`
`
`
`
`‘)‘)
`
`ll
`
`
`
`aa
`
`
`
`.~ ‘.~ ‘
`
`
`
`V‘!V‘!
`
`
`
`..
`
`
`ii
`
`..
`
`
`MATERIONMATERION
`
`EX. 2002-001EX. 2002-001
`
`PGR2019-00017PGR2019-00017
`
`MATERION
`PGR2019-00017
`EX. 2002-001
`
`

`

`Third Edi ti on
`
`•
`SICS
`
`For Scientists and Engineers
`
`Paul A. Tipler
`
`Worth Publishers
`
`j
`
`MATERION
`PGR2019-00017
`EX. 2002-002
`
`

`

`For Claudia
`
`Physics for Scientists and Engineers, Third Edition
`Paul A. Tipler
`
`Copyright © 1991, 1982, 1976 by Worth Publishers, Inc.
`All rights reserved
`
`Printed in the United States of America
`
`Library of Congress Catalog Card Number: 89-52165
`ISBN: 0-87901-430-x
`
`Printing:
`
`5
`
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`Year:
`
`95
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`Development Editors: Valerie Neal and Steven Tenney
`
`Design: Malcolm Grear Designers
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`Art Director: George Touloumes
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`Production Editor: Elizabeth Mastalski
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`Production Supervisor: Sarah Segal
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`Layout: Patricia Lawson
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`Photographs: Steven Tenney, John Schultz of PAR/ NYC,
`and Lana Berkovich
`
`Line Art: York Graphic Services and Demetrios Zangos
`
`Composition: York Graphic Services
`
`Printing and binding: R. R. Donnelley and Sons
`
`Cover: Supersonic Candlelight. A stroboscopic color schlieren
`or shadow picture taken at one-third microsecond exposure
`shows a supersonic bullet passing through the hot air rising
`above a candle. Schlieren pictures make visible the regions
`of nonuniform density in air. Estate of Harold E. Edgerton/
`Courtesy of Palm Press.
`
`Illustration credits begin on p. IC-1 which constitute '!n
`extension of the copyright page.
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`Worth Publishers
`
`33 Irving Place
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`
`MATERION
`PGR2019-00017
`EX. 2002-003
`
`

`

`943
`
`A multiple-exposure view
`showing the 26-m tracking
`antenna at Wallops Station, Vir(cid:173)
`ginia, and a total solar eclipse.
`Electromagnetic radiation at radio
`wavelengths, like that at optical
`wavelengths, is not readily ab(cid:173)
`sorbed by the earth's atmo(cid:173)
`sphere-making it a viable means
`of communication between two
`distant points on the ground or
`between a point on the ground
`and a plane, satellite, or space(cid:173)
`craft. Objects are tracked by
`aiming a continuous radar beam
`at them and receiving the re(cid:173)
`flected beam.
`
`axwell' s Equations and
`Electromagnetic Waves
`
`bout 1860, the great Scottish physicist James Clerk Maxwell found that the
`erimental laws of electricity and magnetism-the laws of Coulomb,
`uss, Biot-Savart, Ampere, and Faraday, which we have studied in Chap-
`18 through 28-could be summarized in a concise mathematical form
`w known as Maxwell's equations. One of the laws, Ampere's law, con(cid:173)
`ed an inconsistency, which Maxwell was able to remove with the inven(cid:173)
`~£ the displacement current (Section 29-1). The new consistent set of
`~~ns p1~edicts t~e possibility of ele~tromagnetic "".av:s.
`th x_weU s equat10ns relate the electnc~ and magnetic-field vectors E and
`eir sources, which are electric charges, currents, and changing fields.
`e equ~tions play a role in classical electromagnetism analogous to that
`i:t~n. slaws in classical mechanics. In principle, all problems in classi(cid:173)
`ctncity and magnetism can be solved using Maxwell's equations, just
`~~1?blems i:1 classical mechanics can be solved using Newton's laws.
`, hos equations are considerably more complicated than Newton's
`Yo ~ever, and their application to most problems involves mathemat(cid:173)
`th~r t~e scope of this book. Nevertheless, Maxwell's equations are of
`w ehcal importance.
`"on fell showed that these equations could be combined to yield a wave
`or the electric- and magnetic-field vectors E and B. Such electro-
`
`MATERION
`PGR2019-00017
`EX. 2002-004
`
`

