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Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 35

The Nature of Light and the Laws of Geometric Optics - all with Video Answers

Educators

Chapter Questions

01:59

Problem 1

The Apollo 11 astronauts set up a highly reflecting panel on the Moon's surface. The speed of light can be found by measuring the time it takes a laser beam to travel from Earth, reflect from the retroreflector, and return to Farth. If this interval is measured to be $2.51 \mathrm{~s}$, what is the measured speed of light? Take the centerto-center distance from the Earth to the Moon to be $3.84 \times 10^{8} \mathrm{~m}$, and do not neglect the sizes of the Earth and the Moon.

Keshav Singh
Keshav Singh
Numerade Educator
01:06

Problem 2

As a result of his observations, Roemer concluded that eclipses of Io by Jupiter were delayed by 22 min during a six-month period as the Earth moved from the point in its orbit where it is closest to Jupiter to the diametrically opposite point where it is farthest from Jupiter. Using $1.50 \times 10^{8} \mathrm{~km}$ as the average radius of the Earth's orbit around the Sun, calculate the speed of light from these data.

Narayan Hari
Narayan Hari
Numerade Educator
06:39

Problem 3

In an experiment to measure the speed of light using the apparatus of Fizeau (see Fig. $35.2$ ), the distance between light source and mirror was $11.45 \mathrm{~km}$ and the wheel had 720 notches. The experimentally determined value of $c$ was $2.998 \times 10^{8} \mathrm{~m} / \mathrm{s} .$ Calculate the minimum angular speed of the wheel for this experiment.

Aatish Gupta
Aatish Gupta
Numerade Educator
03:36

Problem 4

Figure $\mathrm{P} 35.4$ shows an apparatus used to measure the speed distribution of gas molecules. It consists of two slotted rotating disks separated by a distance $d$, with the slots displaced by the angle $\theta$. Suppose that the speed of light is measured by sending a light beam from the left through this apparatus. (a) Show that a light beam will be seen in the detector (that is, will make it through both slots) only if its speed is given by $c=\omega d / \theta$, where $\omega$ is the angular speed of the disks and $\theta$ is measured in radians. (b) What is the measured speed of light if the distance between the two slotted rotating disks is $2.50 \mathrm{~m}$, the slot in the second disk is displaced $1 / 60$ of $1^{\circ}$ from the slot in the first disk, and the disks are rotating at $5555 \mathrm{rev} / \mathrm{s} ?$

Keshav Singh
Keshav Singh
Numerade Educator
01:47

Problem 5

A narrow beam of sodium yellow light, with wavelength $589 \mathrm{~nm}$ in vacuum, is incident from air onto a smooth water surface at an angle $\theta_{1}=35.0^{\circ} .$ Dctermine the angle of refraction $\theta_{2}$ and the wavelength of the light in water.

Keshav Singh
Keshav Singh
Numerade Educator
02:27

Problem 6

The wavelength of red helium-neon laser light in air is $682.8 \mathrm{~nm} .$ (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?

Keshav Singh
Keshav Singh
Numerade Educator
00:40

Problem 7

An underwater scuba diver sees the Sun at an apparent angle of $45.0^{\circ}$ from the vertical. What is the actual direction of the Sun?

Salamat Ali
Salamat Ali
Numerade Educator
03:47

Problem 8

A lascr beam is incident at an angle of $30.0^{\circ}$ from the vertical onto a solution of corn syrup in water. If the beam is refracted to $19.24^{\circ}$ from the vertical, (a) what is the index of refraction of the syrup solution? Suppose that the light is red, with a vacuum wavelength of $632.8 \mathrm{~nm}$. Find its (b) wavelength, (c) frequency, and
(d) speed in the solution.

Keshav Singh
Keshav Singh
Numerade Educator
01:51

Problem 9

Find the speed of light in (a) flint glass, (b) water, and
(c) cubic zirconia.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:33

Problem 10

A light ray initially in water enters a transparent substance at an angle of incidence of $37.0^{\circ}$, and the transmitted ray is refracted at an angle of $25.0^{\circ} .$ Calculate the speed of light in the transparent substance.

Bethany Campbell
Bethany Campbell
Numerade Educator
02:55

Problem 11

. 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.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:08

Problem 12

Light of wavelength $436 \mathrm{~nm}$ in air enters a fishbowl filled with water and then exits through the crown glass wall of the container. What is the wavelength of the light (a) in the water and (b) in the glass?

