University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 31

Electromagnetic Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

Which of the following phenomena can be observed for electromagnetic waves but not for sound waves?
a) interference
c) polarization
e) scattering
b) diffraction
d) absorption

Ajay Singhal

Ajay Singhal

Numerade Educator

Problem 2

Which of the following statements concerning electromagnetic waves are incorrect? (Select all that apply.)
a) Electromagnetic waves in vacuum travel at the speed of light.
b) The magnitudes of the electric field and the magnetic field are equal.
c) Only the electric field vector is perpendicular to the direction of the wave's propagation.
d) Both the electric field vector and the magnetic field vector are perpendicular to the direction of propagation.
e) An electromagnetic wave carries energy only when $E=B$.

Ajay Singhal

Numerade Educator

Problem 3

The international radio station Voice of Slobbovia announces that it is "transmitting to North America on the 49 -meter band." Which frequency is the station transmitting on?
a) $820 \mathrm{kHz}$
d) The information given tells
b) $6.12 \mathrm{MHz}$
nothing about the frequency.
c) $91.7 \mathrm{MHz}$

Ajay Singhal

Numerade Educator

Problem 4

Which of the following exerts the largest amount of radiation pressure?
a) a $1-\mathrm{mW}$ laser pointer on a $2-\mathrm{mm}$ -diameter spot $1 \mathrm{~m}$ away
b) a 200-W light bulb on a 4 -mm-diameter spot $10 \mathrm{~m}$ away
c) a 100 -W light bulb on a 2 -mm-diameter spot 4 m away
d) a 200 -W light bulb on a 2 -mm-diameter spot 5 m away
e) All of the above exert the same pressure.

Linda Winkler

Numerade Educator

Problem 5

What is the direction of the net force on the moving positive charge in the figure?
a) into the page
c) out of the page
b) toward the right
d) toward the left

Ajay Singhal

Numerade Educator

Problem 6

A proton moves perpendicularly to crossed electric and magnetic fields as shown in the figure. What is the direction of the net force on the proton?
a) toward the left
b) toward the right
c) into the page
d) out of the page

Ajay Singhal

Numerade Educator

Problem 7

It is speculated that isolated magnetic “charges" (magnetic monopoles) may exist somewhere in the universe. Which of Maxwell's equations, (1) Gauss's Law for Electric Fields, (2) Gauss's Law for Magnetic Fields, (3) Faraday's Law of Induction, and/or (4) the MaxwellAmpere Law, would be altered by the existence of magnetic monopoles?
a) only (2)
c) (2) and (3)
b) (1) and (2)
d) only (3)

Ajay Singhal

Numerade Educator

Problem 8

According to Gauss's Law for Magnetic Fields, all magnetic field lines form a complete loop. Therefore, the direction of the magnetic field $\vec{B}$ points from $\quad$ pole to $\longrightarrow$ pole outside of an ordinary bar magnet and from pole to pole inside the magnet.
a) north, south, north, south
c) south, north, south, north
b) north, south, south, north
d) south, north, north, south

Ajay Singhal

Numerade Educator

Problem 9

Unpolarized light with intensity $I_{\text {in }}=1.87 \mathrm{~W} / \mathrm{m}^{2}$ passes through two polarizers. The emerging polarized light has intensity $I_{\text {out }}=0.383 \mathrm{~W} / \mathrm{m}^{2}$. What is the angle between the two polarizers?
a) $23.9^{\circ}$
c) $50.2^{\circ}$
e) $88.9^{\circ}$
b) $34.6^{\circ}$
d) $72.7^{\circ}$

Ajay Singhal

Numerade Educator

Problem 10

The average intensity of sunlight at the Earth's surface is approximately $1400 \mathrm{~W} / \mathrm{m}^{2}$, if the Sun is directly overhead. The average distance between the Earth and the Sun is $1.50 \cdot 10^{11} \mathrm{~m}$. What is the average power emitted by the Sun?
a) $99.9 \cdot 10^{25} \mathrm{~W}$
c) $6.3 \cdot 10^{27} \mathrm{~W}$
e) $5.9 \cdot 10^{29} \mathrm{~W}$
b) $4.0 \cdot 10^{26} \mathrm{~W}$
d) $4.3 \cdot 10^{28} \mathrm{~W}$

