Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 33

Electromagnetic Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

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. (a) If $0.20$ of the incident intensity is transmitted by the two sheets, what is $\theta ?$ (b) What percentage of the incident intensity is transmitted if the first sheet's angle is reduced to $0^{\circ}$ ?

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

2 In Fig. 33-26, initially unpolarized light is sent into a system of three polarizing sheets whose polarizing directions make angles of $\theta_{1}=\theta_{2}=\theta_{3}=40^{\circ}$ with the direction of the $y$ axis. (a) What percentage of the initial intensity is transmitted by the system?

Raj Bala

Numerade Educator

Problem 3

In Fig. 33-26, initially unpolarized light is sent into a system of three polarizing sheets whose polarizing directions make angles of $\theta_{1}=40^{\circ}, \theta_{2}=10^{\circ}$, and $\theta_{3}=40^{\circ}$ with the direction of the $y$ axis. (a) What percentage of the light's initial intensity is transmitted by the system? (Hint: Be careful with the angles.) (b) At what angle to the $y$ axis is the emerging light polarized? (Give the angle and indicate either clockwise or counterclockwise.)

Raj Bala

Numerade Educator

Problem 4

In Fig. 33-27, a beam of unpolarized light, with intensity $43 \mathrm{~W} / \mathrm{m}^{2}$, is sent into a system of two polarizing sheets with polarizing directions at angles $\theta_{1}=60^{\circ}$ and $\theta_{2}=90^{\circ}$ to the $y$ axis. What is the intensity of the light transmitted by the system?

Raj Bala

Numerade Educator

Problem 5

In Fig. 33-27, 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 $\theta_{1}=70^{\circ}$ and $\theta_{2}=90^{\circ}$ to the $y$ axis. (a) What is the intensity of the light transmitted by the two-sheet system? (b) What is the transmitted intensity if, instead, the initial polarization is parallel to the $x$ axis?

Raj Bala

Numerade Educator

Problem 6

In Fig. 33-27, 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 $33-28$ 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 system when $\theta_{2}=110^{\circ}$ ?

Raj Bala

Numerade Educator

Problem 7

In Fig. 33-29, 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? (e) What is the incident angle if the index of refraction is $1.30$ ?

Raj Bala

Numerade Educator

Problem 8

In Fig. 33-30, 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 $\theta_{2}$ can be varied. Figure $33-31$ 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 $\theta_{2}=35^{\circ}$ ?

Raj Bala

Numerade Educator

Problem 9

When the rectangular metal tank in Fig. 33-32 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 $L=0.680 \mathrm{~m}$, what is the index of refraction of the liquid?

Raj Bala

Numerade Educator

Problem 10

In Fig. 33-30, unpolarized light
is sent into a system of three polar izing 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 $\theta_{2}$ can be varied. Figure $33-33$ 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 $\theta_{2}=110^{\circ} ?$

Raj Bala

Numerade Educator

Problem 11

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 $65 \%$ of the original intensity?

Raj Bala

Numerade Educator

Problem 12

In Fig. 33-30, unpolarized light is sent into a system of three polarizing sheets, which transmits $0.0450$ 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 $2 ?$

Raj Bala

Numerade Educator

Problem 13

In Fig. 33-34, 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
$$
n=\frac{\sin \frac{1}{2}(\psi+\phi)}{\sin \frac{1}{2} \phi}
$$
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.)

Sheh Lit Chang

University of Washington

Problem 14

Suppose the prism of Fig. 33-34 has apex angle $\phi=60.0^{\circ}$ and index of refraction $n=1.56$. (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?

Raj Bala

Numerade Educator

Problem 15

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. 33-35). 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 $3.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$, 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)?

Raj Bala

Numerade Educator

Problem 16

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 $2.0$ 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?

Raj Bala

Numerade Educator

Problem 17

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 $4.0$ during the rotation, what fraction of the intensity of the original beam is associated with the beam's polarized light?