Mayukh Banik
Mayukh Banik
Numerade Educator
02:06

Problem 13

An opaque cylindrical tank with an open top has a diameter of $3.00 \mathrm{~m}$ and is completely filled with water. When the setting Sun reaches an angle of $28.0^{\circ}$ above the horizon, sunlight ceases to illuminate any part of the bottom of the tank. How deep is the tank?

Keshav Singh
Keshav Singh
Numerade Educator
01:27

Problem 14

The angle between the two mirrors illustrated in Figure P35. 14 is a right angle. The beam of light in the vertical plane $P$ strikes mirror 1 as shown. (a) Determine the distance that the reflected light beam travels before striking mirror 2. (b) In what direction does the light beam travel after being reflected from mirror $2 ?$

Keshav Singh
Keshav Singh
Numerade Educator
05:44

Problem 15

How many times will the incident beam shown in Figure P35.15 be reflected by each of the parallel mirrors?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:25

Problem 16

When the light illustrated in Figure $\mathrm{P} 35.16$ passes through the glass block, it is shifted laterally by the distance $d$. If $n=1.50$, what is the value of $d$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:29

Problem 17

Find the time required for the light to pass through the glass block described in Problem 16 .

Keshav Singh
Keshav Singh
Numerade Educator
04:04

Problem 18

The light beam shown in Figure P35.18 makes an angle of $20.0^{\circ}$ with the normal line $N N^{\prime}$ in the linseed oil. Determine the angles $\theta$ and $\theta^{\prime} .$ (The index of refraction for linseed oil is $1.48 .$ )

Aatish Gupta
Aatish Gupta
Numerade Educator
01:11

Problem 19

Two light pulses are emitted simultaneously from a source. Both pulses travel to a detector, but one first passes through $6.20 \mathrm{~m}$ of ice. Determine the difference in the pulses' times of arrival at the detector.

Keshav Singh
Keshav Singh
Numerade Educator
03:14

Problem 20

When you look through a window, by how much time is the light you see delayed by having to go through glass instead of air? Make an order-of-magnitude estimate on the basis of data you specify. By how many wavelengths is it delayed?

Aatish Gupta
Aatish Gupta
Numerade Educator
03:45

Problem 21

Light passes from air into flint glass. (a) What angle of incidence must the light have if the component of its velocity perpendicular to the interface is to remain constant? (b) Can the component of velocity parallel to the interface remain constant during refraction?

Keshav Singh
Keshav Singh
Numerade Educator
02:46

Problem 22

The reflecting surfaces of two intersecting flat mirrors are at an angle of $\theta\left(0^{\circ}<\theta<90^{\circ}\right)$, as shown in Figure P35.22. If a light ray strikes the horizontal mirror, show that the emerging ray will intersect the incident ray at an angle of $\beta=180^{\circ}-2 \theta$.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
03:52

Problem 23

A light ray enters the atmosphere of a planet and descends vertically $20.0 \mathrm{~km}$ to the surface. The index of refraction where the light enters the atmosphere is $1.000$, and it increases linearly to the surface where it has a value of $1.005 .$ (a) How long does it take the ray to traverse this path? (b) Compare this to the time it takes in the absence of an atmosphere.

Keshav Singh
Keshav Singh
Numerade Educator
03:32

Problem 24

A light ray enters the atmosphere of a planet and descends vertically to the surface a distance $h$. The index of refraction where the light enters the atmosphere is $1.000$, and it increases linearly to the surface where it has a value of $n$. (a) How long does it take the ray to traverse this path? (b) Compare this to the time it takes in the absence of an atmosphere.

Keshav Singh
Keshav Singh
Numerade Educator
02:40

Problem 25

A narrow white light beam is incident on a block of fused quartz at an angle of $30.0^{\circ} .$ Find the angular width of the light beam inside the quartz.

Aatish Gupta
Aatish Gupta
Numerade Educator
03:05

Problem 26

A ray of light strikes the midpoint of one face of an equiangular glass prism $(n=1.50)$ at an angle of incidence of $30.0^{\circ}$. Trace the path of the light ray through the glass, and find the angles of incidence and refraction at each surface.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
02:48

Problem 27

A prism that has an apex angle of $50.0^{\circ}$ is made of cubic zirconia, with $n=2.20$. What is its angle of minimum deviation?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:20

Problem 28

Light with a wavelength of $700 \mathrm{~nm}$ is incident on the face of a fused quartz prism at an angle of $75.0^{\circ}$ (with respect to the normal to the surface). The apex angle of the prism is $60.0^{\circ} .$ Using the value of $n$ from Figure $35.20$, calculate the angle (a) of refraction at this first surface, $(b)$ of incidence at the second surface, $(c)$ of refraction at the second surface, and (d) between the incident and emerging rays.

