Fundamentals of Physics

David Halliday, Robert Resnick

Chapter 33

Electromagnetic Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

A certain helium-neon laser emits red light in a narrow band of wavelengths centered at $632.8 \mathrm{nm}$ and with a "wavelength width" (such as on the scale of Fig. $33-1$ ) of $0.0100 \mathrm{nm}$. What is the corresponding "frequency width" for the emission?

Sheh Lit Chang

Sheh Lit Chang

University of Washington

Problem 2

Project Seafarer was an ambitious program to construct an enormous antenna, buried underground on a site about $10000 \mathrm{~km}^{2}$ in area. Its purpose was to transmit signals to submarines while they were decply submerged. If the effective wavelength were $1.0 \times 10^{4}$ Earth radii, what would be the (a) frequency and (b) period of the radiations emitted? Ordinarily, electromagnetic radiations do not penetrate very far into conductors such as seawater, and so normal signals cannot reach the submarines.

Keshav Singh

Numerade Educator

Problem 3

From Fig. 33-2, approximate the (a) smaller and (b) larger
wavelength at which the eye of a standard observer has half the eye’s
maximum sensitivity. What are the (c) wavelength, (d) frequency,
and (e) period of the light at which the eye is the most sensitive?

Keshav Singh

Numerade Educator

Problem 4

About how far apart must you hold your hands for them to be separated by 1.0 nano-light-second (the distance light travels
in 1.0 ns)?

Sheh Lit Chang

University of Washington

Problem 5

What inductance must be connected to a 17 pF capacitor in an oscillator capable of generating 550 nm (i.e., visible) electromagnetic waves? Comment on your answer.

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 6

What is the wavelength of the clectromagnetic wave emitted by the oscillator -antenna system of Fig. $33-3$ if $L=0.253 \mu \mathrm{H}$ and $C=25.0 \mathrm{pF} ?$

Keshav Singh

Numerade Educator

Problem 7

What is the intensity of a traveling plane electromagnetic wave if $B_{m}$ is $1.0 \times 10^{-4} \mathrm{~T} ?$

Sheh Lit Chang

University of Washington

Problem 8

Assume (unrealistically) that a TV station acts as a point source broadcasting isotropically at $1.0 \mathrm{MW}$. What is the intensity of the transmitted signal reaching Proxima Centauri, the star nearest our solar system, 4.3 ly away? (An alien civilization at that distance might be able to watch $X$ Files. ) A light-ycar (ly) is the distance light travels in one year.

Keshav Singh

Numerade Educator

Problem 9

Some neodymium–glass lasers can provide 100 TW of power in 1.0 ns pulses at a wavelength of $0.26 \mu \mathrm{m}$. How much energy is contained in a single pulse?

Keshav Singh

Numerade Educator

Problem 10

A plane electromagnetic wave has a maximum electric field magnitude of $3.20 \times 10^{-4} \mathrm{~V} / \mathrm{m} .$ Find the magnetic field amplitude.

Sheh Lit Chang

University of Washington

Problem 11

A plane electromagnetic wave traveling in the positive direction of an $x$ axis in vacuum has components $E_{x}=E_{y}=0$ and $E_{z}=(2.0 \mathrm{~V} / \mathrm{m}) \cos \left[\left(\pi \times 10^{15} \mathrm{~s}^{-1}\right)(t-x / c)\right] .$ (a) What is the amplitude
of the magnetic field component? (b) Parallel to which axis does the magnetic field oscillate? (c) When the electric field component is in the positive direction of the $z$ axis at a certain point $P$, what is the direction of the magnetic field component there?

Keshav Singh

Numerade Educator

Problem 12

In a plane radio wave the maximum value of the electric field component is $5.00 \mathrm{~V} / \mathrm{m}$. Calculate (a) the maximum value of the magnetic field component and (b) the wave intensity.

Sheh Lit Chang

University of Washington

Problem 13

Sunlight just outside Earth's atmosphere has an intensity of $1.40 \mathrm{~kW} / \mathrm{m}^{2} .$ Calculate (a) $E_{m}$ and (b) $B_{m}$ for sunlight there, assuming it to be a plane wave.

Sheh Lit Chang

University of Washington

Problem 14

An isotropic point source emits light at wavelength $500 \mathrm{nm},$ at the rate of $200 \mathrm{~W}$. A light detector is positioned $400 \mathrm{~m}$ from the source. What is the maximum rate $\partial B / \partial \mathrm{t}$ at which the magnetic component of the light changes with time at the detector's location?

Sheh Lit Chang

University of Washington

Problem 15

An airplane flying at a distance of $10 \mathrm{~km}$ from a radio transmitter receives a signal of intensity $10 \mu \mathrm{W} / \mathrm{m}^{2}$. What is the amplitude of the (a) electric and (b) magnetic component of the signal at the airplane? (c) If the transmitter radiates uniformly over a hemisphere, what is the transmission power?

Sheh Lit Chang

University of Washington

Problem 16

Frank D. Drake, an investigator in the SETI (Search for Extra-Terrestrial Intelligence) program, once said that the large radio telescope in Arecibo, Puerto Rico (Fig. 33-36), "can detect a signal which lays down on the entire surface of the earth a power of only one picowatt." (a) What is the power that would be received by the Arecibo antenna for such a signal? The antenna diameter is $300 \mathrm{~m}$. (b) What would be the power of an isotropic source at the center of our galaxy that could provide such a signal? The galactic center is $2.2 \times 10^{4}$ ly away. A light-year is the distance light travels in one year.

Sheh Lit Chang

University of Washington

Problem 17

The maximum electric field $10 \mathrm{~m}$ from an isotropic point source of light is $2.0 \mathrm{~V} / \mathrm{m}$. What are (a) the maximum value of the magnetic field and (b) the average intensity of the light there? (c) What is the power of the source?

Sheh Lit Chang

University of Washington

Problem 18

The intensity $I$ of light from an isotropic point source is determined as a function of distance $r$ from the source. Figure $33-37$ gives intensity $I$ versus the inverse square $r^{-2}$ of that distance. The vertical axis scale is set by $I_{s}=200 \mathrm{~W} / \mathrm{m}^{2}$, and the horizontal axis scale is set by $r_{x}^{-2}=8.0 \mathrm{~m}^{-2}$. What is the power of the source?