Ajay Singhal

Numerade Educator

Problem 11

In a polarized light experiment, a setup similar to the one in Figure 31.23 is used. Unpolarized light with intensity $I_{0}$ is incident on polarizer
1. Polarizers 1 and 3 are crossed (at a $90^{\circ}$ angle), and their orientations are fixed during the experiment. Initially, polarizer 2 has its polarizing angle at $45^{\circ} .$ Then, at time $t=0,$ polarizer 2 starts to rotate with angular velocity
$\omega$ about the direction of propagation of light in a clockwise direction as viewed by an observer looking toward the light source. A photodiode is used to monitor the intensity of the light emerging from polarizer $3 .$
a) Determine an expression for this intensity as a function of time.
b) How would the expression from part (a) change if polarizer 2 were rotated about an axis parallel to the direction of propagation of the light but displaced by a distance $d<R,$ where $R$ is the radius of the polarizer?

Keshav Singh

Numerade Educator

Problem 12

A dipole antenna is located at the origin with its axis along the $z$ -axis. As electric current oscillates up and down the antenna, polarized electromagnetic radiation travels away from the antenna along the positive $y$ -axis. What are the possible directions of electric and magnetic fields at point $A$ on the $y$ -axis? Explain.

Ajay Singhal

Numerade Educator

Problem 13

Surface from the Sun?Does the information in Section 31.6 affect the answer to Example 31.2 regarding the root-mean-square magnitude of electric field at the Earth's.

Ajay Singhal

Numerade Educator

Problem 14

Maxwell's equations predict that there are no magnetic monopoles. If these monopoles existed, how would the motion of charged particles change as they approached such a monopole?

Ajay Singhal

Numerade Educator

Problem 15

If two communication signals were sent at the same time to the Moon, one via radio waves and one via visible light, which one would arrive at the Moon first?

Ajay Singhal

Numerade Educator

Problem 16

Show that Ampere's Law is not necessarily consistent if the surface through which the flux is to be calculated is a closed surface, but that the Maxwell-Ampere Law always is. (Hence, Maxwell's introduction of his law of induction and the displacement current are not optional; they are logically necessary.) Show also that Faraday's Law of Induction does not suffer from this consistency problem.

Derrick Danso

Numerade Educator

Problem 17

Maxwell's equations and Newton's laws of motion are mutually inconsistent; the great edifice of classical physics is fatally flawed. Explain why.

Michael Sorola

Numerade Educator

Problem 18

Practically everyone who has studied the electromagnetic spectrum has wondered how the world would appear if we could see over a range of frequencies comparable to the ten octaves over which we can hear rather than the less than one octave over which we can see. (An octave refers to a factor of 2 in frequency.) But this is practically impossible. Why?

Ajay Singhal

Numerade Educator

Problem 19

Isotropic electromagnetic waves expand uniformly outward in all directions in three dimensions. Electromagnetic waves from a small, isotropic source are not plane waves, which have constant maximum amplitudes.
a) How does the maximum amplitude of the electric field of radiation from a small, isotropic source vary with distance from the source?
b) Compare this with the electrostatic field of a point charge.

Ajay Singhal

Numerade Educator

Problem 20

A pair of sunglasses is held in front of a flat-panel computer monitor (which is on) so that the lenses are always parallel to the display. As the lenses are rotated, it is noticed that the intensity of light coming from the display and passing through the lenses is varying. Why?

Michael Sorola

Numerade Educator

Problem 21

Two polarizing filters are crossed at $90^{\circ}$, so when light is shined from behind the pair of filters, no light passes through. A third filter is inserted between the two, initially aligned with one of them. Describe what happens as the intermediate filter is rotated through an angle of $360^{\circ}$

Narayan Hari

Numerade Educator

Problem 22

An electric field of magnitude $200.0 \mathrm{~V} / \mathrm{m}$ is directed perpendicular to a circular planar surface with radius $6.00 \mathrm{~cm} .$ If the electric field increases at a rate of $10.0 \mathrm{~V} /(\mathrm{m} \mathrm{s})$, determine the magnitude and the direction of the magnetic field at a radial distance $10.0 \mathrm{~cm}$ away from the center of the circular area.

