Module 26: Section 13.1 Module 27 and 28: Section 13.2 through 13.6 Module 29 and 30: Section 13.7 through Table of Contents

Transcription

1 Module 6: Section 13.1 Module 7 and 8: Section 13. though 13.6 Module 9 and 3: Section 13.7 though Table of Contents 13.1 The Displacement Cuent Gauss s Law fo Magnetism Maxwell s Equations Plane Electomagnetic Waves One-Dimensional Wave Equation Standing Electomagnetic Waves Poynting Vecto Example 13.1: Sola Constant Example 13.: Intensity of a Standing Wave Enegy Tanspot Momentum and Radiation Pessue Poduction of Electomagnetic Waves Electic Dipole Radiation 1 Animation Electic Dipole Radiation Animation Radiation Fom a Quate-Wave Antenna Animation Plane Waves (link) Sinusoidal Electomagnetic Wave Summay Appendix: Reflection of Electomagnetic Waves at Conducting Sufaces Poblem-Solving Stategy: Taveling Electomagnetic Waves Solved Poblems Plane Electomagnetic Wave One-Dimensional Wave Equation These notes ae excepted Intoduction to Electicity and Magnetism by Sen-Ben Liao, Pete Doumashkin, and John Belche, Copyight 4, ISBN

2 Poynting Vecto of a Chaging Capacito Poynting Vecto of a Conducto Conceptual Questions Additional Poblems Sola Sailing Reflections of Tue Love Coaxial Cable and Powe Flow Supeposition of Electomagnetic Waves Sinusoidal Electomagnetic Wave Radiation Pessue of Electomagnetic Wave Enegy of Electomagnetic Waves Wave Equation Electomagnetic Plane Wave Sinusoidal Electomagnetic Wave

3 Maxwell s Equations and Electomagnetic Waves 13.1 The Displacement Cuent In Chapte 9, we leaned that if a cuent-caying wie possesses cetain symmety, the magnetic field can be obtained by using Ampee s law: B d s = μ I enc (13.1.1) The equation states that the line integal of a magnetic field aound an abitay closed loop is equal to μ I, whee I is the conduction cuent passing though the suface enc enc bound by the closed path. In addition, we also leaned in Chapte 1 that, as a consequence of the Faaday s law of induction, a changing magnetic field can poduce an electic field, accoding to d E d s = dt B da (13.1.) One might then wonde whethe o not the convese could be tue, namely, a changing electic field poduces a magnetic field. If so, then the ight-hand side of Eq. (13.1.1) will have to be modified to eflect such symmety between E and B. To see how magnetic fields can be ceated by a time-vaying electic field, conside a capacito which is being chaged. Duing the chaging pocess, the electic field stength inceases with time as moe chage is accumulated on the plates. The conduction cuent that caies the chages also poduces a magnetic field. In ode to apply Ampee s law to calculate this field, let us choose cuve C shown in Figue to be the Ampeian loop. S Figue Sufaces S 1 and S bound by cuve C. 13-3

4 If the suface bounded by the path is the flat suface S 1, then the enclosed cuent is I enc = I. On the othe hand, if we choose S to be the suface bounded by the cuve, then I enc = since no cuent passes though S. Thus, we see that thee exists an ambiguity in choosing the appopiate suface bounded by the cuve C. Maxwell showed that the ambiguity can be esolved by adding to the ight-hand side of the Ampee s law an exta tem dφ I d = ε E (13.1.3) dt which he called the displacement cuent. The tem involves a change in electic flux. The genealized Ampee s (o the Ampee-Maxwell) law now eads B dφ d s = μ I + μ ε E = μ (I + I d ) (13.1.4) dt The oigin of the displacement cuent can be undestood as follows: Figue Displacement though S In Figue 13.1., the electic flux which passes though S is given by Φ = d = EA = Q E E A (13.1.5) ε S whee A is the aea of the capacito plates. Fom Eq. (13.1.3), we eadily see that the displacement cuent I d is elated to the ate of incease of chage on the plate by dφ I d = ε E dq = dt dt (13.1.6) Howeve, the ight-hand-side of the expession, dq / dt, is simply equal to the conduction cuent, I. Thus, we conclude that the conduction cuent that passes though S 1 is 13-4

5 pecisely equal to the displacement cuent that passes though S, namely I = I d. With the Ampee-Maxwell law, the ambiguity in choosing the suface bound by the Ampeian loop is emoved. 13. Gauss s Law fo Magnetism We have seen that Gauss s law fo electostatics states that the electic flux though a closed suface is popotional to the chage enclosed (Figue 13..1a). The electic field lines oiginate fom the positive chage (souce) and teminate at the negative chage (sink). One would then be tempted to wite down the magnetic equivalent as Q m Φ = d = (13..1) B B A μ S whee Q m is the magnetic chage (monopole) enclosed by the Gaussian suface. Howeve, despite intense seach effot, no isolated magnetic monopole has eve been obseved. Hence, Q m = and Gauss s law fo magnetism becomes Φ = d A = (13..) B B S Figue Gauss s law fo (a) electostatics, and (b) magnetism. This implies that the numbe of magnetic field lines enteing a closed suface is equal to the numbe of field lines leaving the suface. That is, thee is no souce o sink. In addition, the lines must be continuous with no stating o end points. In fact, as shown in Figue 13..1(b) fo a ba magnet, the field lines that emanate fom the noth pole to the south pole outside the magnet etun within the magnet and fom a closed loop Maxwell s Equations We now have fou equations which fom the foundation of electomagnetic phenomena: 13-5

6 Gauss's law fo E Law Equation Physical Intepetation Q E d A = ε S Electic flux though a closed suface is popotional to the chaged enclosed dφ B Faaday's law E d s Changing magnetic flux poduces an = dt electic field Gauss's law fo B B d A = The total magnetic flux though a closed suface is zeo S dφ E Ampee Maxwell law B d s Electic cuent and changing electic = μ I + μ ε dt flux poduces a magnetic field Collectively they ae known as Maxwell s equations. The above equations may also be witten in diffeential foms as E = ε B E = t (13.3.1) B = E B = μ J + με t whee ρ and J ae the fee chage and the conduction cuent densities, espectively. In the absence of souces wheeq =, I =, the above equations become ρ E A d = S d dφ B E s = dt B A d = S dφ B d s = με dt E (13.3.) An impotant consequence of Maxwell s equations, as we shall see below, is the pediction of the existence of electomagnetic waves that tavel with speed of light c =1/ μ ε. The eason is due to the fact that a changing electic field poduces a magnetic field and vice vesa, and the coupling between the two fields leads to the geneation of electomagnetic waves. The pediction was confimed by H. Hetz in

7 13.4 Plane Electomagnetic Waves To examine the popeties of the electomagnetic waves, let s conside fo simplicity an electomagnetic wave popagating in the + x -diection, with the electic field E pointing in the +y-diection and the magnetic field B in the +z-diection, as shown in Figue below. Figue A plane electomagnetic wave What we have hee is an example of a plane wave since at any instant both E and B ae unifom ove any plane pependicula to the diection of popagation. In addition, the wave is tansvese because both fields ae pependicula to the diection of popagation, which points in the diection of the coss poduct E B. Using Maxwell s equations, we may obtain the elationship between the magnitudes of the fields. To see this, conside a ectangula loop which lies in the xy plane, with the left side of the loop at x and the ight at x +Δx. The bottom side of the loop is located at y, and the top side of the loop is located at y +Δy, as shown in Figue Let the unit vecto nomal to the loop be in the positive z-diection, nˆ = k ˆ. Using Faaday s law Figue Spatial vaiation of the electic field E d E d s B da dt = (13.4.1) 13-7

