Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 34

Electromagnetic Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 2

What is the wavelength of the electromagnetic wave emitted by the oscillator-antenna system of Fig. $34-4$ if $L=$ $0.253 \mu \mathrm{H}$ and $C=25.0 \mathrm{pF} ?$

Sheh Lit Chang

University of Washington

Problem 3

The electric field of a certain plane electromagnetic wave is given by $E_{x}=0 ; E_{y}=0 ; E_{z}=(2.0 \mathrm{~V} / \mathrm{m}) \cos [(\pi \times$
$\left.\left.10^{15} \mathrm{~s}^{-1}\right)(t-x / c)\right]$, with $c=3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}$. The wave is propagating
in the positive $x$ direction. Write expressions for the components of the magnetic field of the wave.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 4

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

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 5

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

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 6

Show, by finding the direction of the Poynting vector $\vec{S}$, that the directions of the electric and magnetic fields at all points in Figs. $34-6$ to $34-8$ are consistent at all times with the assumed directions of propagation.

Zachary Warner

Numerade Educator

Problem 7

The radiation emitted by a laser spreads out in the form of a narrow cone with
circular cross section. The angle $\theta$ of the cone (see Fig. $34-20$ ) is called the full-angle beam divergence. An argon laser, radiating at $514.5 \mathrm{~nm}$,
is aimed at the Moon in a ranging experiment. If the beam has a full-angle beam divergence of $0.880$ \murad, what area on the Moon's surface is illuminated by the laser?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

Our closest stellar neighbor, Proxima Centauri, is $4.3$ lightyears away. It has been suggested that TV programs from our planet have reached this star and may have been viewed by the hypothetical inhabitants of a hypothetical planet orbiting it. Suppose a television station on Earth has a power of $1.0 \mathrm{MW}$. What is the intensity of its signal at Proxima Centauri?

Alfjad Alfjad

Numerade Educator

Problem 9

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 10

What is the intensity of a plane traveling electromagnetic wave if $\left|\vec{B}^{\max }\right|$ is $1.0 \times 10^{-4} \mathrm{~T} ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 11

The maximum electric field at a distance of $10 \mathrm{~m}$ from anisotropic point light source 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?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 12

Sunlight just outside the Earth's atmosphere has an intensity of $1.40 \mathrm{~kW} / \mathrm{m}^{2} .$ Calculate $\left|\vec{E}^{t \max }\right|$ and $\left|\vec{B}^{\max }\right|$ for sunlight there, assuming it to be a plane wave.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 13

An airplane flying at a distance of $10 \mathrm{~km}$ from a radio transmitter receives a signal of intensity $10 \mu \mathrm{W} / \mathrm{m}^{2} .$ Calculate
(a) the amplitude of the electric field at the airplane due to this signal, (b) the amplitude of the magnetic field at the airplane, and
(c) the total power of the transmitter, assuming the transmitter to radiate uniformly in all directions.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 14

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, "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 a source at the center of our galaxy that could provide such a signal? The galactic center is $2.2 \times$ $10^{4}$ ly away. Take the source as radiating uniformly in all directions.

Alfjad Alfjad

Numerade Educator

Problem 15

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 $\partial B / \partial t$ at which the magnetic component of the light changes with time at the detector's location?

Sheh Lit Chang

University of Washington

Problem 16

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 17

Magnetic Component Two 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?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 18

An electromagnetic wave with a wavelength of $450 \mathrm{~nm}$ travels through vacuum in the negative direction of a $y$ axis with its electric component directed parallel to the $x$ axis. The $\mathrm{rms}$ value of the electric component is $5.31 \times$ $10^{-6} \mathrm{~V} / \mathrm{m} .$ Write an equation for the magnetic component in the form of Eq. $34-3$, but complete with numbers.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 19

The intensity of direct solar radiation th?t is not absorbed by the atmosphere on a particular summer day is $100 \mathrm{~W} / \mathrm{m}^{2}$. How close would you have to stand to a $1.0 \mathrm{~kW}$ electric heater to feel the same intensity? Assume that the heater radiates uniformly in all directions.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 20

The intensity $I$ of light from an isotropic point light source is determined as a function of the distance $r$ from the
source. Figure $34-21$ gives intensity $\bar{I}$ versus the inverse square $r^{-2}$ of that distance. What is the power of the source?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 21

During a test, a NATO surveillance radar system, $\quad 20 .$ operating at $12 \mathrm{GHz}$ and $180 \mathrm{~kW}$ 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 it 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 the (c) intensity,
(d) maximum value of the electric field vector, and (e) rms value of the magnetic field of the reflected (and now detected) radar beam?