`

`944
`
`Chapter 29 Maxwell's Equations and Electromagnetic Waves
`
`magnetic waves are caused by accelerating charges, for example, the charge
`in an alternating current in an antenna. They were first produced in th s
`laboratory by _Heinrich _Hertz in 1887. Maxwell showed that the sp eed 0~
`electromagnetic waves m free space should be
`
`1
`c =--
`
`-~
`
`
`where Ea, the permittivity of free space, is the constant appearing in Cou(cid:173)
`lomb's and Gauss's laws and fLo, the permeability of free space, is that in th
`Biot- Savart law and Ampere's law. When the measured value of Eo and th:
`defined value of µ, 0 are put into Equation 29-1, the speed of electromagnetic
`waves is found to be about 3 x 108 m/ s, the same as the measured speed of
`light. Maxwell noted this "coincidence" with great excitement and correctly
`surmised that light itself is an electromagnetic wave.
`In this chapter we begin by showing that Ampere's law as stated in
`Chapter 25 does not hold for discontinuous currents. We then show how
`Maxwell generalized Ampere's law by adding a term now called Maxwell's
`displacement current. After stating Maxwell's equations and relating them
`to the laws of electricity and magnetism that we have already studied, we
`will show that these equations imply that electric and magnetic field vectors
`obey a wave equation that describes waves that propagate through free
`space with speed c = 1/~. Finally, we will illustrate how electromag(cid:173)
`netic waves carry energy and momentum, and discuss the electromagnetic
`spectrum.
`
`29-1
`
`29-1 Maxwell's Displacement Current
`
`As we studied in Chapter 25, Ampere's law (Equation 25-15) relates the line
`integral of the magnetic field around some closed curve C to the current that
`passes through any area bounded by that curve:
`
`f B·df = /Lal
`
`for any closed curve C
`
`29-2
`
`c
`We noted that this equation holds only for continuous currents. We can ~ee
`that it does not hold for discontinuous currents by considering the chargmg
`of a capacitor (Figure 29-1). According to Ampere's law, the line integral of
`the magnetic field B around a closed curve equals µ,0 times the total current
`through any surface bounded by the curve. Such a surface need not be a
`plane. Two surfaces bounded by the curve C are indicated in Figure 29-1.
`
`Plates of
`capacitor
`
`Figure 29-1 Two surfaces 51 and 52 bounded by the
`same curve C. The current I passes through surface 51
`but not 52 . Ampere's law, which relates the line inte(cid:173)
`gral of the magnetic field B around the curve C to the
`total current passing through any surface bounded by
`C, is not valid when the current is not continuous, as
`when it stops at the capacitor plate here.
`
`MATERION
`PGR2019-00017
`EX. 2002-005
`
`

`

`Section 29-1 Maxwell's Displacement Current
`
`945
`
`fhe current through surface 51 is I. There is no current through surface 52
`because the charge stops on the capacitor plate. There is thus ambiguity in
`the phrase "the current through any surface. bounded by the curve." For
`continuous currents, we get the same current no matter which surface we
`choose.
`Maxwell recognized this flaw in Ampere's law and showed that the law
`can be generalized to include all situations if the current I in the equation is
`replaced by the sum of the conduction current I and another term lct, called
`Maxwell's displacement current, defined as
`
`dc/Je
`Ict = Eo - -
`'
`dt
`
`29-3
`
`axwell's ...
`ng them
`iied, we
`! vectors
`.igh free
`:tromag(cid:173)
`nagnetic
`
`, can see
`:harging
`tegral of
`I current
`not be a
`ire 29-1.
`
`where cfae is the flux of the electric field through the same surface bounded
`by the curve C. The generalized form of Ampere's law is then
`
`i B·dC = f.Lo(I + Id) = µ,0I + f.LoEo d!e
`
`29-4
`
`We can understand this generalization by considering Figure 29-1 again.
`Let us call the sum I + Ict the generalized current. According to the argu(cid:173)
`ment just stated, the same generalized current must cross any area bounded
`by the curve C. Thus, there can be no net generalized current into or out of
`the closed volume. If there is a net true current I into the volume, there must
`be an equal net displacement current Ict out of the volume. In the volume in
`the figure, there is a net conduction current I into the volume that increases
`the charge within the volume:
`
`I= dQ
`dt
`
`The flux of the electric field out of the volume is related to the charge by
`Gauss's law
`
`cPe,net = J.. En dA = J__Qinside
`1s
`Eo
`The rate of increase of the charge is thus proportional to the rate of increase
`of the net flux out of the volume:
`
`E d<f>e,net _ dQ _ I
`- dt -
`dt
`0
`d
`
`Thus, the net conduction current into the volume equals the net displace(cid:173)
`ment current out of the volume. The generalized current is always continu(cid:173)
`ous.
`It is interesting to compare Equation 29-4 to Faraday's law (Equation
`26-6)
`
`J..
`c = lc E·dC = -dt
`d<f>m
`29-5
`Where c is the induced emf in a circuit and c/Jrn is the magnetic flux through
`the circuit. According to Faraday's law, a changing magnetic flux produces
`an electric field whose line integral around a curve is proportional to the rate
`.of change of magnetic flux through the curve. Maxwell's modification of
`Ampere's law shows that a changing electric flux produces a magnetic field
`Whose line integral around a curve is proportional to the rate of change of
`he electric flux. We thus have the interesting reciprocal result that a chang-
`ng magnetic field produces an electric field (Faraday's law) and a changing
`lectric field produces a magnetic field (generalized form of Ampere's law.)
`ote that there is no magnetic analogue of a conduction current I.
`
`f
`
`MATERION
`PGR2019-00017
`EX. 2002-006
`
`