Keshav Singh
Keshav Singh
Numerade Educator
03:25

Problem 29

The index of refraction for violet light in silica flint glass is $1.66$, and that for red light is $1.62$. What is the angular dispersion of visible light passing through a prism of apex angle $60.0^{\circ}$ if the angle of incidence is $50.0^{\circ}$ ? (See Fig. P35.29.)

Keshav Singh
Keshav Singh
Numerade Educator
04:44

Problem 30

Show that if the apex angle $\Phi$ of a prism is small, an approximate value for the angle of minimum deviation is $\delta_{\min }=(n-1) \Phi$.

Ajay Singhal
Ajay Singhal
Numerade Educator
03:44

Problem 31

A triangular glass prism with an apex angle of $\Phi=60.0^{\circ}$ has an index of refraction $n=1.50$ (Fig. P35.31). What is the smallest angle of incidence $\theta_{1}$ for which a light ray can emerge from the other side?

Shoukat Ali
Shoukat Ali
Other Schools
00:00

Problem 32

A triangular glass prism with an apex angle of $\Phi$ has an index of refraction $n$ (Fig. P35.31). What is the smallest angle of incidence $\theta_{1}$ for which a light ray can emerge from the other side?

Mayukh Banik
Mayukh Banik
Numerade Educator
00:57

Problem 33

An experimental apparatus includes a prism made of sodium chloride. The angle of minimum deviation for light of wavelength $589 \mathrm{~nm}$ is to be $10.0^{\circ}$. What is the required apex angle of the prism?

Amit Srivastava
Amit Srivastava
Numerade Educator
08:26

Problem 34

A triangular glass prism with an apex angle of $60.0^{\circ}$ has an index of refraction of $1.50 .$ (a) Show that if its angle of incidence on the first surface is $\theta_{1}=48.6^{\circ}$, light will pass symmetrically through the prism, as shown in Figure $35.26$. (b) Find the angle of deviation $\delta_{\min }$ for $\theta_{1}=$ $48.6^{\circ} .$ (c) Find the angle of deviation if the angle of incidence on the first surface is $45.6^{\circ} .$ (d) Find the angle of deviation if $\theta_{1}=51.6^{\circ}$.

Aatish Gupta
Aatish Gupta
Numerade Educator
02:17

Problem 35

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

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:13

Problem 36

Repeat Problem 35 for the situation in which the materials are surrounded by water.

Keshav Singh
Keshav Singh
Numerade Educator
01:21

Problem 37

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.

Keshav Singh
Keshav Singh
Numerade Educator
01:58

Problem 38

Determine the maximum angle $\theta$ for which the light rays incident on the end of the pipe shown in Figure P35.38 are subject to total internal reflection along the walls of the pipe. Assume that the pipe has an index of refraction of $1.36$ and the outside medium is air.
3

Keshav Singh
Keshav Singh
Numerade Educator
03:27

Problem 39

A glass fiber $(n=1.50)$ is submerged in water $(n=1.33)$. What is the critical angle for light to stay inside the optical fiber?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:03

Problem 40

A glass cube is placed on a newspaper, which rests on a table. A person reads all of the words the cube covers, through all of one vertical side. Determine the maximum possible index of refraction of the glass.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:58

Problem 41

A large Lucite cube $(n=1.59)$ has a small air bubble (a defect in the casting process) below one surface. When a penny (diameter, $1.90 \mathrm{~cm}$ ) is placed directly over the bubble on the outside of the cube, one cannot see the bubble by looking down into the cube at any angle. However, when a dime (diameter, $1.75 \mathrm{~cm}$ ) is placed directly over it, one can see the bubble by looking down into the cube. What is the range of the possible depths of the air bubble beneath the surface?

Keshav Singh
Keshav Singh
Numerade Educator
03:22

Problem 42

A room contains air in which the speed of sound is $343 \mathrm{~m} / \mathrm{s}$. The walls of the room are made of concrete, in which the speed of sound is $1850 \mathrm{~m} / \mathrm{s}$. (a) Find the critical angle for total internal reflection of sound at the concrete-air boundary. (b) In which medium must the sound be traveling to undergo total internal reflection?
(c) "A bare concrete wall is a highly efficient mirror for sound." Give evidence for or against this statement.