Sheh Lit Chang

University of Washington

Problem 22

Show that in a plane traveling electromagnetic wave the intensity - that is, the average rate of energy transport per unit area - is given by
$$
\langle S\rangle=\frac{\left(E^{\mathrm{max}}\right)^{2}}{2 \mu_{0} c}=\frac{\left(B^{\max }\right)^{2}}{2 \mu_{0}}
$$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 23

High-power lasers are used to compress a plasma (a gas of charged particles) by radiation pressure. A laser generating pulses of radiation of peak power $1.5 \mathrm{GW}$ 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 pulses directly back along their paths.

Alfjad Alfjad

Numerade Educator

Problem 24

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 25

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 26

Radiation from the Sun reaching the Earth (just outside the atmosphere) has an intensity of $1.4 \mathrm{~kW} / \mathrm{m}^{2}$. (a) Assuming that the 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 the Earth due to radiation pressure.
(b) Compare it with the force due to the Sun's gravitational attraction.

Alfjad Alfjad

Numerade Educator

Problem 27

A plane electromagnetic wave, with wavelength $3.0 \mathrm{~m}$, travels in vacuum in the positive $x$ direction with its electric field $\vec{E}$, of amplitude $300 \mathrm{~V} / \mathrm{m}$, directed along the $y$ axis.
(a) What is the frequency $f$ of the wave? (b) What are the direction and amplitude of the magnetic field associated with the wave? (c) What are the values of $k$ and $\omega$ if $\vec{E}=\vec{E}^{\max } \sin (k x-\omega t) ?$ (d) What is the time-averaged rate of energy flow in watts per square meter associated with this wave? (e) If the wave falls on a perfectly absorbing sheet of area $2.0 \mathrm{~m}^{2}$, at what rate is momentum delivered to the sheet and what is the radiation pressure exerted on the sheet?

Alfjad Alfjad

Numerade Educator

Problem 28

A helium-neon laser of the type often found in physics laboratories has a beam power of $5.00 \mathrm{~mW}$ at a wavelength of $633 \mathrm{~nm} .$ The beam is focused by a lens to a circular spot whose effective diameter may be taken to be equal to $2.00$ wavelengths. Calculate (a) the intensity of the focused beam, (b) the radiation pressure exerted on a tiny perfectly absorbing sphere whose diameter is that of the focal spot, (c) the force exerted on this sphere, and (d) the magnitude of the acceleration imparted to it. Assume a sphere density of $5.00 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$.

Alfjad Alfjad

Numerade Educator

Problem 29

Prove, for a plane electromagnetic wave that is normally incident on a plane 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.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

In Fig. 34-22, a laser beam of power $4.60 \mathrm{~W}$ and diameter $2.60 \mathrm{~mm}$ is directed upward at one circular face (of diameter $d<2.60 \mathrm{~mm}$ ) of a perfectly reflecting cylinder, which is made to "hover" by the beam's radiation pressure. The cylinder's density is $1.20$ $\mathrm{g} / \mathrm{cm}^{3}$. What is the cylinder's height $H$ ?

Alfjad Alfjad

Numerade Educator

Problem 31

A small spaceship whose mass is $1.5 \times 10^{3} \mathrm{~kg}$ (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?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 32

Prove that the average pressure of a stream of bullets striking a plane surface perpendicularly is twice the kinetic energy density in the stream outside the surface. Assume that the bullets are completely absorbed by the surface. Contrast this with Problem $29 .$

David Morabito

Numerade Educator

Problem 33

A particle in the solar system in 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 34

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 sail be if the radiation force is to be equal in magnitude in 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 $C$ for needed data. (With a larger sail, the ship is continually driven away from the Sun.)