Keshav Singh

Numerade Educator

Problem 19

High-power lasers are used to compress a plasma (a gas of charged particles) by radiation pressure. A laser generating radiation pulses with peak power $1.5 \times 10^{3} \mathrm{MW}$ is focused onto $1.0 \mathrm{~mm}^{2}$ of high-electron-density plasma. Find the pressure exerted on the plasma if the plasma reflects all the light beams directly back along their paths.

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 20

Radiation from the Sun reaching Earth (just outside the atmosphere) has an intensity of $1.4 \mathrm{~kW} / \mathrm{m}^{2}$. (a) Assuming that Earth (and its atmosphere) behaves like a flat disk perpendicular to the Sun's rays and that all the incident energy is absorbed, calculate the force on Earth due to radiation pressure.
(b) For comparison, calculate the force due to the Sun's gravitational attraction.

Averell Hause

Carnegie Mellon University

Problem 21

What is the radiation pressure $1.5 \mathrm{~m}$ away from a $500 \mathrm{~W}$ lightbulb? Assume that the surface on which the pressure is exerted faces the bulb and is perfectly absorbing and that the bulb radiates uniformly in all directions.

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 22

A black, totally absorbing piece of cardboard of area $A=2.0 \mathrm{~cm}^{2}$ intercepts light with an intensity of $10 \mathrm{~W} / \mathrm{m}^{2}$ from a camera strobe light. What radiation pressure is produced on the cardboard by the light?

Averell Hause

Carnegie Mellon University

Problem 23

Someone plans to float a small, totally absorbing sphere $0.500 \mathrm{~m}$ above an isotropic point source of light, so that the upward radiation force from the light matches the downward gravitational force on the sphere. The sphere's density is $19.0 \mathrm{~g} / \mathrm{cm}^{3},$ and its radius is $2.00 \mathrm{~mm}$. (a) What power would be required of the light source? (b) Even if such a source were made, why would the support of the sphere be unstable?

Sheh Lit Chang

University of Washington

Problem 24

It has been proposed that a spaceship might be propelled in the solar system by radiation pressure, using a large sail made of foil. How large must the surface area of the sail be if the radiation force is to be equal in magnitude to the Sun's gravitational attraction? Assume that the mass of the ship $+$ sail is $1500 \mathrm{~kg}$, that the sail is perfectly reflecting, and that the sail is oriented perpendicular to the Sun's rays. See Appendix $\mathrm{C}$ for needed data. (With a larger sail, the ship is continuously driven away from the Sun.)

Averell Hause

Carnegie Mellon University

Problem 25

Prove, for a plane electromagnetic wave that is normally incident on a flat surface, that the radiation pressure on the surface is equal to the energy density in the incident beam. (This relation between pressure and energy density holds no matter what fraction of the incident energy is reflected.)

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 26

In Fig. 33-38, a laser beam of power $4.60 \mathrm{~W}$ and diameter $D=2.60 \mathrm{~mm}$ is directed upward at one circular face (of diameter $d<2.60 \mathrm{~mm}$ ) of a perfectly reflecting cylinder. The cylinder is levitated because the upward radiation force matches the downward gravitational force. If the cylinder's density is $1.20 \mathrm{~g} / \mathrm{cm}^{3},$ what is its height $H ?$

Sheh Lit Chang

University of Washington

Problem 27

A plane electromagnetic wave, with wavelength $3.0 \mathrm{~m}$, travels in vacuum in the positive direction of an $x$ axis. The electric field, of amplitude $300 \mathrm{~V} / \mathrm{m},$ oscillates parallel to the $y$ axis. What are the (a) frequency, (b) angular frequency, and (c) angular wave number of the wave? (d) What is the amplitude of the magnetic field component? (e) Parallel to which axis does the magnetic field oscillate? (f) What is the timeaveraged rate of energy flow in watts per square meter associated with this wave? The wave uniformly illuminates a surface of area $2.0 \mathrm{~m}^{2} .$ If the surface totally absorbs the wave, what are $(\mathrm{g})$ the rate at which momentum is transferred to the surface and (h) the radiation pressure on the surface?

Keshav Singh

Numerade Educator

Problem 28

The average intensity of the solar radiation that strikes normally on a surface just outside Earth's atmosphere is $1.4 \mathrm{~kW} / \mathrm{m}^{2}$
(a) What radiation pressure $p_{r}$ is exerted on this surface, assuming complete absorption? (b) For comparison, find the ratio of $p,$ to Earth's sea-level atmospheric pressure, which is $1.0 \times 10^{5} \mathrm{~Pa}$.

Sheh Lit Chang

University of Washington

Problem 29

A small spaceship with a mass of only $1.5 \times 10^{3} \mathrm{~kg}$ (including an astronaut) is drifting in outer space with negligible gravitational forces acting on it. If the astronaut turns on a $10 \mathrm{~kW}$ laser beam, what speed will the ship attain in 1.0 day because of the momentum carried away by the beam?

Sheh Lit Chang

University of Washington

Problem 30

A small laser emits light at power $5.00 \mathrm{~mW}$ and wave- Iength $633 \mathrm{nm}$. The laser beam is focused (narrowed) until its diameter matches the $1266 \mathrm{nm}$ diameter of a sphere placed in its path. The sphere is perfectly absorbing and has density $5.00 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$. What are (a) the beam intensity at the sphere's location, (b) the radiation pressure on the sphere, (c) the magnitude of the corresponding force, and (d) the magnitude of the acceleration that force alone would give the sphere?

Keshav Singh

Numerade Educator

Problem 31

As a comet swings around the Sun, ice on the comet's surface vaporizes, releasing trapped dust particles and ions. The ions, because they are electrically charged, are forced by the electrically charged solar wind into a straight ion tail that points radially away from the Sun (Fig. $33-39$ ). The (electrically neutral) dust particles are pushed radially outward from the Sun by the radiation force on them from sunlight. Assume that the dust particles are spherical, have density $3.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3},$ and are totally absorbing. (a) What radius must a particle have in order to follow a straight path, like path 2 in the figure? (b) If its radius is larger, does its path curve away from the Sun (like path 1 ) or toward the Sun (like path 3 )?

Eduard Sanchez

Numerade Educator

Problem 32

In Fig. 33-40, initially unpolarized light is sent into a system
of three polarizing sheets whose polarizing directions make angles of $\theta_{1}=\theta_{2}=\theta_{3}=50^{\circ}$ with the direction of the $y$ axis. What percentage of the initial intensity is transmitted by the system? (Hint: Be careful with the angles.)