Ajay Singhal

Numerade Educator

Problem 23

A parallel plate capacitor has air between disk-shaped plates of radius $4.00 \mathrm{~mm}$ that are coaxial and $1.00 \mathrm{~mm}$ apart. Charge is being accumulated on the plates of the capacitor. What is the displacement current between the plates at an instant when the rate of charge accumulation on the plates is $10.0 \mu \mathrm{C} / \mathrm{s} ?$

Narayan Hari

Numerade Educator

Problem 24

A parallel plate capacitor has circular plates of radius $10.0 \mathrm{~cm}$ that are separated by a distance of $5.00 \mathrm{~mm}$. The potential across the capacitor is increased at a constant rate of $1.20 \mathrm{kV} / \mathrm{s}$. Determine the magnitude of the magnetic field between the plates at a distance $r=4.00 \mathrm{~cm}$ from the center.

Ajay Singhal

Numerade Educator

Problem 25

A parallel plate capacitor has air between disk-shaped plates of radius $4.00 \mathrm{~mm}$ that are coaxial and $1.00 \mathrm{~mm}$ apart. Charge is being accumulated on the plates of the capacitor. What is the displacement current between the plates at an instant when the rate of charge accumulation on the plates is $10.0 \mu \mathrm{C} / \mathrm{s} ?$

Ajay Singhal

Numerade Educator

Problem 26

A parallel plate capacitor has circular plates of radius $10.0 \mathrm{~cm}$ that are separated by a distance of $5.00 \mathrm{~mm}$. The potential across the capacitor is increased at a constant rate of $1.20 \mathrm{kV} / \mathrm{s}$. Determine the magnitude of the magnetic field between the plates at a distance $r=4.00 \mathrm{~cm}$ from the center.

Ajay Singhal

Numerade Educator

Problem 27

The voltage across a cylindrical conductor of radius $r$, length $L$, and resistance $R$ varies with time. The time-varying voltage causes a time-varying current, $i$, to flow in the cylinder. Show that the displacement current equals $\epsilon_{0}$ odildt, where $\rho$ is the resistivity of the conductor.

Ajay Singhal

Numerade Educator

Problem 28

The amplitude of the electric field of an electromagnetic wave is $250 . \mathrm{V} / \mathrm{m}$. What is the amplitude of the magnetic field of the electromagnetic wave?

Ajay Singhal

Numerade Educator

Problem 29

Determine the distance in feet that light can travel in vacuum during $1.00 \mathrm{~ns}$

Ajay Singhal

Numerade Educator

Problem 30

How long does it take light to travel from the Moon to the Earth? From the Sun to the Earth? From Jupiter to the Earth?

Ajay Singhal

Numerade Educator

Problem 31

Alice made a telephone call from her home telephone in New York to her fiancé stationed in Baghdad, about $10,000 \mathrm{~km}$ away, and the signal was carried on a telephone cable. The following day, Alice called her fiancé again from work using her cell phone, and the signal was transmitted via a satellite $36,000 \mathrm{~km}$ above the Earth's surface, halfway between New York and Baghdad. Estimate the time taken for the signals sent by (a) the telephone cable and (b) via the satellite to reach Baghdad, assuming that the signal speed in both cases is the same as the speed of light, $c$. Would there be a noticeable delay in either case?

Ajay Singhal

Numerade Educator

Problem 32

Electric and magnetic fields in many materials can be analyzed using the relationships for these fields in vacuum, but substituting relative

Ajay Singhal

Numerade Educator

Problem 33

The wavelength range for visible light is $400 \mathrm{nm}$ to $700 \mathrm{nm}$ (see Figure 31.10 ) in air. What is the frequency range of visible light?

Ajay Singhal

Numerade Educator

Problem 34

The antenna of a cell phone is a straight rod $8.0 \mathrm{~cm}$ long. Calculate the operating frequency of the signal from this phone, assuming that the

Ajay Singhal

Numerade Educator

Problem 35

Suppose an RLC circuit in resonance is used to produce a radio wave of wavelength $150 \mathrm{~m}$. If the circuit has a 2.0 -pF capacitor, what size inductor is used?