Alfjad Alfjad

Numerade Educator

Problem 35

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 36

Radiation of intensity $I$ is normally incident on an object that absorbs a fraction frac of it and reflects the rest back along the original path. What is the radiation pressure on the object?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 37

A beam of unpolarized light of intensity $10 \mathrm{~mW} / \mathrm{m}^{2}$ is sent through a polarizing sheet as in Fig. $34-13$. (a) Find the maximum value of the electric field of the transmitted beam. (b) What radiation pressure is exerted on the polarizing sheet?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 38

The magnetic field equations for an electromagnetic wave in vacuum are $B_{x}=B \quad \sin (k y+\omega t)$ $B_{y}=B_{z}=0 .$ (a) What is the direction of propagation? (b) Write the electric field equations. (c) Is the wave polarized? If so, in what direction?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 39

In Fig. 34-23, initially unpolarized light is sent through three polarizing sheets whose polarizing axes 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.)

Alfjad Alfjad

Numerade Educator

Problem 40

In Fig. $34-23$, initially unpolarized light is sent through three polarizing sheets whose polarizing axes 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 of the three sheets? (Hint: Be careful with the angles.)

Alfjad Alfjad

Numerade Educator

Problem 41

A horizontal beam of vertically polarized light of intensity $43 \mathrm{~W} / \mathrm{m}^{2}$ is sent through two polarizing sheets. The polarizing axis of the first is at $70^{\circ}$ to the vertical, and that of the second is horizontal. What is the intensity of the light transmitted by the pair of sheets?

Alfjad Alfjad

Numerade Educator

Problem 42

$A$ beam of polarized light is sent through a system of two polarizing sheets. Relative to the polarization axis of that incident light, the polarizing axes 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 $\theta$ ?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

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

Suppose that in Problem 41 the initial beam is unpolarized. What then is the intensity of the transmitted light?

Alfjad Alfjad

Numerade Educator

Problem 45

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 46

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 47

An unpolarized beam of light is sent through 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 48

In Fig. $34-24$, unpolarized light with an intensity of $25 \mathrm{~W} / \mathrm{m}^{2}$ is sent into a system of four polarizing sheets. What is the intensity of the light that emerges from the system?

Alfjad Alfjad

Numerade Educator

Problem 49

A beam of 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?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 50

In Fig. 34 $25 a$, unpolarized light is sent through a system of two polarizing sheets. The angles $\theta_{1}$ and $\theta_{2}$ of the polarizing axes of the sheets are measured counterclockwise from the positive direction of the $y$ axis (they are not drawn to scale in the figure). Angle $\bar{\theta}_{1}$ is fixed but angle $\theta_{2}$ can be varied. Figure $34-25 b$ 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 two-sheet svstem when $\theta_{2}=90^{\circ} ?$

Keshav Singh

Numerade Educator

Problem 51

In Fig. $34-26$, light that is initially unpolarized is sent into a system of three polarizing sheets. What fraction of the initial light intensity emerges from the system?

Alfjad Alfjad

Numerade Educator

Problem 52

In Fig. $34-27 a$, unpolarized light is sent through a system of three polarizing sheets. The angles $\theta_{1}, \theta_{2}$, and $\theta_{3}$ of the polarizing axes of the sheets 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 $\theta_{2}$ can be varied. Figure $34-28$ 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 53

A system of three polarizing sheets is shown in Fig. $34-29 .$ When initially unpolarized light is sent into the system, the intensity of the transmitted light is $5.0 \%$ of the initial intensity. What is the value of $\theta ?$

Alfjad Alfjad

Numerade Educator

Problem 54

In Fig. 34 $27 a$, unpolarized light is sent through a system of three polarizing sheets. The angles $\theta_{1}, \theta_{2}$, and $\theta_{3}$ of the polarizing axes of
the sheets 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 $\theta_{2}$ can be varied. Figure $34-27 b$ 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 $\theta_{2}=30^{\circ}$ ?

