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Monday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu1PHYS 1444 – Section 003Lecture #23Monday, Nov. 28, 2005Dr. Jaehoon Yu• EM Waves from Maxwell’s Equations• Speed of EM Waves• Light as EM Wave• Electromagnetic Spectrum• Energy in EM Waves• Energy Transport• The epilogueToday’s homework is homework #12, noon, next Tuesday, Dec. 6!!Monday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu2Announcements• Reading assignments– CH. 32 – 8 and 32 – 9 • No class this Wednesday, Nov. 30• Final term exam – Time: 11am – 12:30pm, Monday Dec. 5– Location: SH103– Covers: CH 29.3 – CH32– Please do not miss the exam– Two best of the three exams will be used for your gradesMonday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu3Maxwell’s Equations• In the absence of dielectric or magnetic materials, the four equations developed by Maxwell are:0enclQEdAε⋅=∫GGv0BdA⋅=∫GGvBdEdldtΦ⋅=−∫GGv000EencldBdl IdtµµεΦ⋅= +∫GGvGauss’ Law for electricityGauss’ Law for magnetismFaraday’s LawAmpére’s LawA generalized form of Coulomb’s law relating electric field to its sources, the electric chargeA magnetic equivalent ff Coulomb’s law relating magnetic field to its sources. This says there are no magnetic monopoles.An electric field is produced by a changing magnetic fieldA magnetic field is produced by an electric current or by a changing electric fieldMonday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu4vGEM Waves and Their Speeds• Let’s consider a region of free space. What’s a free space?– An area of space where there is no charges or conduction currents– In other words, far from emf sources so that the wave fronts are essentially flat or not distorted over a reasonable area– What are these flat waves called?• Plane waves• At any instance E and B are uniform over a large plane perpendicular to the direction of propagation– So we can also assume that the wave is traveling in the x-direction w/ velocity, v=vi, and that E is parallel to y axis and B is parallel to z axisMonday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu5Maxwell’s Equations w/ Q=Ι=0• In this region of free space, Q=0 and Ι=0, thus the four Maxwell’s equations become0enclQEdAε⋅=∫GGv0BdA⋅=∫GGvBdEdldtΦ⋅=−∫GGv000EencldBdl IdtµµεΦ⋅= +∫GGv0EdA⋅=∫GGv0BdA⋅=∫GGvBdEdldtΦ⋅=−∫GGv00EdBdldtµεΦ⋅=∫GGvQencl=0No ChangesNo ChangesIencl=0One can observe the symmetry between electricity and magnetism. The last equation is the most important one for EM waves and their propagation!!Monday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu6EM Waves from Maxwell’s Equations• If the wave is sinusoidal w/ wavelength λ and frequency f, such traveling wave can be written as–Where– What is v?• It is the speed of the traveling wave– What are E0and B0?• The amplitudes of the EM wave. Maximum values of E and B field strengths.E=k =B=ϖ=fλ=yE =()0sinEkxtω−zB=()0sinBkx tω−2πλ2 fπkω=vThusMonday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu7• Let’s apply Faraday’s law – to the rectangular loop of height ∆y and width dx• along the top and bottom of the loop is 0. Why?– Since E is perpendicular to dl.– So the result of the integral through the loop counterclockwise becomes– For the right-hand side of Faraday’s law, the magnetic flux through the loop changes as –From Faraday’s LawEdl⋅=∫GGvEdl⋅=∫GGvBddtΦ=ThusdEy∆=dEdx=Since E and B depend on x and tEBxt∂∂=−∂∂Edl⋅GGBddtΦ−Edx⋅+GG()EdE y+⋅∆ +GGG'Edx⋅+GG'Ey⋅∆=GG0=()EdEy++∆0−Ey−∆=dEy∆dBdtdxy∆dBdxydt−∆dBdt−Monday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu8• Let’s apply Maxwell’s 4thequation– to the rectangular loop of length ∆z and width dx• along the x-axis of the loop is 0– Since B is perpendicular to dl.– So the result of the integral through the loop counterclockwise becomes– For the right-hand side of the equation is–Bdl⋅=∫GGv00EddtµεΦ=ThusdB z−∆=dBdx=Since E and B depend on x and t00BExtµε∂∂=−∂∂Bdl⋅GGBdl⋅=∫GGvFrom Modified Ampére’s Law00EddtµεΦBZ∆()BdBZ−+∆=dB Z−∆00dEdtµεdx z∆00dEdx zdtµε∆00dEdtµε−Monday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu9• Let’s now use the relationship from Faraday’s law• Taking the derivatives of E and B as given their traveling wave form, we obtain– Since E and B are in phase, we can write• This is valid at any point and time in space. What is v?– The velocity of the wave Relationship between E, B and vEx∂=∂Ex∂=∂Bt∂=∂()0coskE kx tω−=00EB=EBv=Bt∂−∂()()0sinEkxtxω∂−=∂()0coskE kx tω−()()0sinBkxttω∂−=∂()0cosBkxtωω−−()0cosBkxtωω−Since Ex∂=∂Bt∂−∂We obtainThuskω=vMonday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu10• Let’s now use the relationship from Apmere’s law• Taking the derivatives of E and B as given their traveling wave form, we obtain– However, from the previous page we obtain–ThusSpeed of EM WavesBx∂=∂Bx∂=∂Et∂=∂()0coskB kx tω−=00BE=2001vεµ=00EB=()()812 2 2 700113.00 108.85 10 4 10vmsCNm TmAεµπ−−== =××⋅⋅×⋅The speed of EM waves is the same as the speed of light. EM waves behaves like the light. 00Etεµ∂−∂()()0sinBkxtxω∂−=∂()0coskB kx tω−()()0sinEkxttω∂−=∂()0cosEkxtωω−−Since Bx∂=∂00Etεµ∂−∂We obtain()00 0cosEkxtεµω ω−Thus00kεµω=00vεµv=001vεµMonday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu11• Taking the time derivative on the relationship from Ampere’s laws, we obtain• By the same token, we take position derivative on the relationship from Faraday’s law• From these, we obtain• Since the equation for traveling wave is• By correspondence, we obtain• A natural outcome of Maxwell’s equations is that E and B obey the wave equation for waves traveling w/ speed – Maxwell predicted the existence of EM waves based on thisSpeed of Light w/o Sinusoidal Wave Forms2Bxt∂=∂∂2v=and22Ex∂=∂22Et∂=∂22Bt∂=∂22xt∂=∂001vεµ=2002Etεµ∂−∂2Bxt∂−∂∂22001 Exεµ∂∂22001Bxεµ∂∂222xvx∂∂001εµMonday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu12Light as EM Wave• People knew some 60 years before Maxwell that light behaves like a wave, but …– They did not


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