Sheh Lit Chang

University of Washington

Problem 33

In Fig. $33-40,$ initially unpolarized light is sent into a system of three polarizing sheets whose polarizing directions make angles of $\theta_{1}=40^{\circ}, \theta_{2}=20^{\circ},$ and $\theta_{3}=40^{\circ}$ with the direction of the $y$ axis. What percentage of the light's initial intensity is transmitted by the system? (Hint: Be careful with the angles.)

Sheh Lit Chang

University of Washington

Problem 34

In Fig. $33-41,$ a beam of unpolarized light, with intensity $43 \mathrm{~W} / \mathrm{m}^{2}$, is sent into a system of two polarizing sheets with polarizing dircctions at angles $\theta_{1}=70^{\circ}$ and $\theta_{2}=90^{\circ}$ to the $y$ axis. What is the intensity of the light transmitted by the system?

Keshav Singh

Numerade Educator

Problem 35

In Fig. $33-41$, a beam of light, with intensity $43 \mathrm{~W} / \mathrm{m}^{2}$ and polarization parallel to a $y$ axis, is sent into a system of two polarizing sheets with polarizing directions at angles of $\theta_{1}=70^{a}$ and $\theta_{2}=90^{\circ}$ to the $y$ axis. What is the intensity of the light transmitted by the two-sheet system?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 36

At a beach the light is generally partially polarized due to reflections off sand and water. At a particular beach on a particular day near sundown, the horizontal component of the electric field vector is 2.3 times the vertical component. A standing sunbather puts on polarizing sunglasses; the glasses eliminate the horizontal field component.
(a) What fraction of the light intensity received before the glasses were put on now reaches the sunbather's eyes? (b) The sunbather, still wearing the glasses, lies on his side. What fraction of the light intensity received before the glasses were put on now reaches his eyes?

Keshav Singh

Numerade Educator

Problem 37

We want to rotate the direction of polarization of a beam of polarized light through $90^{\circ}$ by sending the beam through one or more polarizing sheets. (a) What is the minimum number of sheets required? (b) What is the minimum number of sheets required if the transmitted intensity is to be more than $60 \%$ of the original intensity?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 38

In Fig. $33-42,$ unpolarized light is sent into a system of three polarizing sheets. The angles $\theta_{1}, \theta_{2},$ and $\theta_{3}$ of the polariring directions are measured counterclock wise from the positive direction of the $y$ axis (they are not drawn to scale). Angles $\theta_{1}$ and $\theta_{1}$ are fixed, but angle $\theta_{2}$ can be varied. Figure $33-43$ gives the intensity of the light emerging from sheet 3 as a function of $\theta_{2}$. (The scale of the intensity axis is not indicated.) What percentage of the light's initial intensity is transmitted by the system when $\theta_{2}=30^{\circ} ?$

Keshav Singh

Numerade Educator

Problem 39

Unpolarized light of intensity $10 \mathrm{~mW} / \mathrm{m}^{2}$ is sent into a polarizing sheet as in Fig. $33-11 .$ What are (a) the amplitude of the electric field component of the transmitted light and (b) the radiation pressure on the sheet due to its absorbing some of the light?

Sheh Lit Chang

University of Washington

Problem 40

In Fig. $33-42,$ unpolarized light is sent into a system of three polarizing sheets. The angles $\theta_{1}, \theta_{2},$ and $\theta_{3}$ of the polarizing directions are measured counterclockwise from the positive direction of the $y$ axis (they are not drawn to scale). Angles $\theta_{1}$ and $\theta_{3}$ are fixed, but angle $\bar{\theta}_{2}$ can be varied. Figure $33-44$ gives the intensity of the light emerging from sheet 3 as a function of $\theta_{2}$ (The scale of the intensity axis is not indicated.) What percentage of the light's initial intensity is transmitted by the three-sheet system when $\theta_{2}=90^{\circ} ?$

Keshav Singh

Numerade Educator

Problem 41

A beam of polarized light is sent into a system of two polarizing sheets. Relative to the polarization direction of that incident light, the polarizing directions of the sheets are at angles $\theta$ for the first sheet and $90^{\circ}$ for the second sheet. If 0.10 of the incident intensity is transmitted by the two sheets, what is $\theta ?$

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 42

In Fig. $33-41$, unpolarized light is sent into a system of two polarizing sheets. The angles $\theta_{1}$ and $\theta_{2}$ of the polarizing directions of the sheets are measured counterclockwise from the positive direction of the $y$ axis (they are not drawn to scale in the figure). Angle $\theta_{1}$ is fixed but angle $\theta_{2}$ can be varied. Figure 33.45 gives the intensity of the light emerging from sheet 2 as a function of $\theta_{2}$. (The scale of the intensity axis is not indicated.) What percentage of the light's initial intensity is transmitted by the twosheet system when $\theta_{2}=90^{\circ} ?$

Keshav Singh

Numerade Educator

Problem 43

A beam of partially polarized light can be considered to be a mixture of polarized and unpolarized light. Suppose we send such a beam through a polarizing filter and then rotate the filter through $360^{\circ}$ while keeping it perpendicular to the beam. If the transmitted intensity varies by a factor of 5.0 during the rotation, what fraction of the intensity of the original beam is associated with the beam's polarized light?

Sheh Lit Chang

University of Washington

Problem 44

In Fig. $33-42,$ unpolarized light is sent into a system of three polarizing sheets, which transmits 0.0500 of the initial light intensity. The polarizing directions of the first and third sheets are at angles $\theta_{1}=0^{\circ}$ and $\theta_{3}=90^{\circ} .$ What are the (a) smaller and (b) larger possible values of angle $\theta_{2}\left(<90^{\circ}\right)$ for the polarizing direction of sheet 27

Keshav Singh

Numerade Educator

Problem 45

When the rectangular metal tank in Fig. $33-46$ is filled to the top with an unknown liquid, observer $O$. with eyes level with the top of the tank, can just see corner $E$. A ray that refracts toward $O$ at the top surface of the liquid is shown. If $D=85.0 \mathrm{~cm}$ and $L=1.10 \mathrm{~m},$ what is the index of refraction of the liquid?