8 the left-hand-side can be witten as E E d s y = E y (x + Δx) Δy E y ( x ) Δy = [E y (x + Δx) E y ( x )] Δy = (Δ x Δ y) (13.4.) x whee we have made the expansion Ey Ey ( x+ Δx) = Ey ( x ) + Δx + K (13.4.3) x On the othe hand, the ate of change of magnetic flux on the ight-hand-side is given by d dt B A B d = t z (Δ Δ x y) (13.4.4) Equating the two expessions and dividing though by the aea Δx Δy yields E y = B z x t (13.4.5) The second condition on the elationship between the electic and magnetic fields may be deduced by using the Ampee-Maxwell equation: d B d s = με E da (13.4.6) dt Conside a ectangula loop in the xz plane depicted in Figue , with a unit nomal nˆ = ĵ. Figue Spatial vaiation of the magnetic field B The line integal of the magnetic field is 13-8

9 B d s = B ( x )Δz B ( x +Δ x) Δ z = [ B ( x ) B ( x +Δ x)] Δz z z z z = B z ( Δ Δ ) x x z (13.4.7) On the othe hand, the time deivative of the electic flux is d E E A= με y με d ( ΔxΔ z ) (13.4.8) dt t Equating the two equations and dividing by ΔxΔz, we have B E z y = με (13.4.9) x t The esult indicates that a time-vaying electic field is geneated by a spatially vaying magnetic field. Using Eqs. (13.4.4) and (13.4.8), one may veify that both the electic and magnetic fields satisfy the one-dimensional wave equation. To show this, we fist take anothe patial deivative of Eq. (13.4.5) with espect to x, and then anothe patial deivative of Eq. (13.4.9) with espect to t: E y z z = B = B = E y E με = με x x t t x t t t y (13.4.1) noting the intechangeability of the patial diffeentiations: x Bz Bz = ( ) t t x Similaly, taking anothe patial deivative of Eq. (13.4.9) with espect to x yields, and then anothe patial deivative of Eq. (13.4.5) with espect to t gives B z = με E y E y B = με = με z B = με z (13.4.1) x x t t x t t t The esults may be summaized as: E x με (, t) y = ( ) x t B ( x, t) z 13-9

10 Recall that the geneal fom of a one-dimensional wave equation is given by 1 ψ ( xt, ) = ( ) x v t whee v is the speed of popagation and ψ ( x, t ) is the wave function, we see clealy that both E y and B z satisfy the wave equation and popagate with the speed of light: v = = = m/s = c ( ) με 7 T m/a)( (4 π 1 C /N m ) Thus, we conclude that light is an electomagnetic wave. The spectum of electomagnetic waves is shown in Figue Figue Electomagnetic spectum One-Dimensional Wave Equation It is staightfowad to veify that any function of the fom ψ (x ± vt) satisfies the onedimensional wave equation shown in Eq. ( ). The poof poceeds as follows: Let x = x± vt which yields x / x =1 and x / t patial deivatives with espect to x ae =± v. Using chain ule, the fist two ψ ( x ) ψ x ψ = = x x x x ( ) 13-1

11 Similaly, the patial deivatives in t ae given by ψ ψ ψ x ψ x = x x = x x = x ( ) ψ ψ x ψ = =± v ( ) t x t x ψ ψ ψ x ψ = t ±v = ± v t x x t = v ( ) x Compaing Eq. ( ) with Eq. ( ), we have ψ ψ 1 ψ = = (13.4.) x ' x v t which shows that ψ (x ± vt) satisfies the one-dimensional wave equation. The wave equation is an example of a linea diffeential equation, which means that if ψ 1 ( x, t ) and ψ ( x, t ) ae solutions to the wave equation, then ψ 1 ( x, t ) ±ψ ( x, t ) is also a solution. The implication is that electomagnetic waves obey the supeposition pinciple. One possible solution to the wave equations is E = E ( x, t ) y ĵ = E cos k( x vt)ĵ = E cos( kx ωt)ĵ (13.4.1) B = B z ( xtkˆ, ) = B cos k( x vt)kˆ = B cos( kx ωt)kˆ whee the fields ae sinusoidal, with amplitudes E and B. The angula wave numbe k is elated to the wavelength λ by and the angula fequency ω is π k = (13.4.) λ ω = kv = π v = π f (13.4.3) λ whee f is the linea fequency. In empty space the wave popagates at the speed of light, v= c. The chaacteistic behavio of the sinusoidal electomagnetic wave is illustated in Figue

12 Figue Plane electomagnetic wave popagating in the +x diection. We see that the E and B fields ae always in phase (attaining maxima and minima at the same time.) To obtain the elationship between the field amplitudes E and B, we make use of Eqs. (13.4.4) and (13.4.8). Taking the patial deivatives leads to and E y = ke sin(kx ωt) (13.4.4) x which implies Ek =ωb, o B z = ωb sin(kx ωt) (13.4.5) t E B ω = =c (13.4.6) k Fom Eqs. (13.4.) and (13.4.1), one may easily show that the magnitudes of the fields at any instant ae elated by E = c (13.4.7) B Let us summaize the impotant featues of electomagnetic waves descibed in Eq. (13.4.1): 1. The wave is tansvese since both E and B fields ae pependicula to the diection of popagation, which points in the diection of the coss poduct E B.. The E and B fields ae pependicula to each othe. Theefoe, thei dot poduct vanishes, EB =. 13-1

13 3. The atio of the magnitudes and the amplitudes of the fields is E E ω = = = c B B k 4. The speed of popagation in vacuum is equal to the speed of light, c =1/ με. 5. Electomagnetic waves obey the supeposition pinciple Standing Electomagnetic Waves Let us examine the situation whee thee ae two sinusoidal plane electomagnetic waves, one taveling in the +x-diection, with E ( x, t ) = E cos( k x ω t), B ( x, t ) = B cos( k x ω t) (13.5.1) 1y and the othe taveling in the x-diection, with 1z E y ( xt, ) = E cos( kx +ω t), B z (x, t ) = B cos( k x +ω t) (13.5.) Fo simplicity, we assume that these electomagnetic waves have the same amplitudes ( E 1 = E = E, B 1 = B = B ) and wavelengths ( k 1 = k = k, ω 1 = ω = ω ). Using the supeposition pinciple, the electic field and the magnetic fields can be witten as and E (, ) = E (, ) + E y x t 1y x t y ( x, t ) = E [cos( kx ωt) cos( kx +ωt) ] (13.5.3) B z ( xt, ) = B 1z ( xt, ) + B z ( xt, ) = B [cos( kx ωt) + cos( kx +ωt)] (13.5.4) Using the identities The above expessions may be ewitten as cos( α ± β ) = cos α cos β m sin α sin β (13.5.5) and E (, ) = E y x t [cos kx cos ωt + sin kxsinωt cos kx cos ωt + sin kxsin ωt] = E sin kxsin ωt (13.5.6) 13-13