Ajay Singhal

Numerade Educator

Problem 36

Three FM radio stations covering the same geographical area broadcast at frequencies 91.1,91.3 , and $91.5 \mathrm{MHz}$, respectively. What is the maximum allowable wavelength width of the band-pass filter in a radio receiver such that the FM station 91.3 can be played free of interference from FM 91.1 or FM 91.5 ? Use $c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}$, and calculate the wavelength to an uncertainty of $1 \mathrm{~mm}$

Ajay Singhal

Numerade Educator

Problem 37

A monochromatic point source of light emits $1.5 \mathrm{~W}$ of electromagnetic power uniformly in all directions. Find the Poynting vector at a point situated at each of the following locations:
a) $0.30 \mathrm{~m}$ from the source
b) $0.32 \mathrm{~m}$ from the source
c) $1.00 \mathrm{~m}$ from the source

Ajay Singhal

Numerade Educator

Problem 38

Consider an electron in a hydrogen atom, which is $0.050 \mathrm{nm}$ from the proton in the nucleus.
a) What electric field does the electron experience?
b) In order to produce an electric field whose root-mean-square magnitude is the same as that of the field in part (a), what intensity must a laser light have?

Ajay Singhal

Numerade Educator

Problem 39

A $3.00-\mathrm{kW}$ carbon dioxide laser is used in laser welding. If the beam is $1.00 \mathrm{~mm}$ in diameter, what is the amplitude of the electric field in the beam?

Ajay Singhal

Numerade Educator

Problem 40

Suppose that charges on a dipole antenna oscillate slowly at a rate of 1.00 cycle/s, and the antenna radiates electromagnetic waves in a region of space. If someone measured the time-varying magnetic field in the region and found its maximum to be $1.00 \mathrm{mT}$, what would be the maximum electric field, $E$, in the region, in units of volts per meter? What is the period of the charge oscillation? What is the magnitude of the Poynting vector?

Ajay Singhal

Numerade Educator

Problem 41

Calculate the average value of the Poynting vector, $S_{\text {ave }}$, for an electromagnetic wave having an electric field of amplitude $100 . \mathrm{V} / \mathrm{m}$
a) What is the average energy density of this wave in $J / \mathrm{m}^{3} ?$
b) How large is the amplitude of the magnetic field?

Ajay Singhal

Numerade Educator

Problem 42

The most intense beam of light that can propagate through dry air must have an electric field whose maximum amplitude is no greater than the breakdown value for air: $E_{\max }^{\text {air }}=3.0 \cdot 10^{6} \mathrm{~V} / \mathrm{m}$, assuming that this value is unaffected by the frequency of the wave.
a) Calculate the maximum amplitude the magnetic field of this wave can have.
b) Calculate the intensity of this wave.
c) What happens to a wave more intense than this?

Ajay Singhal

Numerade Educator

Problem 43

A continuous-wave (cw) argon-ion laser beam has an average power of $10.0 \mathrm{~W}$ and a beam diameter of $1.00 \mathrm{~mm} .$ Assume that the intensity of the beam is the same throughout the cross section of the beam (which is not true, as the actual distribution of intensity is a Gaussian function).
a) Calculate the intensity of the laser beam. Compare this with the average intensity of sunlight at Earth's surface ( $1400 . \mathrm{W} / \mathrm{m}^{2}$ ).
b) Find the root-mean-square electric field in the laser beam.
c) Find the average value of the Poynting vector over time.
d) If the wavelength of the laser beam is $514.5 \mathrm{nm}$ in vacuum, write an expression for the instantaneous Poynting vector, where the instantaneous Poynting vector is zero at $t=0$ and $x=0$.
e) Calculate the root-mean-square value of the magnetic field in the laser beam.