Aatish Gupta
Aatish Gupta
Numerade Educator
01:53

Problem 43

In about 1965 , engineers at the Toro Company invented a gasoline gauge for small engines, diagrammed in Figure $\mathrm{P} 35.43 .$ The gauge has no moving parts. It consists of a flat slab of transparent plastic fitting vertically into a slot in the cap on the gas tank. None of the plastic has a reflective coating. The plastic projects from the horizontal top down nearly to the bottom of the opaque tank. Its lower edge is cut with facets making angles of $45^{\circ}$ with the horizontal. A lawnmower operator looks down from above and sees a boundary between bright and dark on the gauge. The location of the boundary, across the width of the plastic, indicates the quantity of gasoline in the tank. Explain how the gauge works. Explain the design requirements, if any, for the index of refraction of the plastic.

Keshav Singh
Keshav Singh
Numerade Educator
03:29

Problem 44

The shoreline of a lake runs from east to 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 a negligible amount of 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.

Keshav Singh
Keshav Singh
Numerade Educator
01:56

Problem 45

A narrow beam of light is incident from air onto a glass surface with an index of refraction of $1.56$. Find the an-
gle of incidence for which the corresponding angle of refraction is one half the angle of incidence. (Hint: You might want to use the trigonometric identity $\sin 2 \theta=$ $2 \sin \theta \cos \theta .)$

Keshav Singh
Keshav Singh
Numerade Educator
02:41

Problem 46

(a) Consider a horizontal interface between air above and glass with an index of $1.55$ below. Draw a light ray incident from the air at an angle of incidence of $30.0^{\circ}$. Determine the angles of the reflected and refracted rays and show them on the diagram. (b) Suppose instead that the light ray is incident from the glass at an angle of incidence of $30.0^{\circ}$. Determine the angles of the reflected and refracted rays and show all three rays on a new diagram. (c) For rays incident from the air onto the air-glass surface, determine and tabulate the angles of reflection and refraction for all the angles of incidence at $10.0^{\circ}$ intervals from 0 to $90.0^{\circ} .$ (d) Do the same for light rays traveling up to the interface through the glass.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
01:27

Problem 47

A small underwater pool light is $1.00 \mathrm{~m}$ below the surface. The light emerging from the water forms a circle on the water's surface. What is the diameter of this circle?

Keshav Singh
Keshav Singh
Numerade Educator
01:27

Problem 48

One technique for measuring the angle of a prism is shown in Figure P35.48. A parallel beam of light is directed on the angle so that the beam reflects from opposite sides. Show that the angular separation of the two beams is given by $B=2 A$.

Keshav Singh
Keshav Singh
Numerade Educator
04:00

Problem 49

The walls of a prison cell are perpendicular to the four cardinal compass directions. On the first day of spring, light from the rising Sun enters a rectangular window in the eastern wall. The light traverses $2.37 \mathrm{~m}$ horizontally to shine perpendicularly on the wall opposite the window. A young prisoner observes the patch of light moving across this western wall and for the first time forms his own understanding of the rotation of the Earth.
(a) With what speed does the illuminated rectangle move? (b) The prisoner holds a small square mirror flat against the wall at one corner of the rectangle of light. The mirror reflects light back to a spot on the eastern wall close beside the window. How fast does the smaller square of light move across that wall? (c) Seen from a latitude of $40.0^{\circ}$ north, the rising Sun moves through the sky along a line making a $50.0^{\circ}$ angle with the southeastern horizon. In what direction does the rectangular patch of light on the western wall of the prisoner's cell move? (d) In what direction does the smaller square of light on the eastern wall move?

Keshav Singh
Keshav Singh
Numerade Educator
02:52

Problem 50

The laws of refraction and reflection are the same for sound as for light. The speed of sound in air is $340 \mathrm{~m} / \mathrm{s}$, and that of sound in water is $1510 \mathrm{~m} / \mathrm{s}$. If a sound wave approaches a plane water surface at an angle of incidence of $12.0^{\circ}$, what is the angle of refraction?

Julie Farhm
Julie Farhm
Numerade Educator
00:33

Problem 51

Cold sodium atoms (near absolute zero) in a state called a Bose-Einstein condensate can slow the speed of light from its normally high value to a speed approaching that of an automobile in a city. The speed of light in one such medium was recorded as $61.15 \mathrm{~km} / \mathrm{h}$.
(a) Find the index of refraction of this medium.
(b) What is the critical angle for total internal reflection if the condensate is surrounded by vacuum?