14 B (x, t ) = B [cos kx cos ωt + sin kx sinωt + cos kx cosωt sin kx sinωt] z = cos kx cos ωt B (13.5.7) One may veify that the total fields E y ( x, t) and B z ( xt, ) still satisfy the wave equation stated in Eq. ( ), even though they no longe have the fom of functions of kx ± ωt. The waves descibed by Eqs. (13.5.6) and (13.5.7) ae standing waves, which do not popagate but simply oscillate in space and time. Let s fist examine the spatial dependence of the fields. Eq. (13.5.6) shows that the total electic field emains zeo at all times if sin kx =, o nπ nπ nλ x = = = n =,1,,K (nodal planes of E) k π / λ, (13.5.8) The planes that contain these points ae called the nodal planes of the electic field. On the othe hand, when sin kx =±1, o 1 π 1 π n 1 x = n + n λ, n,1,, K (anti-nodal planes of = + = + = E) k π / λ 4 (13.5.9) the amplitude of the field is at its maximum E. The planes that contain these points ae the anti-nodal planes of the electic field. Note that in between two nodal planes, thee is an anti-nodal plane, and vice vesa. Fo the magnetic field, the nodal planes must contain points which meets the condition cos kx =. This yields 1 π n 1 x = n + = + λ, n =,1,,K (nodal planes of B) k 4 Similaly, the anti-nodal planes fo Bcontain points that satisfy cos kx =±1, o (13.5.1) nπ nπ nλ x = = =, n = k π / λ,1,,k (anti-nodal planes of B) ( ) Thus, we see that a nodal plane of Ecoesponds to an anti-nodal plane of vesa. B, and vice Fo the time dependence, Eq. (13.5.6) shows that the electic field is zeo eveywhee when sinωt =, o 13-14

15 nπ nπ nt t = = =, n =,1,, K (13.5.1) ω π /T whee T =1/ f = π / ωis the peiod. Howeve, this is pecisely the maximum condition fo the magnetic field. Thus, unlike the taveling electomagnetic wave in which the electic and the magnetic fields ae always in phase, in standing electomagnetic waves, the two fields ae 9 out of phase. Standing electomagnetic waves can be fomed by confining the electomagnetic waves within two pefectly eflecting conductos, as shown in Figue Figue Fomation of standing electomagnetic waves using two pefectly eflecting conductos Poynting Vecto In Chaptes 5 and 11 we had seen that electic and magnetic fields stoe enegy. Thus, enegy can also be caied by the electomagnetic waves which consist of both fields. Conside a plane electomagnetic wave passing though a small volume element of aea A and thickness dx, as shown in Figue Figue Electomagnetic wave passing though a volume element The total enegy in the volume element is given by 13-15

16 du = uadx = (u E + u B )Adx = 1 ε E + B Adx (13.6.1) μ whee u E = 1 ε E, B u B = μ (13.6.) ae the enegy densities associated with the electic and magnetic fields. Since the electomagnetic wave popagates with the speed of light c, the amount of time it takes fo the wave to move though the volume element is dt = dx / c. Thus, one may obtain the ate of change of enegy pe unit aea, denoted with the symbol S, as S = du = c ε E + B (13.6.3) Adt μ The SI unit of S is W/m. Noting that E = cb and c = 1/ με, the above expession may be ewitten as S = c ε E + B = cb = cε E = EB (13.6.4) μ μ μ In geneal, the ate of the enegy flow pe unit aea may be descibed by the Poynting vecto S (afte the Bitish physicist John Poynting), which is defined as 1 S = E B (13.6.5) μ with S pointing in the diection of popagation. Since the fields E and B ae pependicula, we may eadily veify that the magnitude of S is E B EB S = = = S (13.6.6) μ μ As an example, suppose the electic component of the plane electomagnetic wave is E = E cos(kx ωt)ĵ. The coesponding magnetic component is B = B cos(kx ωt)kˆ, and the diection of popagation is +x. The Poynting vecto can be obtained as S = 1 (E cos(kx ωt)ĵ) (B cos( kx ωt)kˆ )= EB cos ( kx ωt)î (13.6.7) μ μ 13-16

17 Figue Poynting vecto fo a plane wave As expected, S points in the diection of wave popagation (see Figue 13.6.). The intensity of the wave, I, defined as the time aveage of S, is given by EB EB E cb I = S = cos ( kx ωt) = = = (13.6.8) μ cμ μ μ whee we have used 1 cos ( kx ωt) = (13.6.9) To elate intensity to the enegy density, we fist note the equality between the electic and the magnetic enegy densities: B u B = = ( E / c ) μ μ E = = ε E c μ = u E (13.6.1) The aveage total enegy density then becomes u = u E E + u B = ε =ε E = B ( ) 1 = B μ μ Thus, the intensity is elated to the aveage enegy density by Example 13.1: Sola Constant I = S = c u (13.6.1) At the uppe suface of the Eath s atmosphee, the time-aveaged magnitude of the 3 Poynting vecto, S = W m, is efeed to as the sola constant

18 (a) Assuming that the Sun s electomagnetic adiation is a plane sinusoidal wave, what ae the magnitudes of the electic and magnetic fields? (b) What is the total time-aveaged powe adiated by the Sun? The mean Sun-Eath distance is R = m. Solution: (a) The time-aveaged Poynting vecto is elated to the amplitude of the electic field by Thus, the amplitude of the electic field is S = c ε E. E = S cε ( Wm 3 = =1.1 1 V m. 8 1 (3. 1 m s )( C N m ) ) The coesponding amplitude of the magnetic field is E V m 6 B = = = T. 8 c 3. 1 m s Note that the associated magnetic field is less than one-tenth the Eath s magnetic field. (b) The total time aveaged powe adiated by the Sun at the distance R is P = S A = S 4π R = ( W m )4π (1.5 1 m) = W The type of wave discussed in the example above is a spheical wave (Figue a), which oiginates fom a point-like souce. The intensity at a distance fom the souce is P I = S = ( ) 4π which deceases as 1/. On the othe hand, the intensity of a plane wave (Figue b) emains constant and thee is no speading in its enegy

19 Figue (a) a spheical wave, and (b) plane wave. Example 13.: Intensity of a Standing Wave Compute the intensity of the standing electomagnetic wave given by Solution: E ( x, t ) = E y cos kx cos ωt, B ( x, t ) = B sin kx z sin ωt The Poynting vecto fo the standing wave is E B S = μ 1 = (E cos kx μ cos ωt ĵ) ( B sin kx sin ωt kˆ ) EB = 4 (sin kx cos kx sin ωt cos ωt)î μ ( ) EB = (sin kx sin ωt )î μ The time aveage of S is EB S = sin kx μ sin ωt = ( ) The esult is to be expected since the standing wave does not popagate. Altenatively, we may say that the enegy caied by the two waves taveling in the opposite diections to fom the standing wave exactly cancel each othe, with no net enegy tansfe Enegy Tanspot Since the Poynting vecto S epesents the ate of the enegy flow pe unit aea, the ate of change of enegy in a system can be witten as du = S d A ( ) dt 13-19

20 whee da = da nˆ, whee nˆ is a unit vecto in the outwad nomal diection. The above expession allows us to intepet S as the enegy flux density, in analogy to the cuent density J in I = dq = J d A ( ) dt If enegy flows out of the system, then S = S nˆ and du / dt <, showing an oveall decease of enegy in the system. On the othe hand, if enegy flows into the system, then S = S( nˆ ) and du / dt >, indicating an oveall incease of enegy. As an example to elucidate the physical meaning of the above equation, let s conside an inducto made up of a section of a vey long ai-coe solenoid of length l, adius and n tuns pe unit length. Suppose at some instant the cuent is changing at a ate di / dt >. Using Ampee s law, the magnetic field in the solenoid is o B d s = Bl = μ ( ) NI C Thus, the ate of incease of the magnetic field is B = μ ni kˆ ( ) db di = μ n ( ) dt dt Accoding to Faaday s law: ε = d dφ B E s = dt C (13.6.) changing magnetic flux esults in an induced electic field., which is given by o E (π ) = μ n di π dt E = μ n di φˆ (13.6.1) dt 13-