Raj Bala

Numerade Educator

Problem 18

Rainbows from square drops. Suppose that, on some surreal world, raindrops had a square cross section and always fell with one face horizontal. Figure 33-36 shows such a falling drop, with a white beam of sunlight incident at $\theta=65.0^{\circ}$ 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 $(n=1.331)$ 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.)

Raj Bala

Numerade Educator

Problem 19

In Fig. 33-37, a $2.50 \mathrm{~m}$ 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?

Raj Bala

Numerade Educator

Problem 20

In Fig. $33-38 a$, 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 $33-38 b$ 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 $\theta_{3 a}=30.0^{\circ}$ 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 $75^{\circ}$ and the index of refraction of material 3 is $2.4$, what is $\theta_{3}$ ?

Raj Bala

Numerade Educator

Problem 21

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 $16.0^{\circ}$ with the normal. What is the index of refraction of the glass?

Raj Bala

Numerade Educator

Problem 22

A certain type of glass has an index of refraction of $1.62$ for red light. (a) At what angle will light (in air) be completely polarized when reflected by the glass? (b) Will the reflection angle for complete polarization of blue light be larger or smaller?

Raj Bala

Numerade Educator

Problem 23

In Fig. 33-39, 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 $n_{4}=1.75$, what is the value of (a) $\theta_{5}$ in the air and (b) $\theta_{4}$ in the bottom material?

Raj Bala

Numerade Educator

Problem 24

An isotropic point source emits light at wavelength $500 \mathrm{~nm}$, at the rate of $300 \mathrm{~W}$. 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?

Raj Bala

Numerade Educator

Problem 25

In Fig. 33-40, a ray of light is perpendicular to the face $a b$ of a glass prism $(n=1.81)$. 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.

Raj Bala

Numerade Educator

Problem 26

$$
\begin{aligned}
&\text { Dispersion in a window pane. In Fig. } 33-41, \text { a } \\
&\text { beam of white light is incident at angle } \theta=40^{\circ} \text { on } \\
&\text { a common window pane (shown in cross section). } \\
&\text { For the pane's type of glass, the index of refraction } \\
&\text { for visible light ranges from } 1.524 \text { at the blue end } \\
&\text { of the spectrum to } 1.509 \text { at the red end. The two } \\
&\text { sides of the pane are parallel. What is the angular } \\
&\text { spread of the colors in the beam (a) when the light } \\
&\text { enters the pane and (b) when it emerges from the } \\
&\text { opposite side? (Hint: When you look at an object } \\
&\text { through a window pane, are the colors in the light from the object }
\end{aligned}
$$

Raj Bala

Numerade Educator

Problem 27

The maximum electric field $27 \mathrm{~m}$ from an isotropic point source of light is $17 \mathrm{~V} / \mathrm{m}$. 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?

Raj Bala

Numerade Educator

Problem 28

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 ratio of that force to the force due to the Sun's gravitational attraction.

Raj Bala

Numerade Educator

Problem 29

Figure 33-42 shows light reflecting from two perpendicular reflecting surfaces $A$ and $B$. Find the angle between the incoming ray $i$ and the outgoing ray $r^{\prime}$.

Raj Bala

Numerade Educator

Problem 30

In Fig. 33-43a, 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 33-43b 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 $\theta_{2 b}=40.0^{\circ}$. (a) What is the index of refraction of material 1? (b) If the incident angle is changed to $65^{\circ}$ and material 2 has $n_{2}=2.4$, then what is angle $\theta_{2}$ ?

Raj Bala

Numerade Educator

Problem 31

In Fig. 33-44, 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 $n_{1}=1.80, n_{2}=1.40$, and $n_{3}=1.20$.
(a) What is angle $\theta$ ? (b) If $\theta$ is increased, is there refraction of light into material 3 ?

Raj Bala

Numerade Educator

Problem 32

In a plane radio wave the maximum value of the electric field component is 8.00 V/m. Calculate (a)
the maximum value of the magnetic field component and (b) the
wave intensity.