Linda Winkler

Numerade Educator

Problem 44

A voltage, $V$, is applied across a cylindrical conductor of radius $r$ length $L,$ and resistance $R .$ As a result, a current, $i,$ is flowing through the conductor, which gives rise to a magnetic field, $B$. The conductor is placed along the $y$ -axis, and the current is flowing in the positive $y$ -direction. Assume that the electric field is uniform throughout the conductor.
a) Find the magnitude and the direction of the Poynting vector at the surface of the conductor of the static electric and magnetic fields.
b) Show that $\int \vec{S} \cdot d \vec{A}=i^{2} R$

Keshav Singh

Numerade Educator

Problem 45

Radiation from the Sun reaches the Earth at a rate of $1.40 \mathrm{~kW} / \mathrm{m}^{2}$ above the atmosphere and at a rate of $1.00 \mathrm{~kW} / \mathrm{m}^{2}$ on an ocean beach.
a) Calculate the maximum values of $E$ and $B$ above the atmosphere.
b) Find the pressure and the force exerted by the radiation on a person lying flat on the beach who has an area of $0.750 \mathrm{~m}^{2}$ exposed to the Sun.

Ajay Singhal

Numerade Educator

Problem 46

Scientists have proposed using the radiation pressure of sunlight for travel to other planets in the Solar System. If the intensity of the electromagnetic radiation produced by the Sun is about $1.40 \mathrm{~kW} / \mathrm{m}^{2}$ near the Earth, what size would a sail have to be to accelerate a spaceship with a mass of 10.0 metric tons at $1.00 \mathrm{~m} / \mathrm{s}^{2} ?$
a) Assume that the sail absorbs all the incident radiation.
b) Assume that the sail perfectly reflects all the incident radiation.

Ajay Singhal

Numerade Educator

Problem 47

A solar sail is a giant circle (with a radius $R=10.0 \mathrm{~km}$ ) made of a material that is perfectly reflecting on one side and totally absorbing on the other side. In deep space, away from other sources of light, the cosmic microwave background will provide the primary source of radiation incident on the sail. Assuming that this radiation is that of an ideal black body at $T=2.725 \mathrm{~K}$, calculate the net force on the sail due to its reflection and absorption. Also assume that any heat transferred to the sail will be conducted away, and that the photons are incident perpendicular to the surface of the sail.

Ajay Singhal

Numerade Educator

Problem 48

Two astronauts are at rest in outer space, one $20.0 \mathrm{~m}$ from the Space Shuttle and the other $40.0 \mathrm{~m}$ from the shuttle. Using a $100.0-\mathrm{W}$ laser, the astronaut located $40.0 \mathrm{~m}$ away from the shuttle decides to propel the other astronaut toward the Space Shuttle. He focuses the laser on a piece of totally reflecting fabric on her space suit. If her total mass with equipment is $100.0 \mathrm{~kg},$ how long will it take her to reach the Space Shuttle?

Ajay Singhal

Numerade Educator

Problem 49

A laser that produces a spot of light that is $1.00 \mathrm{~mm}$ in diameter is shone perpendicularly on the center of a thin, perfectly reflecting circular ( $2.00 \mathrm{~mm}$ in diameter) aluminum plate mounted vertically on a flat piece of cork that floats on the surface of the water in a large beaker. The mass of this "sailboat" is $0.100 \mathrm{~g}$, and it travels $2.00 \mathrm{~mm}$ in $63.0 \mathrm{~s}$. Assuming that the laser power is constant in the region where the sailboat is located during its motion, what is the power of the laser? (Neglect air resistance and the viscosity of water.)

Ajay Singhal

Numerade Educator

Problem 50

A tiny particle of density $2000 . \mathrm{kg} / \mathrm{m}^{3}$ is at the same distance from the Sun as the Earth is $\left(1.50 \cdot 10^{11} \mathrm{~m}\right)$. Assume that the particle is spherical and perfectly reflecting. What would its radius have to be for the outward radiation pressure on it to be $1.00 \%$ of the inward gravitational attraction of the Sun? (Take the Sun's mass to be $\left.2.00 \cdot 10^{30} \mathrm{~kg} .\right)$

Ajay Singhal

Numerade Educator

Problem 51

Silica aerogel, an extremely porous, thermally insulating material made of silica, has a density of $1.00 \mathrm{mg} / \mathrm{cm}^{3}$. A thin circular slice of aerogel has a diameter of $2.00 \mathrm{~mm}$ and a thickness of $0.10 \mathrm{~mm} .$
a) What is the weight of the aerogel slice (in newtons)?
b) What are the intensity and the radiation pressure of a $5.00-\mathrm{mW}$ laser beam of diameter $2.00 \mathrm{~mm}$ on the sample?
c) How many $5.00-\mathrm{mW}$ lasers with a beam diameter of $2.00 \mathrm{~mm}$ would be needed to make the slice float in the Earth's gravitational field? Use $g=9.81 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 52

Two polarizers are out of alignment by $30.0^{\circ} .$ If light of intensity $1.00 \mathrm{~W} / \mathrm{m}^{2}$ and initially polarized halfway between the polarizing angles of the two filters passes through both filters, what is the intensity of the transmitted light?