21 The diection of E is clockwise, the same as the induced cuent, as shown in Figue Figue Poynting vecto fo a solenoid with di / dt > The coesponding Poynting vecto can then be obtained as E B 1 μ n di ˆ ˆ μ n I di S = = ˆ μ μ dt φ (μ ni k ) = (13.6.) dt which points adially inwad, i.e., along the ˆ diection. The diections of the fields and the Poynting vecto ae shown in Figue Since the magnetic enegy stoed in the inducto is B 1 U B = (π l ) = μ π n I l (13.6.3) μ the ate of change of U B is whee P = du B = μ π n I l di = I ε dt dt (13.6.4) dφ B db nl π di ε = N = ( ) dt dt π = μ n l dt (13.6.5) is the induced emf. One may eadily veify that this is the same as d = μ ni di di S A (π l ) = μ π n I l dt dt (13.6.6) Thus, we have 13-1

22 du B = S d A> (13.6.7) dt The enegy in the system is inceased, as expected when di / dt >. On the othe hand, if di / dt <, the enegy of the system would decease, with du B / dt < Momentum and Radiation Pessue The electomagnetic wave tanspots not only enegy but also momentum, and hence can exet a adiation pessue on a suface due to the absoption and eflection of the momentum. Maxwell showed that if the plane electomagnetic wave is completely absobed by a suface, the momentum tansfeed is elated to the enegy absobed by ΔU Δ p = (complete absoption) (13.7.1) c On the othe hand, if the electomagnetic wave is completely eflected by a suface such as a mio, the esult becomes ΔU Δ p = (complete eflection) (13.7.) c Fo the complete absoption case, the aveage adiation pessue (foce pe unit aea) is given by Since the ate of enegy deliveed to the suface is F 1 dp 1 du P = = = (13.7.3) A A dt Ac dt we aive at du dt = S A = IA I P = (complete absoption) (13.7.4) c Similaly, if the adiation is completely eflected, the adiation pessue is twice as geat as the case of complete absoption: I P = (complete eflection) (13.7.5) c 13-

23 13.8 Poduction of Electomagnetic Waves Electomagnetic waves ae poduced when electic chages ae acceleated. In othe wods, a chage must adiate enegy when it undegoes acceleation. Radiation cannot be poduced by stationay chages o steady cuents. Figue depicts the electic field lines poduced by an oscillating chage at some instant. Figue Electic field lines of an oscillating point chage A common way of poducing electomagnetic waves is to apply a sinusoidal voltage souce to an antenna, causing the chages to accumulate nea the tips of the antenna. The effect is to poduce an oscillating electic dipole. The poduction of electic-dipole adiation is depicted in Figue Figue Electic fields poduced by an electic-dipole antenna. At time t = the ends of the ods ae chaged so that the uppe od has a maximum positive chage and the lowe od has an equal amount of negative chage. At this instant the electic field nea the antenna points downwad. The chages then begin to decease. Afte one-fouth peiod, t= T /4, the chages vanish momentaily and the electic field stength is zeo. Subsequently, the polaities of the ods ae evesed with negative chages continuing to accumulate on the uppe od and positive chages on the lowe until t= T /, when the maximum is attained. At this moment, the electic field nea the od points upwad. As the chages continue to oscillate between the ods, electic fields ae poduced and move away with speed of light. The motion of the chages also poduces a cuent which in tun sets up a magnetic field encicling the ods. Howeve, the behavio 13-3

24 of the fields nea the antenna is expected to be vey diffeent fom that fa away fom the antenna. Let us conside a half-wavelength antenna, in which the length of each od is equal to one quate of the wavelength of the emitted adiation. Since chages ae diven to oscillate back and foth between the ods by the altenating voltage, the antenna may be appoximated as an oscillating electic dipole. Figue depicts the electic and the magnetic field lines at the instant the cuent is upwad. Notice that the Poynting vectos at the positions shown ae diected outwad. Figue Electic and magnetic field lines poduced by an electic-dipole antenna. In geneal, the adiation patten poduced is vey complex. Howeve, at a distance which is much geate than the dimensions of the system and the wavelength of the adiation, the fields exhibit a vey diffeent behavio. In this fa egion, the adiation is caused by the continuous induction of a magnetic field due to a time-vaying electic field and vice vesa. Both fields oscillate in phase and vay in amplitude as 1/. The intensity of the vaiation can be shown to vay as sin θ /, whee θ is the angle measued fom the axis of the antenna. The angula dependence of the intensity I ( θ ) is shown in Figue Fom the figue, we see that the intensity is a maximum in a plane which passes though the midpoint of the antenna and is pependicula to it. Figue Angula dependence of the adiation intensity. 13-4

25 Electic Dipole Radiation 1 Animation Conside an electic dipole whose dipole moment vaies in time accoding to 1 p () t = p 1+ cos 1 π t kˆ (13.8.1) Figue shows one fame of an animation of these fields. Close to the dipole, the field line motion and thus the Poynting vecto is fist outwad and then inwad, coesponding to enegy flow outwad as the quasi-static dipola electic field enegy is being built up, and enegy flow inwad as the quasi-static dipole electic field enegy is being destoyed. T Figue Radiation fom an electic dipole whose dipole moment vaies by 1% (link) Even though the enegy flow diection changes sign in these egions, thee is still a small time-aveaged enegy flow outwad. This small enegy flow outwad epesents the small amount of enegy adiated away to infinity. Outside of the point at which the oute field lines detach fom the dipole and move off to infinity, the velocity of the field lines, and thus the diection of the electomagnetic enegy flow, is always outwad. This is the egion dominated by adiation fields, which consistently cay enegy outwad to infinity Electic Dipole Radiation Animation Figue shows one fame of an animation of an electic dipole chaacteized by π t p () t = p cos kˆ (13.8.) T The equation shows that the diection of the dipole moment vaies between +kˆ and kˆ. 13-5

26 Figue Radiation fom an electic dipole whose dipole moment completely eveses with time (link) Radiation Fom a Quate-Wave Antenna Animation Figue (a) shows the adiation patten at one instant of time fom a quate-wave antenna. Figue (b) shows this adiation patten in a plane ove the full peiod of the adiation. A quate-wave antenna poduces adiation whose wavelength is twice the tip to tip length of the antenna. This is evident in the animation of Figue (b). Figue Radiation patten fom a quate-wave antenna: (a) The azimuthal patten at one instant of time (link), and (b) the adiation patten in one plane ove the full peiod (link) Plane Waves (link) We have seen that electomagnetic plane waves popagate in empty space at the speed of light. Below we demonstate how one would ceate such waves in a paticulaly simple plana geomety. Although physically this is not paticulaly applicable to the eal wold, it is easonably easy to teat, and we can see diectly how electomagnetic plane waves ae geneated, why it takes wok to make them, and how much enegy they cay away with them. 13-6

27 To make an electomagnetic plane wave, we do much the same thing we do when we make waves on a sting. We gab the sting somewhee and shake it, and theeby geneate a wave on the sting. We do wok against the tension in the sting when we shake it, and that wok is caied off as an enegy flux in the wave. Electomagnetic waves ae much the same poposition. The electic field line seves as the sting. As we will see below, thee is a tension associated with an electic field line, in that when we shake it (ty to displace it fom its initial position), thee is a estoing foce that esists the shake, and a wave popagates along the field line as a esult of the shake. To undestand in detail what happens in this pocess will involve using most of the electomagnetism we have leaned thus fa, fom Gauss's law to Ampee's law plus the easonable assumption that electomagnetic infomation popagates at speed c in a vacuum. How do we shake an electic field line, and what do we gab on to? What we do is shake the electic chages that the field lines ae attached to. Afte all, it is these chages that poduce the electic field, and in a vey eal sense the electic field is "ooted" in the electic chages that poduce them. Knowing this, and assuming that in a vacuum, electomagnetic signals popagate at the speed of light, we can petty much puzzle out how to make a plane electomagnetic wave by shaking chages. Let's fist figue out how to make a kink in an electic field line, and then we'll go on to make sinusoidal waves. Suppose we have an infinite sheet of chage located in the yz-plane, initially at est, with suface chage density σ, as shown in Figue Figue Electic field due to an infinite sheet with chage density σ. Fom Gauss's law discussed in Chapte 4, we know that this suface chage will give ise to a static electic field E : +(σ ε ) î, x > E = (σ ε ) î, x < (13.8.3) Now, at t =, we gab the sheet of chage and stat pulling it downwad with constant velocity v = v ˆ j. Let's examine how things will then appea at a late time t = T. In 13-7