Raj Bala

Numerade Educator

Problem 33

A point source of light is $1.20 \mathrm{~m}$ below the surface of a body of water. Find the diameter of the circle at the surface through which light emerges from the water.

Raj Bala

Numerade Educator

Problem 34

About how far apart must you hold your hands for them to be separated by $2.0$ nano-light-second (the distance light travels in $2.0 \mathrm{~ns})$ ?

Raj Bala

Numerade Educator

Problem 35

A plane electromagnetic wave, with wavelength $5.0 \mathrm{~m}$, travels in vacuum in the positive direction of an $x$ axis. The electric field, of amplitude $215 \mathrm{~V} / \mathrm{m}$, 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 time-averaged 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 $(g)$ the rate at which momentum is transferred to the surface and (h) the radiation pressure on the surface?

Raj Bala

Numerade Educator

Problem 36

A black, totally absorbing piece of cardboard of area $A=2.0 \mathrm{~cm}^{2}$ intercepts light with an intensity of $20 \mathrm{~W} / \mathrm{m}^{2}$ from a camera strobe light. What radiation pressure is produced on the cardboard by the light?

Raj Bala

Numerade Educator

Problem 37

What is the intensity of a traveling plane electromagnetic wave if $B_{m}$ is $3.0 \times 10^{-4} \mathrm{~T}$ ?

Raj Bala

Numerade Educator

Problem 38

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 water above the benzene?

Raj Bala

Numerade Educator

Problem 39

Figure 33-45 depicts a simplistic optical fiber: a plastic core $\left(n_{1}=1.58\right)$ is surrounded by a plastic sheath $\left(n_{2}=1.46\right)$. A light ray is incident on one end of the fiber at angle $\theta$. The ray is to undergo total internal reflection at point $A$, where it encounters the core-sheath boundary. (Thus there is no loss of light through that bound-
ary.) What is the maximum value of $\theta$ that allows total internal reflection at $A$ ?

Raj Bala

Numerade Educator

Problem 40

A catfish is $1.50 \mathrm{~m}$ 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?

Raj Bala

Numerade Educator

Problem 41

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.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 42

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. 33-46), "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 9000 ly away that could provide such a signal? A light-year (ly) is the distance light travels in one year.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 43

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?

Sheh Lit Chang

University of Washington

Problem 44

A small laser emits light at power $5.00 \mathrm{~mW}$ and wavelength $633 \mathrm{~nm}$. The laser beam is focused (narrowed) until its diameter matches the $1206 \mathrm{~nm}$ 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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

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

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

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 $1800 \mathrm{~kg}$, 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 continuously driven away from the Sun.)

Raj Bala

Numerade Educator

Problem 47

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 1.71). At what angle of incidence does the reflected light end up fully polarized?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

Assume (unrealistically) that a TV station acts as a point source broadcasting isotropically at $3.0 \mathrm{MW}$. 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-year (ly) is the distance light travels in one year.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 49

In the ray diagram of Fig. 33-47, 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 $\phi=71.0^{\circ}$, 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 ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

A plane electromagnetic wave has a maximum electric field magnitude of $1.80 \times 10^{-4} \mathrm{~V} / \mathrm{m}$. Find the magnetic field amplitude.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

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 $4.5 \times 10^{3} \mathrm{MW}$ is focused onto $0.80 \mathrm{~mm}^{2}$ 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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

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 deeply submerged. If the effective wavelength were $2.0 \times 10^{4}$ 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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 53

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 $25 \mathrm{~kW}$ laser beam, what speed will the ship attain in $45.0$ day because of the momentum carried away by the beam?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 54

$$
\begin{aligned}
&\text { In Fig. } 33-48 \text {, a light ray in }\\
&\begin{aligned}
&\text { air is incident at angle } \theta_{1} \text { on a block } \\
&\text { of transparent plastic with an index } \\
&\text { of refraction of } 1.56 \text {. The dimen- } \\
&\text { sions indicated are } H=2.00 \mathrm{~cm} \\
&\text { and } W=3.00 \mathrm{~cm} \text {. The light passes } \\
&\text { through the block to one of its sides } \\
&\text { and there undergoes reflection (in- } \\
&\text { side the block) and possibly }
\end{aligned}
\end{aligned}
$$