Ajay Singhal

Numerade Educator

Problem 53

A $10.0-\mathrm{mW}$ vertically polarized laser beam passes through a polarizer whose polarizing angle is $30.0^{\circ}$ from the horizontal. What is the power of the laser beam when it emerges from the polarizer?

Ajay Singhal

Numerade Educator

Problem 54

Unpolarized light of intensity $I_{0}$ is incident on a series of five polarizers, with the polarization direction of each rotated $10.0^{\circ}$ from that of the preceding one. What fraction of the incident light will pass through the series?

Ajay Singhal

Numerade Educator

Problem 55

A laser produces light that is polarized in the vertical direction. The light travels in the positive $y$ -direction and passes through two polarizers, which have polarizing angles of $35.0^{\circ}$ and $55.0^{\circ}$ from the vertical, as shown in the figure. The laser beam is collimated (neither converging nor expanding), has a circular cross section with a diameter of $1.00 \mathrm{~mm}$, and has an average power of $15.0 \mathrm{~mW}$ at point $A$. At point $C$, what are the amplitudes of the electric and magnetic fields, and what is the intensity of the laser light?

Ajay Singhal

Numerade Educator

Problem 56

A laser beam takes $50.0 \mathrm{~ms}$ to be reflected back from a totally reflecting sail on a spacecraft. How far away is the sail?

Ajay Singhal

Numerade Educator

Problem 57

A house with a south-facing roof has photovoltaic panels on the roof. The photovoltaic panels have an efficiency of $10.0 \%$ and occupy an area with dimensions $3.00 \mathrm{~m}$ by $8.00 \mathrm{~m}$. The average solar radiation incident on the panels is $300 . \mathrm{W} / \mathrm{m}^{2}$, averaged over all conditions for a year. How many kilowatt-hours of electricity will the solar panels generate in a 30 -day month?

Ajay Singhal

Numerade Educator

Problem 58

What is the radiation pressure due to Betelgeuse (which has a luminosity, or power output, 10,000 times that of the Sun) at a distance from it equal to that of Uranus's orbit from the Sun?

Narayan Hari

Numerade Educator

Problem 59

Amplitude of the electric field in the beam?

Ajay Singhal

Numerade Educator

Problem 60

What is the wavelength of the electromagnetic waves used for cell phone communications at $848.97 \mathrm{MHz} ?$

Ajay Singhal

Numerade Educator

Problem 61

As shown in the figure, sunlight is coming straight down (negative $z$ -direction) on a solar panel (of length $L=1.40 \mathrm{~m}$ and width $W=0.900 \mathrm{~m}$ ) on the Mars rover Spirit. The amplitude of the electric field in the solar radiation is $673 \mathrm{~V} / \mathrm{m}$ and is uniform (the radiation has the same amplitude everywhere). If the solar panel converts solar radiation to electrical power with an efficiency of $18.0 \%$, how much average power can the panel generate?

Ajay Singhal

Numerade Educator

Problem 62

A $14.9-\mu F$ capacitor, a $24.3-\mathrm{k} \Omega$ resistor, a switch, and a $25.0-\mathrm{V}$ battery are connected in series. What is the rate of change of the electric field between the plates of the capacitor at $t=0.3621 \mathrm{~s}$ after the switch is closed? The area of the plates is $1.00 \mathrm{~cm}^{2}$

Narayan Hari

Numerade Educator

Problem 63

A focused 300 -W spotlight delivers $40 \%$ of its light within a circular area with a diameter of $2 \mathrm{~m}$. What is the root-mean-square electric field in this illuminated area?