28 paticula, befoe the sheet stats moving, let's look at the field line that goes though y = fo t <, as shown in Figue (a). Figue Electic field lines (a) though y = at t <, and (b) at t= T (link) The foot of this electic field line, that is, whee it is anchoed, is ooted in the electic chage that geneates it, and that foot must move downwad with the sheet of chage, at the same speed as the chages move downwad. Thus the foot of ou electic field line, which was initially at y = at t =, will have moved a distance y = vt down the y- axis at time t = T. We have assumed that the infomation that this field line is being dagged downwad will popagate outwad fom x = at the speed of light c. Thus the potion of ou field line located a distance x > ct along the x-axis fom the oigin doesn't know the chages ae moving, and thus has not yet begun to move downwad. Ou field line theefoe must appea at time t = T as shown in Figue (b). Nothing has happened outside of x > ct ; the foot of the field line at x = is a distance y = vt down the y-axis, and we have guessed about what the field line must look like fo < x < ct by simply connecting the two positions on the field line that we know about at time T ( x = and x = ct ) by a staight line. This is exactly the guess we would make if we wee dealing with a sting instead of an electic field. This is a easonable thing to do, and it tuns out to be the ight guess. What we have done by pulling down on the chaged sheet is to geneate a petubation in the electic field, E 1 in addition to the static field E. Thus, the total field E fo < x < ct is E= E + E 1 (13.8.4) As shown in Figue (b), the field vecto E must be paallel to the line connecting the foot of the field line and the position of the field line at x = ct. This implies 13-8

29 tan θ = E 1 = vt = v (13.8.5) E ct c whee E 1 = E 1 and E = E ae the magnitudes of the fields, andθ is the angle with the x-axis. Using Eq. (13.8.5), the petubation field can be witten as E 1 = v E ĵ = c v σ ε c ĵ (13.8.6) whee we have used E = σ ε. We have geneated an electic field petubation, and this expession tells us how lage the petubation field E 1 is fo a given speed of the sheet of chage, v. This explains why the electic field line has a tension associated with it, just as a sting does. The diection of E 1 is such that the foces it exets on the chages in the sheet esist the motion of the sheet. That is, thee is an upwad electic foce on the sheet when we ty to move it downwad. Fo an infinitesimal aea da of the sheet containing chage dq = σ da, the upwad tension associated with the electic field is df = dq E e 1 = (σ da ) vσ ĵ = vσ da ĵ (13.8.7) ε c ε c Theefoe, to ovecome the tension, the extenal agent must apply an equal but opposite (downwad) foce vσ da df = df = ext e ĵ (13.8.8) ε c Since the amount of wok done is dw ext = F d ext s, the wok done pe unit time pe unit aea by the extenal agent is dw ext df ext ds vσ ˆ ˆ v σ v j = dadt da dt ε c ε c = = j ( ) (13.8.9) What else has happened in this pocess of moving the chaged sheet down? Well, once the chaged sheet is in motion, we have ceated a sheet of cuent with suface cuent density (cuent pe unit length) K = σv ĵ. Fom Ampee's law, we know that a magnetic field has been ceated, in addition to E 1. The cuent sheet will poduce a magnetic field (see Example 9.4) 13-9

30 +(μ σv ) kˆ, x > B 1 = (μ σv ) kˆ, x < (13.8.1) This magnetic field changes diection as we move fom negative to positive values of x, (acoss the cuent sheet). The configuation of the field due to a downwad cuent is shown in Figue fo x < ct. Again, the infomation that the chaged sheet has stated moving, poducing a cuent sheet and associated magnetic field, can only popagate outwad fom x = at the speed of light c. Theefoe the magnetic field is still zeo, B= fo x > ct. Note that E 1 = vσ /ε c 1 = = c ( ) B 1 μσv / cμε Figue Magnetic field at t = T (link) The magnetic field B 1 geneated by the cuent sheet is pependicula to E 1 with a magnitude B 1 = E 1 / c, as expected fo a tansvese electomagnetic wave. Now, let s discuss the enegy caied away by these petubation fields. The enegy flux associated with an electomagnetic field is given by the Poynting vecto S. Fo x >, the enegy flowing to the ight is 1 1 vσ μσv v σ S = E B = 1 1 ĵ kˆ = î (13.8.1) μ μ ε c 4ε c This is only half of the wok we do pe unit time pe unit aea to pull the sheet down, as given by Eq. (13.8.9). Since the fields on the left cay exactly the same amount of enegy flux to the left, (the magnetic field B 1 changes diection acoss the plane x = wheeas the electic field E 1 does not, so the Poynting flux also changes acoss x = ). So the total enegy flux caied off by the petubation electic and magnetic fields we 13-3

31 have geneated is exactly equal to the ate of wok pe unit aea to pull the chaged sheet down against the tension in the electic field. Thus we have geneated petubation electomagnetic fields that cay off enegy at exactly the ate that it takes to ceate them. Whee does the enegy caied off by the electomagnetic wave come fom? The extenal agent who oiginally shook the chage to poduce the wave had to do wok against the petubation electic field the shaking poduces, and that agent is the ultimate souce of the enegy caied by the wave. An exactly analogous situation exists when one asks whee the enegy caied by a wave on a sting comes fom. The agent who oiginally shook the sting to poduce the wave had to do wok to shake it against the estoing tension in the sting, and that agent is the ultimate souce of enegy caied by a wave on a sting Sinusoidal Electomagnetic Wave How about geneating a sinusoidal wave with angula fequency ω? To do this, instead of pulling the chage sheet down at constant speed, we just shake it up and down with a velocity v () t = v cos ωt ĵ. The oscillating sheet of chage will geneate fields which ae given by: fo x > and, fo x <, cμ σv x μσv x E 1 = cos ω t ĵ, B 1 = cosω t kˆ ( ) c c cμ σv x μσv x E 1 = cos ω t + ĵ, B 1 = cosω t + kˆ ( ) c c In Eqs. ( ) and ( ) we have chosen the amplitudes of these tems to be the amplitudes of the kink geneated above fo constant speed of the sheet, with E 1 / B 1 = c, but now allowing fo the fact that the speed is vaying sinusoidally in time with fequency ω. But why have we put the (t x/ c) and (t+ x/ c ) in the aguments fo the cosine function in Eqs. ( ) and ( )? Conside fist x >. If we ae sitting at some x > at time t, and ae measuing an electic field thee, the field we ae obseving should not depend on what the cuent sheet is doing at that obsevation time t. Infomation about what the cuent sheet is doing takes a time x / c to popagate out to the obseve at x >. Thus what the obseve at x > sees at time t depends on what the cuent sheet was doing at an ealie time, namely t x/ c. The electic field as a function of time should eflect that time delay due to the finite speed of popagation fom the oigin to some x >, and this is the eason the (t x/ c )appeas in Eq. ( ), and not t itself. Fo x <, the agument is exactly the same, except if x <, t+ x / c is the expession fo the ealie time, and not t x / c. This 13-31