`

`946
`
`Chapter 29 Maxwell's Equations and Electromagnetic Waves
`
`Example 29-1
`A parall.el-plat~ capacitor has c~~sely spaced circular plates of radius R.
`Charge is flowing onto the positive plate and off of the negative plate
`the rate I = dQ/ dt = 2.5 A. Compute the displacement current betwee~
`the plates.
`
`Since the plates are closely spaced, the electric field between them .
`uniform in the direction from the positive plate to the n egative pl :s
`with magnitude E = a/ E0, where (J' is the magnitude of the charge a e .
`unit area on either plate. Consider any plane between the plates ap~
`parallel to them. Since E is perpendicular to the plates and therefore~
`0
`the plane, and is uniform between the plates and zero outside th
`plates, the electric flux through the plane is
`e
`</Je = 7TR2E = (7TR2)((J'/ Eo) = Q/ Eo
`2
`where Q = 7TR
`(J' is the magnitude of the total charge on either plate.
`The displacement current is then
`lct = Eod<f>e/dt = dQ/ dt = 2.5 A
`
`Example 29-2
`
`The circular plates in Example 29-1 have a radius of R = 3. 0 cm . Find the
`magnetic field at a point between the plates at a distance r = 2.0 cm
`from the axis of the plates when the current into the positive plate is
`2.5 A.
`
`We find B from the generalized form of Ampere's law (Equation 29-4).
`In Figure 29-2, we have chosen a circular path of radius r = 2. 0 cm about
`the center line joining the plates to compute f B·df . By symm~try, Bis
`tangent to this circle and has the same magnitude everywhere on it.
`Then
`
`f B·df = B(2m')
`
`The electric flux through the area bounded by this curve is
`
`Figure 29-2 Curve C for comput(cid:173)
`ing the displacement current in
`Example 29-2.
`
`r2Q
`Q
`= (7TY2)- - = -2-
`7TR2Eo
`R Eo
`f
`current
`where we have used (J' = Q/ 7TR2. Since there is no con uc 1011 .
`d
`t the
`between the plates of the capacitor, the generalized current is JUS
`displacement current
`
`r 2 dQ
`B(21Tr) = /Lo R2 dt
`
`- dQ
`B = EQ_ - 1
`- - - = (2 x 10- 7 T·m/ A)
`27T R2 dt
`.
`
`- 1 11 >< tO
`0 02 m
`.
`.
`2 (2.5 A) -
`(0.03 m)
`
`s
`
`MATERION
`PGR2019-00017
`EX. 2002-007
`
`

`

`Maxwell's Equations
`
`ell's equations are
`MaxW
`
`Section 29-2 Maxwell's Equations
`
`947
`
`11
`I
`
`f Bn dA = 0
`
`s
`
`th E·df = _!!__ ( Bn dA
`t
`dt JS
`th B·df = /LoI + /LoEo !!__ ( En dA
`t
`dt JS
`
`29-6a
`
`29-6b
`
`29-6c
`
`29-6d
`
`Maxwell's equations
`
`s
`
`1t
`e
`
`Equation 29-6a is Gauss's law; it states that the flux of the electric field
`through any closed surface equals 1/ Eo times the net charge inside the sur(cid:173)
`face . As discussed in Chapter 19, Gauss's law implies that the electric field
`due to a point charge varies inversely as the square of the distance from the
`charge. This law describes how electric-field lines diverge from a positive
`charge and converge on a negative charge. Its experimental basis is Cou(cid:173)
`lomb's law . .
`Equation 29-6b, sometimes called Gauss's law for magnetism, states that
`the flux of the magnetic-field vector B is zero through any closed surface.
`This equation describes the experimental observation that magnetic-field
`lines do not diverge from any point in space or converge on any point; that
`is, it implies that isolated magnetic poles do not exist.
`Equation 29-6c is Faraday's law; it states that the integral of the electric
`field around any closed curve C, which is the emf, equals the (negative) rate
`of change of the magnetic flux through any surface S bounded by the curve.
`(This is not a closed surface, so the magnetic flux through Sis not necessarily
`zero. ) Faraday's law describes how electric-field lines encircle any area
`through which the magnetic flux is changing, and it relates the electric-field
`vector E to the rate of change of the magnetic-field vector B.
`Equation 29-6d, Ampere's law with Maxwell's displacement-current
`modification, states that the line integral of the magnetic field B around any
`closed curve C equals µ, 0 times the current through any surface bounded by
`the curve plus /LoEo times the rate of change of the electric flux through the
`surface. This law describes how the magnetic-field lines encircle an area
`through which a current is passing or the electric flux is changing.
`
`29-3 The Wave Equation for Electromagnetic
`Waves (Optional)
`In Section 13-8, we saw that the harmonic wave functions for waves on a
`string obey a partial differential equation called the wave equation:
`a2y(x, t)
`1 a2y(x, t)
`ax2
`= ~ at 2
`In this equation, y(x, t) is the wave function, which for string waves is the
`displacement of the string. The derivatives are partial derivatives because
`
`29-7
`
`MATERION
`PGR2019-00017
`EX. 2002-008
`
`