Sheh Lit Chang

University of Washington

Problem 46

In Fig. $33-47 a,$ a light ray in an underlying material is incident at angle $\theta_{1}$ on a boundary with water, and some of the light refracts into the water. There are two choices of underlying material. For each, the angle of refraction $\theta_{2}$ versus the incident angle $\theta_{1}$ is given in Fig. $33-47 b$. The horizontal axis scale is set by $\theta_{1 y}=90^{\circ} .$ Without calculation, determine whether the index of refraction of (a) material 1 and (b) material 2 is greater or less than the index of water $(n=1.33) .$ What is the index of refraction of (c) material 1 and (d) material $2 ?$

Keshav Singh

Numerade Educator

Problem 47

Light in vacuum is incident on the surface of a glass slab. In the vacuum the beam makes an angle of $32.0^{\circ}$ with the normal to the surface, while in the glass it makes an angle of $21.0^{\circ}$ with the normal. What is the index of refraction of the glass?

Sheh Lit Chang

University of Washington

Problem 48

In Fig. $33-48 a,$ a light ray in water is incident at angle $\theta_{1}$ on a boundary with an underlying material, into which some of the light refracts. There are two choices of underlying material. For each, the angle of refraction $\theta_{2}$ versus the incident angle $\theta_{1}$ is given in Fig. $33-48 b$. The vertical axis scale is set by $\theta_{2 x}=90^{\circ}$. Without calculation, determine whether the index of refraction of
(a) material 1 and (b) material 2 is greater or less than the index of water $(n=1.33) .$ What is the index of refraction of (c) material 1 and (d) material $2 ?$

Keshav Singh

Numerade Educator

Problem 49

Figure $33-49$ shows light reflecting from two perpendicular reflecting surfaces $A$ and $B .$ Find the angle between the incoming ray $i$ and the outgoing ray $r^{\prime}$

Keshav Singh

Numerade Educator

Problem 50

In Fig. $33-50 a,$ a beam of light in material 1 is incident on a boundary at an angle $\theta_{1}=40^{\circ} .$ Some of the light travels through material $2,$ and then some of it emerges into material 3 . The two boundaries between the three materials are parallel. The final direction of the beam depends, in part, on the index of refraction $n_{3}$ of the third material. Figure $33-50 b$ gives the angle of refraction $\theta_{3}$ in that material versus $n_{3}$ for a range of possible $n_{3}$ values. The vertical axis scale is set by $\theta_{3 q}=30.0^{\circ}$ and $\theta_{3 b}=50.0^{\circ} .$ (a) What is the index of refraction of material $1,$ or is the index impossible to calculate without more information? (b) What is the index of refraction of material $2,$ or is the index impossible to calculate without more information? (c) If $\theta_{1}$ is changed to $70^{\circ}$ and the index of refraction of material 3 is $2.4,$ what is $\theta_{3} ?$

Keshav Singh

Numerade Educator

Problem 51

In Fig. 33-51, light is incident at angle $\theta_{1}=40.1^{\circ}$ on a boundary between two transparent materials. Some of the light travels down through the next three layers of transparent materials, while some of it reflects upward and then escapes into the air. If $n_{1}=1.30, n_{2}=1.40$ $n_{3}=1.32,$ and $n_{4}=1.45,$ what is the value of (a) $\theta_{\mathrm{s}}$ in the air and (b) $\theta_{4}$ in the bottom material?

Keshav Singh

Numerade Educator

Problem 52

In Fig. 33-52a, a beam of light in material 1 is incident on a boundary at an angle of $\theta_{1}=30^{\circ} .$ The extent of refraction of the light into material 2 depends, in part, on the index of refraction $n_{2}$ of material $2 .$ Figure $33-52 b$ gives the angle of refraction $\theta_{2}$ versus $n_{2}$ for a range of possible $n_{2}$ values. The vertical axis scale is set by $\theta_{2 a}=20.0^{\circ}$ and $\theta_{2 \Delta}=40.0^{\circ} .$ (a) What is the index of refraction of material 17 (b) If the incident angle is changed to $60^{\circ}$ and material 2 has $n_{2}=2.4,$ then what is angle $\theta_{2} ?$

Keshav Singh

Numerade Educator

Problem 53

In Fig. 33-53, a ray is incident on one face of a triangular glass prism in air. The angle of incidence $\theta$ is chosen so that the emerging ray also makes the same angle $\theta$ with the normal to the other face. Show that the index of refraction $n$ of the glass prism is given by
$$
n=\frac{\sin \{(\psi+\phi)}{\sin \lfloor\phi}
$$
where $\phi$ is the vertex angle of the prism and $\psi$ is the deviation angle, the total angle through which the beam is turned in passing through the prism. (Under these conditions the deviation angle $\psi$ has the smallest possible value, which is called the angle of minimum deviation.)

Keshav Singh

Numerade Educator

Problem 54

In Fig. $33-54,$ a beam of white light is incident at angle $\theta=50^{\circ}$ on a common window pane (shown in cross section). For the pane's type of glass, the index of refraction for visible light ranges from 1.524 at the blue end of the spectrum to 1.509 at the red end. The two sides of the pane are parallel. What is the angular spread of the colors in the beam (a) when the light enters the pane and (b) when it emerges from the opposite side? (Hint: When you look at an object through a window pane, are the colors in the light from the object dispersed as shown in, say, Fig. $33-20 ?$ )

Keshav Singh

Numerade Educator

Problem 55

In Fig. 33-55, a 2.00-m-long vertical pole extends from the bottom of a swimming pool to a point $50.0 \mathrm{~cm}$ above the water. Sunlight is incident at angle $\theta=55.0^{\circ} .$ What is the length of the shadow of the pole on the level bottom of the pool?

Sheh Lit Chang

University of Washington

Problem 56

Suppose that, on some surreal world, raindrops had a square cross section and always fell with one face horizontal. Figure $33-56$ shows such a falling drop, with a white beam of sunlight incident at $\theta=70.0^{\circ}$ at point $P$. The part of the light that enters the drop then travels to point $A,$ where some of it refracts out into the air and the rest reflects. That reflected light then travels to point $B$. where again some of the light refracts out into the air and the rest reflects. What is the difference in the angles of the red light $(n=1.331)$ and the blue light $(n=1.343)$ that emerge at (a) point $A$ and (b) point $B ?$ (This angular difference in the light emerging at, say, point $A$ would be the rainbow's angular width.)

Sheh Lit Chang

University of Washington

Problem 57

A point source of light is $80.0 \mathrm{~cm}$ below the surface of a body of water. Find the diameter of the circle at the surface through which light emerges from the water.

Sheh Lit Chang

University of Washington

Problem 58

The index of refraction of benzene is $1.8 .$ What is the critical angle for a light ray traveling in benzene toward a flat layer of air above the benzene?