32 is exactly the time-delay effect one gets when one measues waves on a sting. If we ae measuing wave amplitudes on a sting some distance away fom the agent who is shaking the sting to geneate the waves, what we measue at time t depends on what the agent was doing at an ealie time, allowing fo the wave to popagate fom the agent to the obseve. If we note that cosω(t x / c ) = cos(ωt kx ) whee k = ω c is the wave numbe, we see that Eqs. ( ) and ( ) ae pecisely the kinds of plane electomagnetic waves we have studied. Note that we can also easily aange to get id of ou static field E by simply putting a stationay chaged sheet with chage pe unit aea σ at x =. That chaged sheet will cancel out the static field due to the positive sheet of chage, but will not affect the petubation field we have calculated, since the negatively-chaged sheet is not moving. In eality, that is how electomagnetic waves ae geneated--with an oveall neutal medium whee chages of one sign (usually the electons) ae acceleated while an equal numbe of chages of the opposite sign essentially emain at est. Thus an obseve only sees the wave fields, and not the static fields. In the following, we will assume that we have set E to zeo in this way. Figue Electic field geneated by the oscillation of a cuent sheet (link) The electic field geneated by the oscillation of the cuent sheet is shown in Figue , fo the instant when the sheet is moving down and the petubation electic field is up. The magnetic fields, which point into o out of the page, ae also shown. What we have accomplished in the constuction hee, which eally only assumes that the feet of the electic field lines move with the chages, and that infomation popagates at c is to show we can geneate such a wave by shaking a plane of chage sinusoidally. The wave we geneate has electic and magnetic fields pependicula to one anothe, and tansvese to the diection of popagation, with the atio of the electic field magnitude to the magnetic field magnitude equal to the speed of light. Moeove, we see diectly whee the enegy flux S= E B / μ caied off by the wave comes fom. The agent who shakes the chages, and theeby geneates the electomagnetic wave puts the enegy in. If we go to moe complicated geometies, these statements become much moe complicated in detail, but the oveall pictue emains as we have pesented it. 13-3

33 Let us ewite slightly the expessions given in Eqs. ( ) and ( ) fo the fields geneated by ou oscillating chaged sheet, in tems of the cuent pe unit length in the sheet, K( t ) = σ v()ĵ. t Since v () t = v cos ωt ĵ, it follows that K( t ) = σv cos ωt ĵ. Thus, c μ x t E 1 xt / ), 1 xt = î E (, ) 1 (, ) = K(t x c B (, ) ( ) c fo x >, and cμ x t (, ) = K (t + x c B (, ) ˆ E 1 (, ) E 1 xt / ), 1 xt = i ( ) c fo x <. Note that B 1 ( x, t ) eveses diection acoss the cuent sheet, with a jump of K( t ) at the sheet, as it must fom Ampee's law. Any oscillating sheet of cuent must μ geneate the plane electomagnetic waves descibed by these equations, just as any stationay electic chage must geneate a Coulomb electic field. Note: To avoid possible futue confusion, we point out that in a moe advanced electomagnetism couse, you will study the adiation fields geneated by a single oscillating chage, and find that they ae popotional to the acceleation of the chage. This is vey diffeent fom the case hee, whee the adiation fields of ou oscillating sheet of chage ae popotional to the velocity of the chages. Howeve, thee is no contadiction, because when you add up the adiation fields due to all the single chages making up ou sheet, you ecove the same esult we give in Eqs. ( ) and ( ) (see Chapte 3, Section 7, of Feynman, Leighton, and Sands, The Feynman Lectues on Physics, Vol 1, Addison-Wesley, 1963) Summay The Ampee-Maxwell law eads whee B dφ d s = μ I + μ ε E = μ (I + I d ) dt I d = ε dφ E dt is called the displacement cuent. The equation descibes how changing electic flux can induce a magnetic field

34 Gauss s law fo magnetism is Φ = d = B B A S The law states that the magnetic flux though a closed suface must be zeo, and implies the absence of magnetic monopoles. Electomagnetic phenomena ae descibed by the Maxwell s equations: d A = Q dφ E d s = B ε dt d = d s dφ = μ I + μ ε E dt E S B A B S In fee space, the electic and magnetic components of the electomagnetic wave obey a wave equation: E y ( x, t ) με = x t B (, ) z x t The magnitudes and the amplitudes of the electic and magnetic fields in an electomagnetic wave ae elated by E E = ω 1 = = c = m/s B B k με A standing electomagnetic wave does not popagate, but instead the electic and magnetic fields execute simple hamonic motion pependicula to the wouldbe diection of popagation. An example of a standing wave is E ( x, t ) = E y sin kx sin ωt, B ( x, t ) = B cos kx z cos ωt The enegy flow ate of an electomagnetic wave though a closed suface is given by whee du = dt S d A 1 S = E B μ 13-34

35 is the Poynting vecto, and S points in the diection the wave popagates. The intensity of an electomagnetic wave is elated to the aveage enegy density by I = S = c u The momentum tansfeed is elated to the enegy absobed by ΔU c Δ p = ΔU c (complete absoption) (complete eflection) The aveage adiation pessue on a suface by a nomally incident electomagnetic wave is I c P = I c (complete absoption) (complete eflection) 13.1 Appendix: Reflection of Electomagnetic Waves at Conducting Sufaces How does a vey good conducto eflect an electomagnetic wave falling on it? In wods, what happens is the following. The time-vaying electic field of the incoming wave dives an oscillating cuent on the suface of the conducto, following Ohm's law. That oscillating cuent sheet, of necessity, must geneate waves popagating in both diections fom the sheet. One of these waves is the eflected wave. The othe wave cancels out the incoming wave inside the conducto. Let us make this qualitative desciption quantitative. Figue Reflection of electomagnetic waves at conducting suface 13-35

36 Suppose we have an infinite plane wave popagating in the +x-diection, with E = E cos(ωt kx) ĵ, B = B cos(ωt kx)kˆ (13.1.1) as shown in the top potion of Figue We put at the oigin ( x = ) a conducting sheet with width D, which is much smalle than the wavelength of the incoming wave. This conducting sheet will eflect ou incoming wave. How? The electic field of the incoming wave will cause a cuent J = E ρ to flow in the sheet, whee ρ is the esistivity (not to be confused with chage pe unit volume), and is equal to the ecipocal of conductivity σ (not to be confused with chage pe unit aea). Moeove, the diection of J will be in the same diection as the electic field of the incoming wave, as shown in the sketch. Thus ou incoming wave sets up an oscillating sheet of cuent with cuent pe unit length K = J D. As in ou discussion of the geneation of plane electomagnetic waves above, this cuent sheet will also geneate electomagnetic waves, moving both to the ight and to the left (see lowe potion of Figue ) away fom the oscillating sheet of chage. Using Eq. ( ) fo x > the wave geneated by the cuent will be μ E ( x, t ) = c JD cos (ωt kx ) ĵ (13.1.) 1 whee J = J. Fo x <, we will have a simila expession, except that the agument will be (ωt+ kx) (see Figue ). Note the sign of this electic field E 1 at x = ; it is down ( ĵ ) when the sheet of cuent is up (and E is up, +ĵ ), and vice-vesa, just as we saw befoe. Thus, fo x >, the geneated electic field E 1 will always be opposite the diection of the electic field of the incoming wave, and it will tend to cancel out the incoming wave fo x >. Fo a vey good conducto, we have (see next section) K = K = JD = E (13.1.3) cμ so that fo x > we will have E 1 ( x, t ) = E ˆ cos (ωt kx ) j (13.1.4) That is, fo a vey good conducto, the electic field of the wave geneated by the cuent will exactly cancel the electic field of the incoming wave fo x >! And that's what a vey good conducto does. It suppots exactly the amount of cuent pe unit length K = E / cμ needed to cancel out the incoming wave fo x >. Fo x <, this same cuent geneates a eflected wave popagating back in the diection fom which the 13-36

