of yourself as others see you, with no apparent right-left reversal. (b) What If? Now assume that the paper is replaced with a third flat mirror, touching edges with the other two and perpendicular to both. The set of three mirrors is called a corner-cube reflector. A ray of light is incident from any direction within the octant of space bounded by the reflecting surfaces. Argue that the ray will reflect once from each mirror and that its final direction will be opposite to its original direction. The Apollo 11 astronauts placed a panel of corner cube retro reflectors on the Moon. Analysis of timing data taken with it reveals that the radius of the Moon's orbit is increasing at the rate of \(3.8 \mathrm{cm} / \mathrm{yr}\) as it loses kinetic energy because of tidal friction.

A ray of light strikes a flat block of glass \((n=1.50)\) of thickness \(2.00 \mathrm{cm}\) at an angle of \(30.0^{\circ}\) with the normal. Trace the light beam through the glass, and find the angles of incidence and refraction at each surface.

Consider a common mirage formed by super-heated air just above a roadway. A truck driver whose eyes are \(2.00 \mathrm{m}\) above the road, where \(n=1.0003,\) looks forward. She perceives the illusion of a patch of water ahead on the road, where her line of sight makes an angle of \(1.20^{\circ}\) below the horizontal. Find the index of refraction of the air just above the road surface. (Suggestion: Treat this as a problem in total internal reflection.)

The wavelength of red helium-neon laser light in air is \(632.8 \mathrm{nm} .\) (a) What is its frequency? (b) What is its wavelength in glass that has an index of refraction of \(1.50 ?\).(c) What is its speed in the glass?

The shoreline of a lake runs east and west. A swimmer gets into trouble \(20.0 \mathrm{m}\) out from shore and \(26.0 \mathrm{m}\) to the east of a lifeguard, whose station is \(16.0 \mathrm{m}\) in from the shoreline. The lifeguard takes negligible time to accelerate. He can run at \(7.00 \mathrm{m} / \mathrm{s}\) and swim at \(1.40 \mathrm{m} / \mathrm{s} .\) To reach the swimmer as quickly as possible, in what direction should the lifeguard start running? You will need to solve a transcendental equation numerically.
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