Adnan Gill
Adnan Gill
Numerade Educator
05:07

Problem 52

A narrow beam of white light is incident at $25.0^{\circ}$ onto a slab of heavy flint glass $5.00 \mathrm{~cm}$ thick. The indices of refraction of the glass at wavelengths of $400 \mathrm{~nm}$ and $700 \mathrm{~nm}$ are $1.689$ and $1.642$, respectively. Find the width of the visible beam as it emerges from the slab.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:33

Problem 53

A hiker stands on a mountain peak near sunset and observes a rainbow caused by water droplets in the air $8.00 \mathrm{~km}$ away. The valley is $2.00 \mathrm{~km}$ below the mountain peak and entirely flat. What fraction of the complete circular arc of the rainbow is visible to the hiker? (See Fig. $35.25 .$ )

Keshav Singh
Keshav Singh
Numerade Educator
00:31

Problem 54

A fish is at a depth $d$ under water. Take the index of refraction of water as $4 / 3$. Show that when the fish is viewed at an angle of refraction $\theta_{1}$, the apparent depth $z$ of the fish is
$$z=\frac{3 d \cos \theta_{1}}{\sqrt{7+9 \cos ^{2} \theta_{1}}}$$

Prem Bijarniya
Prem Bijarniya
Numerade Educator
03:17

Problem 55

A laser beam strikes one end of a slab of material, as shown in Figure P35.55. The index of refraction of the slab is $1.48 .$ Determine the number of internal reflections of the beam before it emerges from the opposite end of the slab.

Zachary Warner
Zachary Warner
Numerade Educator
03:30

Problem 56

When light is normally incident on the interface between two transparent optical media, the intensity of the reflected light is given by the expression
$$S_{1}^{\prime}=\left(\frac{n_{2}-n_{1}}{n_{2}+n_{1}}\right)^{2} S_{1}$$
In this equation, $S_{1}$ represents the average magnitude of the Poynting vector in the incident light (the incident intensity), $S_{1}^{\prime}$ is the reflected intensity, and $n_{1}$ and $n_{2}$ are the refractive indices of the two media. (a) What fraction of the incident intensity is reflected for $589-\mathrm{nm}$ light normally incident on an interface between air and crown glass? (b) In part (a), does it matter whether the light is in the air or in the glass as it strikes the interface? (c) A Bose-Einstein condensate (see Problem 51 ) has an index of refraction of $1.76 \times 10^{7}$. Find the percent reflection for light falling perpendicularly on its surface. What would the condensate look like?

Keshav Singh
Keshav Singh
Numerade Educator
04:21

Problem 57

Refer to Problem 56 for a description of the reflected intensity of light normally incident on an interface between two transparent media. (a) When light is normally incident on an interface between vacuum and a transparent medium of index $n$, show that the intensity $S_{2}$ of the transmitted light is given by the expression $S_{2} / S_{1}=4 n /(n+1)^{2}$. (b) Light travels perpendicularly through a diamond slab, surrounded by air, with parallel surfaces of entry and exit. Apply the transmission fraction in part (a) to find the approximate overall transmission through the slab of diamond as a percentage. Ignore light reflected back and forth within the slab.

Keshav Singh
Keshav Singh
Numerade Educator
04:36

Problem 58

This problem builds upon the results of Problems 56 and $57 .$ Light travels perpendicularly through a diamond slab, surrounded by air, with parallel surfaces of entry and exit. What fraction of the incident intensity is the intensity of the transmitted light? Include the effects of light reflected back and forth inside the slab.

Keshav Singh
Keshav Singh
Numerade Educator
02:29

Problem 59

The light beam shown in Figure $\mathrm{P} 35.59$ strikes surface 2 at the critical angle. Determine the angle of incidence, $\theta_{1}$.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
02:58

Problem 60

A (0. A $4.00$ -m-long pole stands vertically in a lake having a depth of $2.00 \mathrm{~m}$. When the Sun is $40.0^{\circ}$ above the horizontal, determine the length of the pole's shadow on the bottom of the lake. Take the index of refraction for
water to be $1.33$.

Keshav Singh
Keshav Singh
Numerade Educator
03:19

Problem 61

A light ray of wavelength $589 \mathrm{~nm}$ is incident at an angle $\theta$ on the top surface of a block of polystyrene, as shown in Figure $\mathrm{P} 35.61 .$ (a) Find the maximum value of $\theta$ for which the refracted ray undergoes total internal reflection at the left vertical face of the block. Repeat the calculation for the case in which the polystyrene block is immersed in (b) water and (c) carbon disulfide.