Raj Bala

Numerade Educator

Problem 55

(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 $1.5 \times 10^{8}$
$\mathrm{km}$. (c) What is the round-trip travel time for light between the Earth and a spaceship or-
biting 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?

Sheh Lit Chang

University of Washington

Problem 56

Project Seafarer was an ambitious proposal 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 deeply submerged. If the effective wavelength were $1.0 \times 10^{4}$ Earth radii, what would be (a) the frequency and (b) the period of the radiations emitted? Ordinarily, electromagnetic radiations do not penetrate very far into conductors such as seawater.

Sheh Lit Chang

University of Washington

Problem 57

(A) At what wavelengths does the eye of a standard observer have half its maximum sensitivity? (b) What are the wavelength, frequency, and period of the light for which the eye is the most sensitive?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 58

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. $34-18$ ) of $0.0100 \mathrm{~nm}$. What is the corresponding "frequency width" for the emission?

Sheh Lit Chang

University of Washington

Problem 59

One method for measuring the speed of light, based on observations by Roemer in 1676, consisted of observing the apparent times of revolution of one of the moons of Jupiter. The true period of revolution is $42.5 \mathrm{~h}$. (a) Taking into account the finite speed of light, how would you expect the apparent time for one revolution to change as the Earth moves in its orbit from point $x$ to point $y$ in Fig. $34-30 ?$ (b) What observations would be needed to compute the speed of light? Neglect the motion of Jupiter in its orbit. Figure $34-30$ is not drawn to scale.

Ajay Singhal

Numerade Educator

Problem 60

An electromagnetic wave with frequency 400 terahertz travels through vacuum in the positive direction of an $x$ axis. The wave is polarized, with its electric field directed parallel to the $y$ axis, with amplitude $E^{\max }$. At time $t=0$, the electric field at point $P$ on the $x$ axis has a value of $+E^{\max } / 4$ 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 61

At the Earth's surface, what intensity of light is needed to suspend a totally absorbing spherical particle against its own weight if the mass of the particle is $2.0 \times 10^{-13} \mathrm{~kg}$ and its radius is $2.0 \mu \mathrm{m}$ ?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 62

An oscillating current in an antenna is producing an electromagnetic wave. The region shown in Fig. 34-31 enclosed by a dashed box (not to scale) is far from the antenna. In it, the field produced is well approximated by a plane wave traveling in the $z$ direction and having its $E$ -field pointed along the $x$ direction (using the coordinate system shown).
(a) You perform a series of measurements of the electric field at the origin of your coordinate system and obtain a result that points in the $y$ direction and is well represented by the function
$$
E(t)=E_{0} \cos (\omega t)
$$
What result would you find if instead of at the origin, you repeated the experiment at a point with coordinates $\{0,0, z\} ?$ Explain how you know.
(b) What result would you get if you made your measurements at a point in the box with coordinates $\{2,3, z\} \mathrm{cm} ?$ (The point is still well within the dashed box.)
(c) For what values of $z$ would you find exactly the same result as you found at the origin?

Sheh Lit Chang

University of Washington

Problem 63

After completing the construction of his equations for electromagnetism, Maxwell proposed that visible light was actually an electromagnetic wave. Discuss whether or not this is plausible and what evidence there is for his hypothesis.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 64

Compare and contrast the propagation of a pulse on a string and an electromagnetic pulse shown in Fig. 34-32. In particular, address the similarities and differences for
(a) how the pulse "knows" to move from one position to the next;
(b) what will happen if the wave is passed through a slit. (See the figure.)

Suzanne W.

Numerade Educator

Problem 65

Galileo tried to measure the speed of light by having two people stand on hills about $5 \mathrm{~km}$ apart. Each would hold a shuttered lantern. The first would open his lantern and when the second saw the light, he would open his lantern. The first person would then measure how much time it took between the time he first opened his lantern and the time he saw the light returning.
(a) How much time would it take the light to travel between the two hills?
(b) Is this a good way to measure the speed of light? Support your argument with a brief explanation that includes some quantitative discussion of the uncertainty in the measurement.