37 oiginal incoming wave came, with the same amplitude as the oiginal incoming wave. This is how a vey good conducto totally eflects electomagnetic waves. Below we shall show that K will in fact appoach the value needed to accomplish this in the limit the esistivity ρ appoaches zeo. In the pocess of eflection, thee is a foce pe unit aea exeted on the conducto. This is just the v B foce due to the cuent J flowing in the pesence of the magnetic field of the incoming wave, o a foce pe unit volume of J B. If we calculate the total foce df acting on a cylindical volume with aea da and length D of the conducto, we find that it is in the +x - diection, with magnitude so that the foce pe unit aea, E BdA df = D da = J B DJB da = (13.1.5) cμ df EB = = S (13.1.6) da cμ c o adiation pessue, is just twice the Poynting flux divided by the speed of light c. We shall show that a pefect conducto will pefectly eflect an incident wave. To appoach the limit of a pefect conducto, we fist conside the finite esistivity case, and then let the esistivity go to zeo. Fo simplicity, we assume that the sheet is thin compaed to a wavelength, so that the entie sheet sees essentially the same electic field. This implies that the cuent density J will be unifom acoss the thickness of the sheet, and outside of the sheet we will see fields appopiate to an equivalent suface cuent K( t ) = DJ( t ). This cuent sheet will geneate additional electomagnetic waves, moving both to the ight and to the left, away fom the oscillating sheet of chage. The total electic field, E( x, t ), will be the sum of the incident electic field and the electic field geneated by the cuent sheet. Using Eqs. ( ) and ( ) above, we obtain the following expessions fo the total electic field: cμ x c ), x > E xt = E (, ) + 1 xt = E (xt, ) K(t (, ) xt E (, ) (13.1.7) cμ E (x, t) K(t + x c ), x < We also have a elation between the cuent density J and E fom the micoscopic fom of Ohm's law: J( t ) = E(, t ) ρ, whee E(, t is the total electic field at the position of 13-37

38 the conducting sheet. Note that it is appopiate to use the total electic field in Ohm's law -- the cuents aise fom the total electic field, iespective of the oigin of that field. Thus, we have D E(, t ) K () t = D J () t = ρ At x =, eithe expession in Eq. (13.1.7) gives (13.1.8) E (,) t = E (,) t + E (,) t = E (,) t cμ 1 K () t = E (, t ) Dcμ E (13.1.9) (, t ) ρ whee we have used Eq. (13.1.9) fo the last step. Solving fo E(, t ), we obtain E E (, t ) = (, t ) 1+ Dcμ ρ (13.1.1) Using the expession above, the suface cuent density in Eq. (13.1.8) can be ewitten as D E K () t = D J () t = (, t ) ρ + Dcμ ( ) In the limit whee ρ (no esistance, a pefect conducto), E(, t ) =, as can be seen fom Eq. (13.1.8), and the suface cuent becomes E (, t ) E B K( t ) = = cos ωt c ĵ = cos ωt cμ μ ĵ (13.1.1) μ In this same limit, the total electic fields can be witten as (E E )cos ωt kx ) =, x > E( xt, ) = ( ĵ E [cos( ωt kx ) cos( ωt + kx )] ĵ = E ˆ sin ωt sin kx j, x < ( ) Similaly, the total magnetic fields in this limit ae given by 13-38

39 fo x >, and B (, ) = B ( xt, ) + B (, ) = B cos (ωt kx)kˆ 1 + î E ( x, t ) 1 xt xt c ( ) = B cos (ωt kx)kˆ B cos (ωt kx)kˆ = B ( x, t ) = B [cos( t k ) + cos( t + kx)] kˆ ω x ω = B ˆ cos ωt cos kxk ( ) fo x <. Thus, fom Eqs. ( ) - ( ) we see that we get no electomagnetic wave fo x >, and standing electomagnetic waves fo x <. Note that at x =, the total electic field vanishes. The cuent pe unit length at x =, B K () t = cos ωt μ ĵ ( ) is just the cuent pe length we need to bing the magnetic field down fom its value at x < to zeo fo x >. You may be petubed by the fact that in the limit of a pefect conducto, the electic field vanishes at x =, since it is the electic field at x = that is diving the cuent thee! In the limit of vey small esistance, the electic field equied to dive any finite cuent is vey small. In the limit whee ρ =, the electic field is zeo, but as we appoach that limit, we can still have a pefectly finite and well detemined value of J = E ρ, as we found by taking this limit in Eqs. (13.1.8) and (13.1.1) above Poblem-Solving Stategy: Taveling Electomagnetic Waves This chapte exploes vaious popeties of the electomagnetic waves. The electic and the magnetic fields of the wave obey the wave equation. Once the functional fom of eithe one of the fields is given, the othe can be detemined fom Maxwell s equations. As an example, let s conside a sinusoidal electomagnetic wave with E (,) zt = E sin( kz ωt)ˆ i The equation above contains the complete infomation about the electomagnetic wave: 1. Diection of wave popagation: The agument of the sine fom in the electic field can be ewitten as (kz ωt) = k( z v t ), which indicates that the wave is popagating in the +z-diection.. Wavelength: The wavelength λ is elated to the wave numbe k by λ = π / k

40 3. Fequency: The fequency of the wave, f, is elated to the angula fequency ω by f = / ω π. 4. Speed of popagation: The speed of the wave is given by π ω ω v = λ f = = k π k In vacuum, the speed of the electomagnetic wave is equal to the speed of light, c. 5. Magnetic field B : The magnetic field B is pependicula to both E which points in the +x-diection, and +kˆ, the unit vecto along the +z-axis, which is the diection of popagation, as we have found. In addition, since the wave popagates in the same diection as the coss poduct E B, we conclude that B must point in the +ydiection (since ˆ i ˆ j = kˆ ). Since B is always in phase with E, the two fields have the same functional fom. Thus, we may wite the magnetic field as B( zt,) = B sin( kz ω t)ˆj whee B is the amplitude. Using Maxwell s equations one may show that B = E ( k /ω) = E / c in vacuum. 6. The Poytning vecto: Using Eq. (13.6.5), the Poynting vecto can be obtained as 1 1 EB sin ( kz ωt) ˆ ˆ S = E B = E ˆ sin(kz ωt) i B sin( kz ωt)j = k μ μ μ 7. Intensity: The intensity of the wave is equal to the aveage of S : EB EB E cb I = S = sin ( kz ωt) = = = μ cμ μ μ 8. Radiation pessue: If the electomagnetic wave is nomally incident on a suface and the adiation is completely eflected, the adiation pessue is I EB E B P = = = = c cμ c μ μ 13-4

41 13.1 Solved Poblems Plane Electomagnetic Wave Suppose the electic field of a plane electomagnetic wave is given by Find the following quantities: (a) The diection of wave popagation. (b) The coesponding magnetic field B. E (,) zt = E cos (kz ωt)i ˆ (13.1.1) Solutions: (a) By witing the agument of the cosine function as kz ωt = k( z ct ) whee ω = ck, we see that the wave is taveling in the + z diection. (b) The diection of popagation of the electomagnetic waves coincides with the diection of the Poynting vecto which is given by S= E B / μ. In addition, E and B ae pependicula to each othe. Theefoe, if E = E( zt,) î and S = Skˆ, then B = B( zt,) ĵ. That is, B points in the +y-diection. Since E and B ae in phase with each othe, one may wite To find the magnitude of B, we make use of Faaday s law: B (,) zt = B cos( kz ωt)ĵ (13.1.) which implies dφ B E d s = (13.1.3) dt E x = B y z t (13.1.4) Fom the above equations, we obtain o E k sin( kz ω t ) = B ω sin( kz ωt) (13.1.5) E B ω = = c (13.1.6) k 13-41