Sheh Lit Chang

University of Washington

Problem 59

In Fig. $33-57,$ a ray of light is perpendicular to the face $a b$ of a glass prism $(n=1.52)$ Find the largest value for the angle $\phi$ so that the ray is totally reflected at face $a c$ if the prism is immersed (a) in air and (b) in water.

Sheh Lit Chang

University of Washington

Problem 60

In Fig. 33-58, light from ray $A$ refracts from material $1\left(n_{1}=1.60\right)$ into a thin layer of material 2 $\left(n_{2}=1.80\right),$ crosses that layer, and is then incident at the critical angle on the interface between materials 2 and $3\left(n_{3}=1.30\right) .$ (a) What is the value of incident angle $\theta_{A} ?$ (b) If $\theta_{A}$ is decreased, does part of the light refract into material 3 ?
Light from ray $B$ refracts from material 1 into the thin layer, crosses that layer, and is then incident at the critical angle on the interface between materials 2 and 3 . (c) What is the value of incident angle $\theta_{B} ?$ (d) If $\theta_{B}$ is decreased, does part of the light refract into material $3 ?$

Keshav Singh

Numerade Educator

Problem 61

In Fig. 33-59, light initially in material 1 refracts into material 2 , crosses that material, and is then incident at the critical angle on the interface between materials 2
and $3 .$ The indexes of refraction are $n_{1}=1.60, n_{2}=1.40,$ and $n_{3}=1.20$
(a) What is angle $\theta ?$ (b) If $\theta$ is increased, is there refraction of light into material $3 ?$

Sheh Lit Chang

University of Washington

Problem 62

A catfish is 2.00 m below the surface of a smooth lake. (a) What is the diameter of the circle on the surface through which the fish can see the world outside the water? (b) If the fish descends, does the diameter of the circle increase, decrease, or remain the same?

Sheh Lit Chang

University of Washington

Problem 63

In Fig. $33-60,$ light enters a $90^{\circ}$ triangular prism at point $P$ with incident angle $\theta,$ and then some of it refracts at point $Q$ with an angle of refraction of $90^{\circ} .$ (a) What is the index of refraction of the prism in terms of $\theta ?$ (b) What, numerically, is the maximum value that the index of refraction can have? Does light emerge at $Q$ if the incident angle at $P$ is (c) increased slightly and (d) decreased slightly?

Keshav Singh

Numerade Educator

Problem 64

Suppose the prism of Fig. $33-53$ has apex angle $\phi=60.0^{\circ}$ and index of refraction $n=1.60$. (a) What is the smallest angle of incidence $\theta$ for which a ray can enter the left face of the prism and exit the right face? (b) What angle of incidence $\theta$ is required for the ray to exit the prism with an identical angle $\theta$ for its refraction, as it does in Fig. $33-53 ?$

Keshav Singh

Numerade Educator

Problem 65

Figure $33-61$ depicts a simplistic optical fiber: a plastic core $\left(n_{1}=1.58\right)$ is surrounded by a plastic sheath $\left(n_{2}=1.53\right) .$ A light ray is incident on one end of the fiber at angle $\theta .$ The ray is to undergo total internal reflection at point $\bar{A}$, where it encounters the core-sheath boundary. (Thus there is no loss of light through that boundary.) What is the maximum value of $\theta$ that allows total internal reflection at $\bar{A} ?$

Sheh Lit Chang

University of Washington

Problem 66

In Fig. $33-62,$ a light ray in air is incident at angle $\theta_{1}$ on a block of transparent plastic with an index of refraction of 1.56 . The dimensions indicated are $H=2.00 \mathrm{~cm}$ and $W=3.00 \mathrm{~cm} .$ The light passes through the block to one of its sides and there undergoes reflection (inside the block) and possibly refraction (out into the air). This is the point of first reflection. The reflected light then passes through the block to another of its sides - a point of second reflection. If $\theta_{1}=40^{\circ},$ on which side is the point of (a) first reflection and (b) second reflection? If there is refraction at the point of (c) first reflection and
(d) second reflection, give the angle of refraction; if not, answer "none." If $\theta_{1}=70^{\circ},$ on which side is the point of (e) first reflection and (f) second reflection? If there is refraction at the point of ( $g$ ) first reflection and (h) second reflection, give the angle of refraction; if not, answer "none."

Averell Hause

Carnegie Mellon University

Problem 67

In the ray diagram of Fig. $33-63,$ where the angles are not drawn to scale, the ray is incident at the critical angle on the interface between materials 2 and 3 . Angle $\phi=60.0^{\circ}$, and two of the indexes of refraction are $n_{1}=1.70$ and $n_{2}=1.60 .$ Find (a) index of refraction $n_{3}$ and (b) angle $\theta .$ (c) If $\theta$ is decreased, does light refract into material $3 ?$

Sheh Lit Chang

University of Washington

Problem 68

(a) At what angle of incidence will the light reflected from
water be completely polarized? (b) Does this angle depend on the wavelength of the light?

Sheh Lit Chang

University of Washington

Problem 69

Light that is traveling in water (with an index of refraction of 1.33) is incident on a plate of glass (with index of refraction 1.53). At what angle of incidence does the reflected light end up fully polarized?

Sheh Lit Chang

University of Washington

Problem 70

In Fig. $33-64,$ a light ray in air is incident on a flat layer of material 2 that has an index of refraction $n_{2}=1.5 .$ Beneath material 2 is material 3 with an index of refraction $n_{3}$. The ray is incident on the air-material 2 interface at the Brewster angle for that interface. The ray of light refracted into material 3 happens to be incident on the material 2 - material 3 interface at the Brewster angle for that interface. What is the value of $n_{3} ?$

Keshav Singh

Numerade Educator

Problem 71

(a) How long does it take a radio signal to travel $150 \mathrm{~km}$ from a transmitter to a receiving antenna? (b) We see a full Moon by reflected sunlight. How much earlier did the light that enters our eye leave the Sun? The Earth-Moon and Earth-Sun distances are $3.8 \times 10^{5} \mathrm{~km}$ and $1.5 \times 10^{8} \mathrm{~km},$ respectively. (c) What is the round-trip travel time for light between Earth and a spaceship orbiting Saturn, $1.3 \times 10^{9} \mathrm{~km}$ distant?
(d) The Crab nebula, which is about 6500 light-years (ly) distant, is thought to be the result of a supernova explosion recorded by Chinese astronomers in A.D. $1054 .$ In approximately what year did the explosion actually occur? (When we look into the night sky, we are effectively looking back in time.)