`

`I
`
`I I
`I
`
`l '· '
`
`I
`
`948
`
`Chapter 29 Maxwell's Equations and Electromagnetic Waves
`
`the wave func.tion depends on both x ~nd t. The quantity v is the velocity of
`the wave, which depends on the medmm (and on the frequency if the 11
`<limn is dispersive) . We also saw that the wave equation for shing waves cle(cid:173)
`be derived by applyirig Newton's laws of motion to a string under tensioan
`11
`and we found that the velocity of the waves is VF[µ,, where Fis the tensi
`'
`and µ, the linear mass density. The solutions of this equation are harmo~·n
`wave functions of the form
`ic
`
`y(x, t) = Yo sin (kx - wt)
`
`where k = 2n/ A is the wave number and w = 21Tf is the angular frequency.
`
`A double rainbow over the radio
`telescope at Socorro, New Mexico.
`The telescope consists of a Very
`Large Array (VLA) of antenna
`dishes. The direction of incoming
`radio waves from distant galaxies
`can be determined by the interfer(cid:173)
`ence of signals detected in th e
`array.
`
`y
`
`z
`
`1- .:lx- 1
`
`: -
`
`-
`
`-
`
`-
`
`-
`
`- i ---~f
`
`Ey (xi)
`
`Ey(xz)
`
`Xz
`
`il(
`
`x
`
`Figure 29-3 A rectangular curve
`in the xy plane for the derivation
`of Equ ation 29-8.
`
`In this section, we will use Maxwell's equations to derive the wave equa(cid:173)
`tion for electromagnetic waves. We will not consider how such waves arise
`from the motion of charges but merely show that the laws of electricity and
`magnetism imply a wave equation, which in turn implies the existence of
`electric and magnetic fields E and B that propagate through space with the
`velocity of light c. We will consider only free space, in which there are no
`charges or currents. We will assume that the electric and magnetic fields .E
`and B are functions of time and one space coordinate only, which we ~"'1~
`take to be the x coordinate. Such a wave is called a plane wave, because fie!
`quantities are constant across any plane perpendicular to the x ~xis:
`f
`To obtain the wave equation relating the time and space denvatives 0
`either the electric field E or the magn etic field B, we first relate the sp~:
`derivative of one of the field vectors to the time derivative of the other. n
`do this by applying Equations 29-6c and 29-6d to appropriat.ely ~hoi~a­
`curves in space. We first relate the space derivative of E_,1 to the time e;tan·
`tive of B2 by aprlying Equation 2:-6c. (which is Faraday's la~) t? t~.: ~~-3. If
`gular curve of sides Ax and Ay lymg m the xy plane shown m Figu ·
`roxi·
`Ax and Ay are very small, the line integral of E aro und this curve 15 app
`mately
`
`f E·dC = Ey(x2) Ay - E_,/x1) Ay
`
`f £ at
`· h value 0
`~
`where E.1/ x 1) is the value of Ey at the point x1 and Ey(x2) is t e d bottofll
`the point x2. The contributions of the type Ex Ax from the top anend on y(
`this curve cancel because we have assumed that E does not dep
`
`MATERION
`PGR2019-00017
`EX. 2002-009
`
`