42 Thus, the magnetic field is given by B (,) zt = (E / c )cos( kz ωt) ˆ j (13.1.7) One-Dimensional Wave Equation Veify that, fo ω = kc, Ext (, ) = E cos B( xt, ) = B cos (kx ωt) (kx ωt) (13.1.8) satisfy the one-dimensional wave equation: 1 E( xt, ) x t = (13.1.9) c B ( xt, ) Solution: Diffeentiating E = E cos (kx ωt) with espect to x gives E E = ke sin (kx ωt), = k E cos (kx ωt) (13.1.1) x x Similaly, diffeentiating E with espect to t yields E = ωe sin (kx ωt), t E = ω E cos (kx ωt) ( ) t Thus, E 1 x c E = k + ω E t c cos (kx ωt) = (13.1.1) whee we have made used of the elation ω = kc. One may follow a simila pocedue to veify the magnetic field. 13-4

43 Poynting Vecto of a Chaging Capacito A paallel-plate capacito with cicula plates of adius R and sepaated by a distance h is chaged though a staight wie caying cuent I, as shown in the Figue : Figue Paallel plate capacito (a) Show that as the capacito is being chaged, the Poynting vecto S points adially inwad towad the cente of the capacito. (b) By integating S ove the cylindical bounday, show that the ate at which enegy entes the capacito is equal to the ate at which electostatic enegy is being stoed in the electic field. Solutions: (a) Let the axis of the cicula plates be the z-axis, with cuent flowing in the +zdiection. Suppose at some instant the amount of chage accumulated on the positive plate is +Q. The electic field is σ Q E = kˆ = kˆ ( ) ε π R ε Accoding to the Ampee-Maxwell s equation, a magnetic field is induced by changing electic flux: d B d s = μ I + μ ε E da dt S enc Figue

44 Fom the cylindical symmety of the system, we see that the magnetic field will be cicula, centeed on the z-axis, i.e., B=Bφˆ (see Figue ) Conside a cicula path of adius < R between the plates. Using the above fomula, we obtain o d Q ( ) μ dq B π = + με π = ( ) dt πr ε R dt The Poynting S vecto can then be witten as B = μ dq φˆ ( ) π R dt 1 1 Q = kˆ μ dq S = E B ˆ φ μ μ πr ε π R dt Q dq = ˆ R 4 π ε dt ( ) Note that fo dq / dt > S points in the ˆ diection, o adially inwad towad the cente of the capacito. (b) The enegy pe unit volume caied by the electic field is u E = ε E /. The total enegy stoed in the electic field then becomes = Q h U = u V π R h = 1 ε Q π R E E h R π ε πr ε = ε E ( ) ( ) Diffeentiating the above expession with espect to t, we obtain the ate at which this enegy is being stoed: du E d Q h = = Qh dq dt dt πr ε πr ε dt ( ) On the othe hand, the ate at which enegy flows into the capacito though the cylinde at = R can be obtained by integating S ove the suface aea: 13-44

45 S d A= SA R Q dq Qh (π Rh ) dq = 4 = π εr o dt επ R dt ( ) which is equal to the ate at which enegy stoed in the electic field is changing Poynting Vecto of a Conducto A cylindical conducto of adius a and conductivity σ caies a steady cuent I which is distibuted unifomly ove its coss-section, as shown in Figue Figue (a) Compute the electic field E inside the conducto. (b) Compute the magnetic field B just outside the conducto. (c) Compute the Poynting vecto S at the suface of the conducto. In which diection does S point? (d) By integating S ove the suface aea of the conducto, show that the ate at which electomagnetic enegy entes the suface of the conducto is equal to the ate at which enegy is dissipated. Solutions: (a) Let the diection of the cuent be along the z-axis. The electic field is given by whee R is the esistance and l is the length of the conducto. (b) The magnetic field can be computed using Ampee s law: E= J = I kˆ (13.1.) σ σπ a 13-45

46 B d s = μ I enc (13.1.1) Choosing the Ampeian loop to be a cicle of adius, we have B(π ) = μ I, o μ I B = φˆ (13.1.) π (c) The Poynting vecto on the suface of the wie ( = a) is E B 1 I μ I S = = kˆ φˆ = I 3 ˆ (13.1.3) μ μ σπ a π a π σ a Notice that S points adially inwad towad the cente of the conducto. (d) The ate at which electomagnetic enegy flows into the conducto is given by du I I l P = = S d A= 3 π al = (13.1.4) dt S σπ a σπ a Howeve, since the conductivity σ is elated to the esistance R by The above expession becomes σ = 1 l l = = (13.1.5) ρ AR π a R P= I R (13.1.6) which is equal to the ate of enegy dissipation in a esisto with esistance R Conceptual Questions 1. In the Ampee-Maxwell s equation, is it possible that both a conduction cuent and a displacement ae non-vanishing?. What causes electomagnetic adiation? 3. When you touch the indoo antenna on a TV, the eception usually impoves. Why? 13-46

47 4. Explain why the eception fo cellula phones often becomes poo when used inside a steel-famed building. 5. Compae sound waves with electomagnetic waves. 6. Can paallel electic and magnetic fields make up an electomagnetic wave in vacuum? 7. What happens to the intensity of an electomagnetic wave if the amplitude of the electic field is halved? Doubled? Additional Poblems Sola Sailing It has been poposed that a spaceship might be popelled in the sola system by adiation pessue, using a lage sail made of foil. How lage must the sail be if the adiation foce is to be equal in magnitude to the Sun's gavitational attaction? Assume that the mass of the ship and sail is 165 kg, that the sail is pefectly eflecting, and that the sail is oiented at ight angles to the Sun s ays. Does you answe depend on whee in the sola system the spaceship is located? Reflections of Tue Love (a) A light bulb puts out 1 W of electomagnetic adiation. What is the time-aveage intensity of adiation fom this light bulb at a distance of one mete fom the bulb? What ae the maximum values of electic and magnetic fields, E and B, at this same distance fom the bulb? Assume a plane wave. (b) The face of you tue love is one mete fom this 1 W bulb. What maximum suface cuent must flow on you tue love's face in ode to eflect the light fom the bulb into you adoing eyes? Assume that you tue love's face is (what else?) pefect--pefectly smooth and pefectly eflecting--and that the incident light and eflected light ae nomal to the suface Coaxial Cable and Powe Flow A coaxial cable consists of two concentic long hollow cylindes of zeo esistance; the inne has adius a, the oute has adius b, and the length of both is l, with l >> b. The cable tansmits DC powe fom a battey to a load. The battey povides an electomotive foce ε between the two conductos at one end of the cable, and the load is a esistance R connected between the two conductos at the othe end of the cable. A 13-47

48 cuent I flows down the inne conducto and back up the oute one. The battey chages the inne conducto to a chage Q and the oute conducto to a chage +Q. Figue (a) Find the diection and magnitude of the electic field E eveywhee. (b) Find the diection and magnitude of the magnetic field B eveywhee. (c) Calculate the Poynting vecto S in the cable. (d) By integating S ove appopiate suface, find the powe that flows into the coaxial cable. (e) How does you esult in (d) compae to the powe dissipated in the esisto? Supeposition of Electomagnetic Waves Electomagnetic wave ae emitted fom two diffeent souces with E ωt) ĵ, ω φ 1 ( xt, )= E 1 cos( kx E ( xt, )= E cos( kx t + )ĵ (a) Find the Poynting vecto associated with the esultant electomagnetic wave. (b) Find the intensity of the esultant electomagnetic wave (c) Repeat the calculations above if the diection of popagation of the second electomagnetic wave is evesed so that E ωt) ĵ, ω φ 1 ( xt, )= E 1 cos( kx E ( xt, )= E cos( kx + t + )ĵ Sinusoidal Electomagnetic Wave The electic field of an electomagnetic wave is given by 13-48