Raj Bala

Numerade Educator

Problem 55

What inductance must be connected to a $25 \mathrm{pF}$ capacitor in an oscillator capable of generating $410 \mathrm{~nm}$ (i.e., visible) electromagnetic waves? Comment on your answer.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 56

In Fig. 33-49, light from ray $A$ refracts from material $1\left(n_{1}=1.50\right)$ 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 $\theta_{B} ?(\mathrm{~d})$ If $\theta_{B}$ is decreased, does part of the light refract into material 3?

Raj Bala

Numerade Educator

Problem 57

What is the radiation pressure 2.7 m away from a 315 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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 58

The intensity $I$ of light from an isotropic point source is determined as a function of distance $r$ from the source. Figure 33-50 gives intensity $I$ versus the inverse square $r^{-2}$ of that distance. The vertical axis scale is set by $I_{s}=400 \mathrm{~W} / \mathrm{m}^{2}$, and the horizontal axis scale is set by $r_{s}^{-2}=8.0 \mathrm{~m}^{-2}$. What is the power of the source?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

An airplane flying at a distance of $10 \mathrm{~km}$ from a radio transmitter receives a signal of intensity $28 \mu \mathrm{W} / \mathrm{m}^{2}$. 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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 60

In Fig. 33-51a, 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. $33-51 b$. The vertical axis scale is set by $\theta_{2 s}=80^{\circ}$. 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?

Raj Bala

Numerade Educator

Problem 61

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 $5.00 \mathrm{pm}$. What is the corresponding "frequency width" for the emission?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 62

In Fig. 33-52, a light ray in air is incident on a flat layer of material 2 that has an index of refraction $n_{2}=1.7$. 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 $n_{3}$ ?

Raj Bala

Numerade Educator

Problem 63

Someone plans to float a small, totally absorbing sphere $0.200 \mathrm{~m}$ 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 $0.500 \mathrm{~mm}$. (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?

Raj Bala

Numerade Educator

Problem 64

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 $p_{r}$ to Earth's sea-level atmospheric pressure, which is $1.0 \times 10^{5} \mathrm{~Pa}$.

Sheh Lit Chang

University of Washington

Problem 65

A plane electromagnetic wave traveling in the positive direction of an $x$ axis in vacuum has components $E_{x}=E_{y}=0$ and $E_{z}=(4.0 \mathrm{~V} / \mathrm{m}) \cos \left[\left(\pi \times 10^{15} \mathrm{~s}^{-1}\right)(t-x / c)\right] .$ (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? (d) In what direction is the wave moving?

Raj Bala

Numerade Educator

Problem 66

What is the wavelength of the electromagnetic wave emitted by the oscillator-antenna system of Fig. 33-3 if $L=0.253 \mu \mathrm{H}$ and $C=30.0 \mathrm{pF} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

Unpolarized light of intensity $6.5 \mathrm{~mW} / \mathrm{m}^{2}$ 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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

In Fig. 33-53, a laser beam of power $4.60 \mathrm{~W}$ and diameter $D=3.00 \mathrm{~mm}$ 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 $H ?$

Raj Bala

Numerade Educator

Problem 69

Some neodymium-glass lasers can provide $100 \mathrm{TW}$ of power in $2.2 \mathrm{~ns}$ pulses at a wavelength of $0.26 \mu \mathrm{m}$. How much energy is contained in a single pulse?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

In Fig. 33-54a, 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. 33-54b. The horizontal axis scale is set by $\theta_{1 s}=80^{\circ}$. 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?

Raj Bala

Numerade Educator