`

`Section 29-3 The Wave Equation for Electromagnetic Waves (Optional)
`
`949
`
`e !),,xis very small, we can approximate the difference in Ey at the
`[3ecatts
`)· jrttS x1 and Xz by
`
`aEy
`Ey(xz) - Ey(x1) = 6.E = -
`ax
`
`,
`
`6.x
`
`f
`
`aEy
`E·d€ = -
`ax
`
`6.x 6.y
`
`fhe flux of the magnetic field through this curve is approximately
`
`L B11 dA = Bz 6.x 6. y
`
`Faraday's law then gives
`
`aEy
`-
`ax
`
`aBz
`6.x 6.y = - - 6.x 6.y
`at
`
`aEy
`ax
`
`29-8
`
`Equation 29-8 implies that if there is a component of the electric field Ey that
`depends on x, there must be a component of the magnetic field Bz that
`depends on time or, conversely, that if there is a component of the magnetic
`field Bz that depends on time, there must be a component of the electric field
`Ey that depends on x. We can get a similar equation relating the space deriv(cid:173)
`ative of the magnetic field Bz to the time derivative of the electric field Ey by
`applying Equation 29-6d to the curve of sides 6.x and 6.z in the xz plane
`shown in Figure 29-4. For the case of no conduction currents, Equation
`29-6d is
`
`j, B·d( = µoEo .!!:.__ l En dA
`
`J
`dt s
`The details of this calculation are similar to those for Equation 29-8. The
`result is
`
`aEy
`aBz
`- -= - µoEo - -
`ax
`at
`
`29-9
`
`y
`
`We can eliminate either Bz or Ey from Equations 29-8 and 29-9 by differ(cid:173)
`entiating either equation with respect to x or t. If we differentiate both sides
`of Equation 29-8 with respect to x, we obtain
`
`aax_ ( aa;) = -
`
`aax ( a!z )
`
`or
`
`z
`
`a (aBz)
`- at --a;
`Where the order of the time and space derivatives on the right side have
`been interchanged. We now use Equation 29-9 for aBz/ ax:
`
`a2Ey _
`ax 2
`
`-
`
`azEy = _ J_ ( - µoEo aEy)
`ax 2
`at
`at
`
`x
`
`t:i. z
`~----- //-/---~~
`4-1::i. x--/
`Figure 29-4 A rectangular curve
`in the xz plane for the derivation
`of Equation 29-9.
`
`MATERION
`PGR2019-00017
`EX. 2002-0010
`
`

`

`950
`
`Chapter 29 Maxwell's Equations and Electromagnetic Waves
`
`which yields the wave equation
`a2Ey
`a2Ey
`ax2 = /J-oEo 7
`
`Comparing this equation with Equation 29-7, we see that E ob
`·
`f
`· l
`d
`eys a , ..
`Y
`equation or waves wit 1 spee
`··a~
`
`1
`c=---
`~
`
`which is Equation 29-1.
`If we had instead chosen to eliminate Ey from Equations 29-8
`d
`(by differentiating Equation 29-8 with respect to t, for example) wan 29.g
`have obtained an equation identical to Equation 29-10 except with Be Would
`ing Ey· We can thus see that both the electric field Ey and the magn;~e~ao­
`Bz obey a wave equation for waves traveling with the velocity l/~
`which is the velocity of light.
`1-toEo,
`By following the same line of reasoning as used above, we can re dil
`show (as in ~roblem 29-49) that if ~qua.tion 29-6c (Faraday's law) is ap;liJ
`to the curve m the xz plane shown m Figure 29-4, the spatial variation in E
`is related to the time variation in By by
`
`aEz = aBy
`ax
`at
`Similarly, the application of Equation 26-6d to the curve in the xy plane of
`Figure 29-3 gives
`
`29-11
`
`aEz
`a By
`- - = /J-oEo--
`ax
`at
`
`29-12
`
`We can use these results to show that, for a wave propagating in the x
`direction, the components Ez and By also obey the wave equation.
`So far we have considered only they and z components of the electric
`and magnetic fields. The same type of analysis may be applied to a rectangu(cid:173)
`lar loop in the yz plane, similar to the loops in Figure 29-3 and 29-4, to obtain
`equations analogous to Equations 29-8, 29-9, 29-11, and 29-12 in which the
`time derivatives of Ex and Bx are proportional to the y and z derivatives of
`the field quantities. In the plane wave under consideration, the y and z
`derivatives of the field quantities are zero, so the time derivatives of Ex a~d
`Bx must be zero. In an electromagnetic wave, we are interested only m
`time-varying fields, and so we may subtract out any x components of the
`fields that are constant in time.
`.
`al-
`We have shown that in any plane electromagnetic wave traveling par d
`lel to the x axis the x components of the fields are zero, so the vectors Ea~
`B are perpendicular to the x axis. They are also perpendicular to each ot er
`and each obeys the wave equation:
`
`Wave equation for E
`
`Wave equation for B
`
`ax2
`
`c2 at2
`
`ax2
`
`c2 at2
`
`z9-13a
`
`29.13b
`
`ortant
`.
`.
`As we noted in discussing harmonic waves, a particularly unp func-
`solution to a wave equation like Equation 29-10 is the harmonic wave
`
`MATERION
`PGR2019-00017
`EX. 2002-0011
`
`