Keshav Singh
Keshav Singh
Numerade Educator
03:04

Problem 62

A ray of light passes from air into water. For its deviation angle $\delta=\left|\theta_{1}-\theta_{2}\right|$ to be $10.0^{\circ}$, what must be its angle of incidence?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:51

Problem 63

A shallow glass dish is $4.00 \mathrm{~cm}$ wide at the bottom, as shown in Figure $\mathrm{P} 35.63$. When an observer's eye is positioned as shown, the observer sees the edge of the bottom of the empty dish. When this dish is filled with water, the observer sees the center of the bottom of the dish. Find the height of the dish.

Keshav Singh
Keshav Singh
Numerade Educator
04:47

Problem 64

A material having an index of refraction $n$ is surrounded by a vacuum and is in the shape of a quarter circle of radius $R$ (Fig. P35.64). A light ray parallel to the base of the material is incident from the left at a dis-
tance of $L$ above the base and emerges out of the material at the angle $\theta$. Determine an expression for $\theta$.

Keshav Singh
Keshav Singh
Numerade Educator
02:58

Problem 65

Derive the law of reflection (Eq. $35.2$ ) from Fermat's principle of least time. (See the procedure outlined in Section $35.9$ for the derivation of the law of refraction from Fermat's principle.)

Keshav Singh
Keshav Singh
Numerade Educator
03:36

Problem 66

A transparent cylinder of radius $R=2.00 \mathrm{~m}$ has a mirrored surface on its right half, as shown in Figure P35.66. A light ray traveling in air is incident on the left side of the cylinder. The incident light ray and exiting light ray are parallel and $d=2.00 \mathrm{~m} .$ Determine the index of refraction of the material.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:15

Problem 67

A. H. Pfund's method for measuring the index of refraction of glass is illustrated in Figure P35.67. One face of a slab of thickness $t$ is painted white, and a small hole scraped clear at point $P$ serves as a source of diverging rays when the slab is illuminated from below. Ray $P B B^{\prime}$ strikes the clear surface at the critical angle and is totally reflected, as are rays such as $P C C^{\prime}$. Rays such as $P A A^{\prime}$ emerge from the clear surface. On the painted surface there appears a dark circle of diameter $d$, surrounded by an illuminated region, or halo. (a) Derive a formula for $n$ in terms of the measured quantities $d$ and
t. (b) What is the diameter of the dark circle if $n=1.52$ for a slab $0.600 \mathrm{~cm}$ thick? (c) If white light is used, the critical angle depends on color caused by dispersion. Is the inner edge of the white halo tinged with red light or violet light? Explain.

Keshav Singh
Keshav Singh
Numerade Educator
03:58

Problem 68

A light ray traveling in air is incident on one face of a right-angle prism with an index of refraction of $n=1.50$, as shown in Figure $\mathrm{P} 35.68$, and the ray follows the path shown in the figure. If $\theta=60.0^{\circ}$ and the base of the prism is mirrored, what is the angle $\phi$ made by the outgoing ray with the normal to the right face of the prism?

Keshav Singh
Keshav Singh
Numerade Educator
04:51

Problem 69

A light ray enters a rectangular block of plastic at an angle of $\theta_{1}=45.0^{\circ}$ and emerges at an angle of $\theta_{2}=76.0^{\circ}$, as shown in Figure P35.69. (a) Determine the index of refraction for the plastic. (b) If the light ray enters the plastic at a point $L=50.0 \mathrm{~cm}$ from the bottom edge, how long does it take the light ray to travel through the plastic?

Salamat Ali
Salamat Ali
Numerade Educator
01:33

Problem 70

Students allow a narrow beam of laser light to strike a water surface. They arrange to measure the angle of refraction for selected angles of incidence and record the data shown in the accompanying table. Use the data to verify Snell's law of refraction by plotting the sine of the angle of incidence versus the sine of the angle of refraction. Use the resulting plot to deduce the index of refraction of water.
$$\begin{array}{cc}\begin{array}{c}\text { Angle of Incidence } \\\text { (degrees) }\end{array} & \begin{array}{c}\text { Angle of Refraction } \\\text { (degrees) }\end{array} \\\hline 10.0 & 7.5 \\20.0 & 15.1 \\
30.0 & 22.3 \\40.0 & 28.7 \\50.0 & 35.2 \\60.0 & 40.3 \\70.0 & 45.3 \\80.0 & 47.7\end{array}$$

Zachary Warner
Zachary Warner
Numerade Educator