Sheh Lit Chang

University of Washington

Problem 72

An electromagnetic wave with frequency $4.00 \times 10^{14} \mathrm{~Hz}$ travels through vacuum in the positive direction of an $x$ axis. The wave has its electric field oscillating parallel to the $y$ axis, with an amplitude $E_{m^{*}}$ At time $t=0,$ the electric field at point $P$ on the $x$ axis has a value of $+E_{m} / 4$ and is decreasing with time. What is the distance along the $x$ axis from point $P$ to the first point with $E=0$ if we search in (a) the negative direction and (b) the positive direction of the $x$ axis?

Eduard Sanchez

Numerade Educator

Problem 73

The electric component of a beam of polarized light is
$$E_{y}=(5.00 \mathrm{~V} / \mathrm{m}) \sin \left[\left(1.00 \times 10^{6} \mathrm{~m}^{-1}\right) z+\omega t\right]$$ (a) Write an expression for the magnetic field component of the wave, including a value for $\omega$, What are the (b) wavelength,
(c) period, and (d) intensity of this light? (e) Parallel to which axis does the magnetic field oscillate? (f) In which region of the electromagnetic spectrum is this wave?

Sheh Lit Chang

University of Washington

Problem 74

A particle in the solar system is under the combined influence of the Sun's gravitational attraction and the radiation force due to the Sun's rays. Assume that the particle is a sphere of density $1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and that all the incident light is absorbed.
(a) Show that, if its radius is less than some critical radius $R,$ the particle will be blown out of the solar system. (b) Calculate the critical radius.

Bettina Hanlon

Numerade Educator

Problem 75

In Fig. $33-65,$ a light ray enters a glass slab at point $A$ at incident angle $\theta_{1}=45.0^{\circ}$ and then undergoes total internal reflection at point $B$. (The reflection at $A$ is not shown.) What minimum value for the index of refraction of the glass can be inferred from this information?

Sheh Lit Chang

University of Washington

Problem 76

In Fig. $33-66,$ unpolarized light with an intensity of $25 \mathrm{~W} / \mathrm{m}^{2}$ is sent into a system of four polarizing sheets with polarizing directions at angles $\theta_{1}=40^{\circ}, \theta_{2}=20^{\circ}, \theta_{3}=20^{\circ},$ and $\theta_{4}=30^{\circ} .$ What is the intensity of the light that emerges from the system?

Keshav Singh

Numerade Educator

Problem 77

Figure $33-67$ shows a light ray entering and then leaving a falling, spherical raindrop after one internal reflection (see Fig. $33-21 a$ ). The final direction of travel is deviated (turned) from the initial direction of travel by angular deviation $\theta_{\mathrm{dov}}$ (a) Show that $\theta_{\mathrm{dev}}$ is
$$
\theta_{\text {dev }}=180^{\circ}+2 \theta_{i}-4 \theta_{r}
$$
where $\theta_{i}$ is the angle of incidence of the ray on the drop and $\theta_{r}$ is the angle of refraction of the ray within the drop. (b) Using Snell's law, substitute for $\theta_{r}$ in terms of $\theta_{i}$ and the index of refraction $n$ of the water. Then, on a graphing calculator or with a computer graphing package, graph $\theta_{\text {dev }}$ versus $\theta_{i}$ for the range of possible $\theta_{i}$ values and for $n=1.331$ for red light (at one end of the visible spectrum) and $n=1.343$ for blue light (at the other end). The red-light curve and the blue-light curve have different minima, which means that there is a different angle of minimum deviation for each color. The light of any given color that leaves the drop at that color's angle of minimum deviation is especially bright because rays bunch up at that angle. Thus, the bright red light leaves the drop at one angle and the bright blue light leaves it at another angle. Determine the angle of minimum deviation from the $\theta_{\text {dv}}$ curve for (c) red light and (d) blue light. (e) If these colors form the inner and outer edges of a rainbow (Fig. $33-21 a$ ), what is the angular width of the rainbow?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 78

The primary rainbow described in Problem 77 is the type commonly seen in regions where rainbows appear. It is produced by light reflecting once inside the drops. Rarer is the secondary rainbow described in Module $33-5,$ produced by light reflecting twice inside the drops (Fig. $33-68 a$ ). (a) Show that the angular deviation of light entering and then leaving a spherical water drop is
$$\theta_{\text {dev }}=\left(180^{\circ}\right) k+2 \theta_{i}-2(k+1) \theta_{n}$$
where $k$ is the number of internal reflections. Using the procedure of Problem $77,$ find the angle of minimum deviation for (b) red light and (c) blue light in a secondary rainbow. (d) What is the angular width of that rainbow (Fig. $33-21 d$ )?
The tertiary rainbow depends on three internal reflections (Fig. $33-68 b$ ). It probably occurs but, as noted in Module $33-5$, cannot be seen with the eye because it is very faint and lies in the bright sky surrounding the Sun. What is the angle of minimum deviation for (e) the red light and (f) the blue light in this rainbow?
(g) What is the rainbow's angular width?

Keshav Singh

Numerade Educator

Problem 79

(a) Prove that a ray of light incident on the surface of a sheet of plate glass of thickness $t$ emerges from the opposite face parallel to its initial direction but displaced sideways, as in Fig. $33-69 .$
(b) Show that, for small angles of incidence $\theta,$ this displacement is given by
$$x=t \theta \frac{n-1}{n}$$
where $n$ is the index of refraction of the glass and $\theta$ is measured in radians.

Sheh Lit Chang

University of Washington

Problem 80

An electromagnetic wave is traveling in the negative direction of a $y$ axis. At a particular position and time, the electric field is directed along the positive direction of the $z$ axis and has a magnitude of $100 \mathrm{~V} / \mathrm{m}$. What are the (a) magnitude and (b) direction of the corresponding magnetic field?

Sheh Lit Chang

University of Washington

Problem 81

The magnetic component of a polarized wave of light is
$$B_{x}=\left(4.0 \times 10^{-6} \mathrm{~T}\right) \sin \left[\left(1.57 \times 10^{7} \mathrm{~m}^{-1}\right) y+\omega t\right]$$
(a) Parallel to which axis is the light polarized? What are the
(b) frequency and (c) intensity of the light?

Sheh Lit Chang

University of Washington

Problem 82

In Fig. 33-70, unpolarized light is sent into the system of three polarizing sheets, where the polarizing directions of the first and third sheets are at angles $\theta_{1}=30^{\circ}$ (counterclockwise) and $\theta_{3}=30^{\circ}$ (clockwise). What fraction of the initial light intensity emerges from the system?