`

`Section 29-3 The Wave Equation for Electromagnetic Waves (Optional)
`
`951
`
`Ey = Eyo sin (kx - wt)
`
`29-14
`
`ubstitute this solution into either Equation '29-8 or 29-9, we can see
`If wethse magnetic field B2 is in phase with the electric field Ey. From Equation
`111at
`9 8 we have
`2 - '
`
`= - kE 0 cos (kx - wt)
`Y
`
`aB2
`-
`at
`
`aEy
`= - -
`ax
`
`Solving for B2 gives
`
`.
`k
`B2 = -Eyo sm (kx - wt)
`w
`= B2 0 sin (kx - wt)
`
`29-15
`
`where
`
`_ Eyo
`_ k
`Bzo - -Eyo - - -
`c
`w
`and c = w/k is the velocity of the wave.* Since the electric and magnetic
`fields oscillate in phase with the same frequency, we have the general result
`that the magnitude of the electric field is c times the magnitude of the mag(cid:173)
`netic field for an electromagnetic wave:
`
`E = cB
`
`29-16
`
`Suppose the electric field vector E is confined to the y direction, as ex(cid:173)
`emplified by Equation 29-14. Then £2 = 0, and, according to Equation 29-11,
`dBy/dt = 0. Thus, if E is in the y direction, then the time-varying part (the
`only part we are interested in) of B is in the z direction, as shown in Figure
`29-5. Such a wave is said to be linearly polarized, because if we plot E (or B)
`as a function of time in any plane perpendicular to the x axis, we obtain a
`straight line.
`We see that Maxwell's equations imply wave equations 29-13a and b for
`the electric and magnetic fields; and that if Ey varies harmonically, as in
`Equation 29-14, the magnetic field B2 is in phase with Ey and has an ampli(cid:173)
`tude related to the amplitude of Ey by Equation 29-16. The electric and mag(cid:173)
`netic fields are perpendicular to each other and to the direction of the wave
`propagation, as shown in Figure 29-5. In general, the direction of propaga(cid:173)
`tion of an electromagnetic wave is the direction of the cross product E x B.
`
`'In obtaining Equation 29-15 by the integration of the previous equation, an arbitrary constant of
`integration arises. We have omitted this constant magnetic field from Equation 29-15 because it
`plays no part in the electromagnetic waves we are interested in. Note that if any constant
`electric field is added to Equation 29-14, the new electric field still sa tisfies the wave equation.
`
`B
`
`Figure 29-5 The electric- and
`magnetic-field vectors in a plane(cid:173)
`polarized electromagnetic wave.
`The fields are in phase, perpen(cid:173)
`dicular to each other, and perpen(cid:173)
`dicular to the direction of propa(cid:173)
`gation of the wave.
`
`10
`
`Ve
`
`-9
`Id
`ac(cid:173)
`e Id
`
`11
`
`of
`
`-12
`
`e x
`
`tric
`gu(cid:173)
`ain
`the
`; of
`d z
`ind
`in
`the
`
`ral(cid:173)
`md
`her
`
`I3a
`
`l3b
`
`ant
`nc-
`
`MATERION
`PGR2019-00017
`EX. 2002-0012
`
`