Ajay Singhal

Numerade Educator

Problem 64

What is the electric field amplitude of an electromagnetic wave whose magnetic field amplitude is $5.00 \cdot 10^{-3} \mathrm{~T} ?$

Ajay Singhal

Numerade Educator

Problem 65

What is the distance between successive heating antinodes in a microwave oven's cavity? A microwave oven typically operates at a frequency of $2.4 \mathrm{GHz}$.

Ajay Singhal

Numerade Educator

Problem 66

The solar constant measured by Earth satellites is roughly $1400 \mathrm{~W} / \mathrm{m}^{2}$
a) Find the maximum electric field of the electromagnetic radiation from the Sun.
b) Find the maximum magnetic field of these electromagnetic waves.

Ajay Singhal

Numerade Educator

Problem 67

The peak electric field at a distance of $2.25 \mathrm{~m}$ from a light bulb is $21.2 \mathrm{~V} / \mathrm{m}$
a) What is the peak magnetic field there?
b) What is the power output of the bulb?

Ajay Singhal

Numerade Educator

Problem 68

If the peak electric field due to a star whose radius is twice that of he Sun is $44.0 \mathrm{~V} / \mathrm{m}$ at a distance of $15 \mathrm{AU}$, what is its temperature? Treat the star as a blackbody.

Ajay Singhal

Numerade Educator

Problem 69

A $5.00-\mathrm{mW}$ laser pointer has a beam diameter of $2.00 \mathrm{~mm}$
a) What is the root-mean-square value of the electric field in this laser beam?
b) Calculate the total electromagnetic energy in $1.00 \mathrm{~m}$ of this laser beam.

Ajay Singhal

Numerade Educator

Problem 70

At the surface of the Earth, the Sun delivers an estimated $1.00 \mathrm{~kW} / \mathrm{m}^{2}$ of energy. Suppose sunlight hits a $10.0 \mathrm{~m}$ by $30.0 \mathrm{~m}$ roof at an angle of $90.0^{\circ} .$
a) Estimate the total power incident on the roof.
b) Find the radiation pressure on the roof.

Ajay Singhal

Numerade Educator

Problem 71

The National Ignition Facility has the most powerful laser in the world; it uses 192 beams to aim $500 .$ TW of power at a spherical pellet of diameter $2.00 \mathrm{~mm} .$ How fast would a pellet of density $2.00 \mathrm{~g} / \mathrm{cm}^{3}$ accelerate if only one of the laser beams hits it for $1.00 \mathrm{~ns}$ and $2.00 \%$ of the light is absorbed?

Keshav Singh

Numerade Educator

Problem 72

A resistor consists of a solid cylinder of radius $r$ and length $L$. The resistor has resistance $R$ and is carrying current $i$. Use the Poynting vector to calculate the power radiated out of the surface of the resistor

Ajay Singhal

Numerade Educator

Problem 73

A radio tower is transmitting $30.0 \mathrm{~kW}$ of power equally in all directions. Assume that the radio waves that hit the Earth are reflected.
a) What is the magnitude of the Poynting vector at a distance of $12.0 \mathrm{~km}$ from the tower?
b) What is the root-mean-square value of the electric force on an electron at this location?

Ajay Singhal

Numerade Educator

Problem 74

Quantum theory says that electromagnetic waves actually consist of discrete packets-photons-each with energy $E=\hbar \omega,$ where $\hbar=1.054572 \cdot 10^{-34} \mathrm{~J} \mathrm{~s}$ is Planck's reduced constant and $\omega$ is the angular frequency of the wave.
a) Find the momentum of a photon.
b) Find the magnitude of angular momentum of a photon. Photons are circularly polarized; that is, they are described by a superposition of two plane-polarized waves with equal field amplitudes, equal frequencies, and perpendicular polarizations, one-quarter of a cycle $\left(90^{\circ}\right.$ or $\left.\pi / 2 \mathrm{rad}\right)$ out of phase, so the electric and magnetic field vectors at any fixed point rotate in a circle with the angular frequency of the waves. It can be shown that a circularly polarized wave of energy $U$ and angular frequency $\omega$ has an angular momentum of magnitude $L=U / \omega$. (The direction of the angular momentum is given by the thumb of the right hand, when the fingers are curled in the direction in which the field vectors circulate.)
c) The ratio of the angular momentum of a particle to $\hbar$ is its spin quantum number. Determine the spin quantum number of the photon.