Sheh Lit Chang

University of Washington

Problem 83

A ray of white light traveling through fused quartz is incident at a quartz-air interface at angle $\theta_{1}$. Assume that the index of refraction of quartz is $n=1.456$ at the red end of the visible range and $n=1.470$ at the blue end. If $\theta_{1}$ is (a) $42.00^{\circ},$ (b) $43.10^{\circ}$, and
(c) $44.00^{\circ},$ is the refracted light white, white dominated by the red end of the visible range, or white dominated by the blue end of the visible range, or is there no refracted light?

Sheh Lit Chang

University of Washington

Problem 84

Three polarizing sheets are stacked. The first and third are crossed; the one between has its polarizing direction at $45.0^{\circ}$ to the polarizing directions of the other two. What fraction of the intensity of an originally unpolarized beam is transmitted by the stack?

Sheh Lit Chang

University of Washington

Problem 85

In a region of space where gravitational forces can be neglected, a sphere is accelerated by a uniform light beam of intensity $6.0 \mathrm{~mW} / \mathrm{m}^{2}$. The sphere is totally absorbing and has a radius of $2.0 \mu \mathrm{m}$ and a uniform density of $5.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3} .$ What is the magnitude of the sphere's acceleration due to the light?

Sheh Lit Chang

University of Washington

Problem 86

An unpolarized beam of light is sent into a stack of four polarizing sheets, oriented so that the angle between the polarizing directions of adjacent sheets is $30^{\circ} .$ What fraction of the incident intensity is transmitted by the system?

Sheh Lit Chang

University of Washington

Problem 87

During a test, a NATO surveillance radar system, operating at $12 \mathrm{GHz}$ at $180 \mathrm{~kW}$ of power, attempts to detect an incoming stealth aircraft at $90 \mathrm{~km}$. Assume that the radar beam is emitted uniformly over a hemisphere. (a) What is the intensity of the beam when the beam reaches the aircraft's location? The aircraft reflects radar waves as though it has a cross-sectional area of only $0.22 \mathrm{~m}^{2}$.
(b) What is the power of the aircraft's reflection? Assume that the beam is reflected uniformly over a hemisphere. Back at the radar site, what are (c) the intensity, (d) the maximum value of the electric field vector, and (e) the rms value of the magnetic field of the reflected radar beam?

Sheh Lit Chang

University of Washington

Problem 88

The magnetic component of an electromagnetic wave in vacuum has an amplitude of $85.8 \mathrm{nT}$ and an angular wave number of $4.00 \mathrm{~m}^{-1} .$ What are (a) the frequency of the wave, (b) the rms value of the electric component, and (c) the intensity of the light?

Sheh Lit Chang

University of Washington

Problem 89

Calculate the (a) upper and (b) lower limit of the Brewster angle for white light incident on fused quartz. Assume that the wavelength limits of the light are 400 and $700 \mathrm{nm}$.

Sheh Lit Chang

University of Washington

Problem 90

In Fig. $33-71$, two light rays pass from air through five layers of transparent plastic and then back into air. The layers have parallel interfaces and unknown thicknesses; their indexes of refraction are $\overline{n_{1}}=1.7, n_{2}=1.6, n_{3}=1.5, n_{4}=1.4,$ and $n_{5}=1.6 .$ Ray $b$ is incident at angle $\theta_{b}=20^{\circ} .$ Relative to a normal at the last interface, at what angle do (a) ray $a$ and (b) ray $b$ emerge? (Hint: Solving the problem algebraically can save time.) If the air at the left and right sides in the figure were, instead, glass with index of refraction $1.5,$ at what angle would (c) ray $a$ and (d) ray $b$ emerge?

Keshav Singh

Numerade Educator

Problem 91

A helium-neon laser, radiating at $632.8 \mathrm{nm}$, has a power output of $3.0 \mathrm{~mW}$. The beam diverges (spreads) at angle $\theta=0.17$ mrad (Fig. $33-72$ ). (a) What is the intensity of the beam $40 \mathrm{~m}$ from the laser? (b) What is the power of a point source providing that intensity at that distance?

Sheh Lit Chang

University of Washington

Problem 92

In about A.D. $150,$ Claudius Ptolemy gave the following measured values for the angle of incidence $\theta_{1}$ and the angle of refraction $\theta_{2}$ for a light beam passing from air to water:
$$ \begin{array}{lcll} \hline \theta_{1} & \theta_{2} & \theta_{1} & \theta_{2} \\ \hline 10^{\circ} & 8^{\circ} & 50^{\circ} & 35^{\circ} \\ 20^{\circ} & 15^{\circ} 30^{\prime} & 60^{\circ} & 40^{\circ} 30^{\prime} \\ 30^{\circ} & 22^{\circ} 30^{\prime} & 70^{\circ} & 45^{\circ} 30^{\prime} \\ 40^{\circ} & 29^{\circ} & 80^{\circ} & 50^{\circ} \\
\hline\end{array}$$
Assuming these data are consistent with the law of refraction, use them to find the index of refraction of water. These data are interesting as perhaps the oldest recorded physical measurements.

Keshav Singh

Numerade Educator

Problem 93

A beam of initially unpolarized light is sent through two polarizing sheets placed one on top of the other. What must be the angle between the polarizing directions of the sheets if the intensity of the transmitted light is to be one-third the incident intensity?

Sheh Lit Chang

University of Washington

Problem 94

In Fig. $33-73,$ a long. straight copper wire (diameter $2.50 \mathrm{~mm}$ and resistance $1.00 \Omega$ per $300 \mathrm{~m}$ ) carries a uniform current of 25.0 A in the positive $x$ direction. For point $P$ on the wire's surface, calculate the magnitudes of (a) the electric field $\vec{E}$, (b) the mag- netic field $\vec{B}$, and (c) the Poynting vector $\vec{S}$, and (d) determine the direction of $\vec{S}$

Keshav Singh

Numerade Educator

Problem 95

Figure $33-74$ shows a cylindrical resistor of length $l,$ radius $a$, and resistivity $\rho,$ carrying current $i$ (a) Show that the Poynting vector $\vec{S}$ at the surface of the resistor is every where directed normal to the surface, as shown. (b) Show that the rate $P$ at which energy flows into the resistor through its cylindrical surface, calculated by integrating the Poynting vector over this surface, is cqual to the rate at which thermal energy is produced:
$$\int \vec{S} \cdot d \vec{A}=i^{2} R$$
where $d \vec{A}$ is an element of area on the cylindrical surface and $R$ is the resistance.