`

`952
`
`Chapter 29 Maxwell's Equations and Electromagnetic Waves
`
`A radar image of the south coast
`of New Guinea. A radar sys tem
`operates by transmitting radio
`waves toward objects, sensing the
`echo of waves reflected back, and
`determining the distance to the
`object from the intervening time
`interval.
`
`2
`
`In
`th
`ar
`u
`el
`
`Example 29-3
`
`The electric field vector of an electromagnetic wave is given by E(x, t) =
`Ea sin (kx - wt) j + Ea cos (kx - wt) k. (a) Find the corresponding mag(cid:173)
`netic field. (b) Compute E·B and Ex B.
`
`(a) We can use either Equation 29-11 or 29-12 to find By. From Equa(cid:173)
`tion 29-11, we obtain
`
`aBy
`-
`at
`
`aE 2
`= -
`ax
`
`a
`= -
`ax
`
`.
`[Ea cos (kx - wt)] = - kE0 sm (kx - wt)
`
`Then, neglecting the arbitrary constant of integration, we obtain
`By = [kEo cos (kx - wt)]( - 1/ w) = - B0 cos (kx - wt)
`where B0 = kE0/w = E0/c. We can find B2 from either Equation 29-8 or
`29-9. Using Equation 29-8, we obtain
`.
`a Ev
`a
`t)
`= - - · = - - [E0 sm (kx - wt)] = - kE0 cos (kx - w
`ax
`ax
`
`aB 2
`-
`at
`
`and
`
`B2 = [- kE0 sin (kx - wt)]( - 1/ w) = Bo sin (kx - wt)
`where again B0 = kE0/ w = E0/ c. The magn etic field is thus given by
`B(x, t) = - B0 cos (kx - wt) j + Bo sin (lex - wt) k.
`This type of electromagnetic wave is said to be circularly polarize.d. B~~
`E and B are constant in magnitude, as can be seen by co~pu~~g(kx
`or B·B. For example E·E = E2 + E2 = E2
`1 ne
`0 sin2 (kx - wt)+ Ea co
`the pa
`.
`.
`Y
`'
`x
`wt) = Ea. At a fixed point x, both vectors rotate in a orcle m
`perpendicular to x with angular frequency w.
`..._
`.
`we ov-
`(b) Computing E·B, withe = kx - wt to simplify the notation,
`tain
`
`E·B = [Ea sin 8 j + E0 cos 8 k]·[ -Bo cos 8 j +Bo sine k]
`. ok
`= - E0B0 sine cos e j-j + EoBo sin 2 8 j·k
`s e sin
`E0B0 cos2 e k·j + EaBo co
`. e == D
`= - E0B0 sin fJ cos e + 0 - 0 + E0B0 cos 8 sin
`
`MATERION
`PGR2019-00017
`EX. 2002-0013
`
`

`

`Section 29-4 Energy and Momentum in an Electromagnetic Wave
`
`953
`
`The electric and magnetic fields are perpendicular to each other as well
`as to the direction of propagation. Computing E x B and using j x j =
`k x k = 0, j x k = i, and k x j = - i, we obtain
`'
`E x B = [ E0 sin () j + E0 cos () k] x [ - B0 cos () j + B0 sin () k]
`() j x k + (- E0B0 cos2 () k x j)
`= EoBo sin2
`() i + EoBo cos2 () i = E0B0i
`= E0B0 sin2
`We note that E x B is in the direction of propagation of the wave.
`
`29-4 Energy and Momentum 1n an
`Electromagnetic Wave
`In our discussion of the transport of energy by a wave of any kind, we saw
`that the intensity of the wave (the average energy per unit time per unit
`area) is equal to the product of the average energy density (the energy per
`unit volume) and the speed of the wave . The energy density stored in the
`electric field is (Equation 21-17)
`
`'°f/e = !EoE2
`and the energy density stored in the magnetic field is (Equation 26-34)
`B2
`T/m = 2µ,o
`
`Jn an electromagnetic wave in free space, E = cB, so we can express the
`magnetic en ergy density in terms of the electric field :
`E2
`B2
`(E/ c)2
`1
`= - EoE2
`T/ = -
`- = - - - = -
`-
`2µ,oc 2
`2
`2µ,o
`2µ,o
`m
`where we have used c2 = 1/ EoJ.Lo· Thus, the electric and magnetic energy
`densities are equal. The total energy density T/ in the wave is the sum of the
`electric and magnetic energy densities. Using E = cB, we may express the
`total energy density in several useful ways:
`
`B2
`1
`1
`T/ = T/e + T/m = -EoE2 + - EoE2 = E0E2 = -
`/Lo
`2
`2
`
`EB
`= - -
`J.LoC
`
`29-17
`
`Energy density in an
`electromagnetic wave
`
`~Section 14-3, we saw that the intensity of a wave (the average power
`owing across an area per unit area) is equal to the product of the average
`ergy density and the speed of the wave. The instantaneous intensity is the
`stantaneous power flowing across an area per unit area. It equals the prod(cid:173)
`ct
`of th
`·
`. e Instantaneous energy density and the wave speed. For an electro-
`gnetic wave in free space, the instantaneous intensity is therefore
`
`B2
`EB
`]ins ta nta n eo us = TJC = CEoE2 = c-
`= -
`/Lo
`/,lo
`Uati
`on 29-18 can be generalized to a vector expression:
`EXB
`S = - -
`J.Lo
`
`29-18
`
`29-19
`
`g-
`
`a-
`
`r
`
`y
`
`h
`£
`
`e
`
`MATERION
`PGR2019-00017
`EX. 2002-0014
`
`