Mayukh Banik

Numerade Educator

Problem 66

The amount of energy from the sun that reaches the ground is on the order of $1 \mathrm{~kW} / \mathrm{m}^{2} .$ Use this information to estimate the area you would need for a solar energy collector to provide all the electricity in your house. Explain carefully your assumptions and reasoning.

Rashmi Sinha

Numerade Educator

Problem 67

Most of you have had the experience of using a microwave oven to boil a cup of water. [If you have not, ask a friend or roommate to help you estimate the time in part (a)]. According to a Pyrex measuring cup that is marked in both English and SI units, one cup contains about $230 \mathrm{ml}$.
(a) From the amount of time it takes to heat one cup of water from room temperature to boiling in a microwave oven, estimate the power that the oven delivers to the water in watts (joules/second).
(b) Assuming that electromagnetic radiation is flowing into the cup in the microwave from all sides, estimate the electromagnetic energy flux, $S$, in $\mathrm{W} / \mathrm{m}^{2}$.
(c) From the flux you calculated in (b), estimate the strength of the electric and magnetic fields in a microwave oven.

Lottie Adams

Numerade Educator

Problem 68

Although light appears to travel at a speed that is for all practical purposes infinite, for some modern purposes the time delay due to light travel time is of great importance. The Global Positioning System (GPS) allows you to determine your position from comparison of the time delays between radio signals from 4 satellites at a height of $20,000 \mathrm{~km}$ above the surface of the earth. (There are actually 24 of these satellites. Your GPS picks out the closest 4 to your current position.) In order to get some idea of how important the speed of light is in establishing your position with one of these gadgets, make some simple assumptions. Assume that a satellite is almost directly overhead. Then figure out how far the satellite will move in the time it takes light (the radio signal) to get from the satellite to your GPS receiver. This
estimates how far off the reading of your position would be if your device didn't include the speed of light in its calculations. To do this:
(a) Figure out what speed the satellite must be traveling to be in a circular orbit.
(b) Estimate the time it would take for a radio signal to get from the satellite to your receiver.
(c) Estimate how far the satellite would move in that time. If you ignore light travel time, this tells about how wrong you would get the satellite's position (and therefore how wrong you would get your position).

Chai Santi

Numerade Educator

Problem 69

Laser eye surgery is carried out by delivering highly intense bursts of energy using electromagnetic waves. A typical laser used in such surgery has a wavelength of $190 \mathrm{~nm}$ (ultraviolet light) and produces bursts of light that last for $1 \mathrm{~ms}$. The laser delivers an energy of $0.5 \mathrm{~mJ}$ to a circular spot on the cornea with a diameter of $1 \mathrm{~mm}$. (The light is well approximated by a plane wave for the short distance between the laser and the cornea.)
(a) Assuming that the energy of a single pulse is delivered to a volume of the cornea about $1 \mathrm{~mm}^{3}$, and assuming that the pulses are delivered so quickly that the energy deposited has no time to
flow out of that volume, how many pulses are required to raise the temperature of that volume from $20^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ ? (Assume that the cornea has a heat capacity similar to that of water.)
(b) Estimate the maximum strength of the electric field in one of these pulses.

Ummatul Choudary

Ummatul Choudary

Numerade Educator

Problem 70

The power radiated by the sun is $3.9 \times$ $10^{26} \mathrm{~W}$. The Earth orbits the Sun in a nearly circular orbit of radius $1.5 \times 10^{11} \mathrm{~m} .$ The Earth's axis of rotation is tilted by $23^{\circ}$ relative to the plane of the orbit (see Fig. 34 -
33) so sunlight does not strike the equator perpendicularly.
(a) At the time of year depicted in Fig. $34-33$, what power strikes a $1 \mathrm{~m}^{2}$ patch of horizontal flat land at the equator at the point $P ?
(b) Will a $1 \mathrm{~m}^{2}$ patch of horizontal flat land at the point $R$ or $S$ receive more radiation?
(c) Explain how your answer to part (b) tells you at which of the points $R$ or $S$ it is summer or winter.

Robert Hackett

Numerade Educator