Keshav Singh

Numerade Educator

Problem 96

A thin, totally absorbing sheet of mass $m,$ face area $A,$ and specific heat $c_{s}$, is fully illuminated by a perpendicular beam of a plane electromagnetic wave. The magnitude of the maximum electric field of the wave is $E_{m}$. What is the rate $d T / d t$ at which the sheet's temperature increases due to the absorption of the wave?

Sheh Lit Chang

University of Washington

Problem 97

Two polarizing sheets, one directly above the other, transmit $p \%$ of the initially unpolarized light that is perpendicularly incident on the top sheet. What is the angle between the polarizing directions of the two sheets?

Sheh Lit Chang

University of Washington

Problem 98

A laser beam of intensity $I$ reflects from a flat, totally reflecting surface of area $A,$ with a normal at angle $\theta$ with the beam. Write an expression for the beam's radiation pressure $p_{r}(\theta)$ on the surface in terms of the beam's pressure $p_{r \perp}$ when $\theta=0^{\circ}$

Keshav Singh

Numerade Educator

Problem 99

A beam of intensity $I$ reflects from a long. totally reflecting cylinder of radius $R ;$ the beam is perpendicular to the central axis of the cylinder and has a diamcter larger than $2 R .$ What is the beam's force per unit length on the cylinder?

Keshav Singh

Numerade Educator

Problem 100

In Fig. 33-75, unpolarized light is sent into a system of three polarizing sheets, where the polarizing directions of the first and second sheets are at angles $\theta_{1}=20^{\circ}$ and $\theta_{2}$ $=40^{\circ} .$ What fraction of the initial light intensity emerges from the system?

Sheh Lit Chang

University of Washington

Problem 101

In Fig. 33-76, unpolarized light is sent into a system of three polarizing sheets with polarizing directions at angles $\theta_{1}=20^{\circ}, \theta_{2}=$ $60^{\circ},$ and $\theta_{3}=40^{\circ} .$ What fraction of the initial light intensity emerges from the system?

Sheh Lit Chang

University of Washington

Problem 102

A square, perfectly reflecting surface is oriented in space to
be perpendicular to the light rays from the Sun. The surface has an edge length of 2.0 m and is located $3.0 \times 10^{11} \mathrm{~m}$ from the Sun's center. What is the radiation force on the surface from the light rays?

Sheh Lit Chang

University of Washington

Problem 103

The rms value of the electric field in a certain light wave is
0.200 V/m. What is the amplitude of the associated magnetic field?

Sheh Lit Chang

University of Washington

Problem 104

In Fig. 33-77, an albatross glides at a constant $15 \mathrm{~m} / \mathrm{s}$ horizon tally above level ground, moving in a vertical plane that contains the Sun. It glides toward a wall of height $h=$ $2.0 \mathrm{~m},$ which it will just barely clear. At that time of day, the angle of the Sun relative to the ground is $\theta=30^{\circ}$ At what speed does the shadow of the albatross move (a) across the level ground and then (b) up the wall? Suppose that later a hawk happens to glide along the same path, also at $15 \mathrm{~m} / \mathrm{s}$. You see that when its shadow reaches the wall, the speed of the shadow noticeably increases. (c) Is the Sun now higher or lower in the sky than when the albatross flew by earlier? (d) If the speed of the hawk's shadow on the wall is $45 \mathrm{~m} / \mathrm{s},$ what is the angle $\theta$ of the Sun just then?

Keshav Singh

Numerade Educator

Problem 105

The magnetic component of a polarized wave of light is given by $B_{x}=(4.00 \mu \mathrm{T}) \sin \left[k y+\left(2.00 \times 10^{15} \mathrm{~s}^{-1}\right) t\right] .$ (a) In which direction does the wave travel, (b) parallel to which axis is it polarized, and (c) what is its intensity?
(d) Write an expression for the electric field of the wave, including a value for the angular wave number. (e) What is the wavelength? (f) In which region of the electromagnetic spectrum is this electromagnetic wave?

Keshav Singh

Numerade Educator

Problem 106

In Fig. $33-78,$ where $n_{1}=1.70$ $n_{2}=1.50,$ and $n_{3}=1.30,$ light refracts from material 1 into material 2 . If it is incident at point $A$ at the critical angle for the interface between materials 2 and 3 , what are (a) the angle of refraction at point $B$ and
(b) the initial angle $\theta ?$ If, instead, light is incident at $B$ at the critical angle for the interface between materials 2 and $3,$ what are $(\mathrm{c})$ the angle of refraction at point $A$ and (d) the initial angle $\theta ?$ If, instead of all that, light is incident at point $A$ at Brewster's angle for the interface between materials 2 and $3,$ what are (e) the angle of refraction at point $B$ and (f) the initial angle $\theta ?$

Keshav Singh

Numerade Educator

Problem 107

When red light in vacuum is incident at the Brewster angle on a certain glass slab, the angle of refraction is $32.0^{\circ},$ What are
(a) the index of refraction of the glass and (b) the Brewster angle?

Sheh Lit Chang

University of Washington

Problem 108

Start from Eqs. $33-11$ and $33-17$ and show that $E(x, t)$ and $B(x, t),$ the electric and magnetic field components of a plane traveling electromagnetic wave, must satisfy the "wave equations"
$$\frac{\partial^{2} E}{\partial r^{2}}=c^{2} frac{\partial^{2} E}{\partial x^{2}} \text { and } \frac{\partial^{2} B}{\partial r^{2}}=c^{2} \frac{\partial^{2} B}{\partial x^{2}}$$

Keshav Singh

Numerade Educator

Problem 109

(a) Show that Eqs. $33-1$ land $33-2$ satisfy the wave equations displayed in Problem $108 .$ (b) Show that any expressions of the form $E=E_{m} f(k x \pm \omega t)$ and $B=B_{m} f(k x \pm \omega t),$ where $f(k x \pm \omega t)$ denotes an arbitrary function, also satisfy these wave equations.

Keshav Singh

Numerade Educator

Problem 110

A point source of light emits isotropically with a power of $200 \mathrm{~W}$. What is the force due to the light on a totally absorbing sphere of radius $2.0 \mathrm{~cm}$ at a distance of $20 \mathrm{~m}$ from the source?

Sheh Lit Chang

University of Washington