Ajay Singhal

Numerade Educator

Problem 75

A microwave operates at $250 .$ W. Assuming that the waves emerge from a point source emitter on one side of the oven, how long does it take to melt an ice cube $2.00 \mathrm{~cm}$ on a side that is $10.0 \mathrm{~cm}$ away from the emitter if $10.0 \%$ of the photons that strike the cube are absorbed by it? How many photons of wavelength $10.0 \mathrm{~cm}$ hit the ice cube per second? Assume a cube density of $0.960 \mathrm{~g} / \mathrm{cm}^{3}$

Keshav Singh

Numerade Educator

Problem 76

An industrial carbon dioxide laser produces a beam of radiation with average power of $6.00 \mathrm{~kW}$ at a wavelength of $10.6 \mu \mathrm{m}$. Such a laser can be used to cut steel up to $25 \mathrm{~mm}$ thick. The laser light is polarized in the $x$ -direction, travels in the positive $z$ -direction, and is collimated (neither diverging or converging) at a constant diameter of $100.0 \mu \mathrm{m}$. Write the equations for the laser light's electric and magnetic fields as a function of time and of position $z$ along the beam. Recall that $\vec{E}$ and $\vec{B}$ are vectors. Leave the overall phase unspecified, but be sure to check the relative phase between $\vec{E}$ and $\vec{B}$.

Linda Winkler

Numerade Educator

Problem 77

During the testing of a new light bulb, a sensor is placed $31.9 \mathrm{~cm}$ from the bulb. It records an intensity of $182.9 \mathrm{~W} / \mathrm{m}^{2}$ for the radiation emitted by the bulb. What is the root-mean-square value of the electric field at the sensor's location?

Ajay Singhal

Numerade Educator

Problem 78

During the testing of a new light bulb, a sensor is placed $42.1 \mathrm{~cm}$ from the bulb. It records an intensity of $191.4 \mathrm{~W} / \mathrm{m}^{2}$ for the radiation emitted by the bulb. What is the root-mean-square value of the magnetic field at the sensor's location?

Ajay Singhal

Numerade Educator

Problem 79

During the testing of a new light bulb, a sensor is placed $52.5 \mathrm{~cm}$ from the bulb. It records a root-mean-square value of $9.142 \cdot 10^{-7} \mathrm{~T}$ for the magnetic field of the radiation emitted by the bulb. What is the intensity of that radiation at the sensor's location?

Ajay Singhal

Numerade Educator

Problem 80

During the testing of a new light bulb, a sensor is placed $17.7 \mathrm{~cm}$ from the bulb. It records a root-mean-square value of $279.9 \mathrm{~V} / \mathrm{m}$ for the electric field of the radiation emitted from the bulb. What is the intensity of that radiation at the sensor's location?

Ajay Singhal

Numerade Educator

Problem 81

To visually examine sunspots through a telescope, astronomers have to reduce the intensity of the sunlight to avoid harming their retinas. They

Ajay Singhal

Numerade Educator

Problem 82

To visually examine sunspots through a telescope, astronomers have to reduce the intensity of the sunlight to avoid harming their retinas. They accomplish this intensity reduction by mounting two polarizers on the telescope. The first polarizer has a polarizing angle of $38.3^{\circ}$ relative to the horizontal. If the astronomers want to reduce the intensity of the sunlight by a factor of 0.7584 , what polarizing angle should the second polarizer have with the horizontal? Assume that this angle is greater than that of the first polarizer.

Ajay Singhal

Numerade Educator

Problem 83

To visually examine sunspots through a telescope, astronomers have to reduce the intensity of the sunlight to avoid harming their retinas. They accomplish this intensity reduction by mounting two linear polarizers on the telescope. The second polarizer has a polarizing angle of $110.6^{\circ}$ relative to the horizontal. If the astronomers want to reduce the intensity of the sunlight by a factor of $0.7645,$ what polarizing angle should the first polarizer have with the horizontal? Assume that this angle is smaller than that of the second polarizer.

Ajay Singhal

Numerade Educator