`

`954
`
`Chapter 29 Maxwell's Equations and Electromagnetic Waves
`
`The vector S is called the Poynting vector after its discoverer Sir J h
`ing. Since E and Bare perpendicular in an electromagnetic wave ~h n Poynt.
`tude of S is the instantaneous intensity of the wave and the dir: t' e 111agn.i.
`the direction of propagation of the wave.
`c ion of Sis
`In a harmonic plane wave of angular frequency w and wave
`
`the ;nstanta;:~ :~g:::c fiel:s~·:: :::e(:b~ wt) nuinbe,k,
`Using these results fo~Equation 29-17, we obtain for the .
`.
`instanta
`neous energy density
`•
`
`E0B0 sin2 (kx - wt)
`EB
`77 = -=
`fLoC
`fLoC
`
`When we average the sine squared function over space or time, we obt .
`factor of l The average energy density is therefore
`am a
`1 EoBo
`ErmsBrms
`Y/a v = 2 fLoC =
`fLoC
`where we have used Enns = Eo/V2 and Brms = Bo/V2. The intensity is thus
`
`29-20
`
`I! I
`
`Intensity of an electromagnetic
`wave
`
`1 EoBo
`I = 'T/avC = 2 -;;;; =
`
`Er;rt, 5B
`/J;O
`
`s
`
`I I
`= S av
`
`29-21
`
`We will now show by a simple example that an electromagnetic wave
`carries momentum. In this example, we will calculate the momentum and
`energy absorbed from the wave by a free charged particle. Consider an elec(cid:173)
`tromagnetic wave moving along the x axis with the electric field in the y
`direction and the magnetic field in the z direction that is incident on a sta(cid:173)
`tionary charge on the x axis as shown in Figure 29-6. For simplicity, we will
`neglect the time dependence of the electric and magnetic fields. The particle
`experiences a force qE in they direction and is thus accelerated by the electric
`field . At any time t, the velocity in the y direction is
`
`v =at = qE t
`m
`Y
`
`After a short time ti the charge has acquired a velocity in the y direction
`given by
`
`The energy acquired by the charge after time ti is
`1 mq 2E2t/
`1 q2E2
`1
`2
`K = - mv 2 = -
`= - - - t i
`m2
`2
`2
`2 m
`Y
`
`29-22
`
`q
`
`~·
`
`(b)
`
`Figure 29-6 An electromagnetic
`wave incident on a point charge
`that is initially at rest on the x
`axis. (a) The electric. force qE ac(cid:173)
`celerates the charge in the up(cid:173)
`ward direction. (b) When the
`charge has acquired a velocity v
`upward, the magnetic force
`qv x B accelerates the charge in
`the direction of the wave.
`
`E
`
`B
`
`(a)
`
`MATERION
`PGR2019-00017
`EX. 2002-0015
`
`

`

`Section 29-4 Energy and Momentum in an Electromagnetic Wave
`
`955
`
`the charge is moving in they direction, it experiences a magnetic force
`Whe~ which is in the positive x direction (the direction of propagation of
`qv >< w;ve) when B is in the z direction. The magnetic force at any time t is
`the
`q2 EB
`Fx = qvyB = - - t
`m
`
`I I
`
`e impulse of this force equals the momentum transferred by the wave to
`1
`: particle. Setting the impulse equal to the momentum pXI we obtain
`11
`
`;
`
`f x dt
`
`0
`
`Iii
`Px =
`=If' q2EB t dt = _!_ q2EB ty
`
`o m
`
`2 m
`
`If we use B = E/c, this becomes
`
`. _ 1(1 q2E2 2)
`
`Px - - - - - t i
`c 2 m
`
`29-23
`
`Comparing Equations 29-22 and 29-23, we see that the momentum acquired
`by the charge in the direction of the wave is 1/ c times the energy. Although
`our simple calculation was not rigorous, the results are correct. In general,
`
`The magnitude of the momentum carried by an electromagnetic
`wave is 1/ c times the energy carried by the wave:
`
`u
`p=-
`c
`
`29-24
`
`Momentum and energy in an
`electromagnetic wave
`
`Since the intensity of a wave is the energy per unit time per unit area, the
`intensity divided by c is the momentum carried by the wave per unit time
`per unit area. The momentum carried per unit time is a force . The intensity
`divided by c is thus a force per unit area, which is a pressure. This pressure
`is called radiation pressure Pr:
`
`p = _!__
`r
`c
`
`29-25
`
`We can relate the radiation pressure to the electric or magnetic fields by
`using Equations 29-21 and 29-16:
`
`I
`EoBo
`p = -= - -
`2µ,oc
`c
`r
`
`29-26
`
`Radiation pressure
`
`Consider an electromagnetic wave incident normally on some surface. If
`the surface absorbs energy U from the electromagnetic wave, it also absorbs
`momentum p given by Equation 29-24, and the pressure exerted on the
`surface equals the radiation pressure. If the wave is reflected, the momen(cid:173)
`tum transferred is twice the energy incident on the surface because the wave
`now carries momentum in the opposite direction. The pressure exerted on
`the surface by the wave is then twice the radiat