PHYS 1444 – Section 003 Lecture #23AnnouncementsMaxwell’s EquationsEM Waves and Their SpeedsMaxwell’s Equations w/ Q=I=0EM Waves from Maxwell’s EquationsSlide 7Slide 8Slide 9Slide 10Slide 11Light as EM WaveSlide 13Electromagnetic SpectrumExample 32 – 2EM Wave in the Transmission LinesEnergy in EM WavesEnergy TransportSlide 19Average Energy TransportExample 32 – 4You have worked very hard and well !!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:0enclQE dAe� =�rr�0B dA� =�rr�BdE dldtF� =-�rr�0 0 0EencldB dl Idtm meF� = +�rr�Gauss’ 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 Yu4vrEM 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==0•In this region of free space, Q=0 and =0, thus the four Maxwell’s equations become0enclQE dAe� =�rr�0B dA� =�rr�BdE dldtF� =-�rr�0 0 0EencldB dl Idtm meF� = +�rr�0E dA� =�rr�0B dA� =�rr�BdE dldtF� =-�rr�0 0EdB dldtmeF� =�rr�Qencl=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 and frequency f, such traveling wave can be written as–Where–What is v?•It is the speed of the traveling wave–What are E0 and B0?•The amplitudes of the EM wave. Maximum values of E and B field strengths.E =k =B =v =f l =yE =( )0sinE kx tw-zB =( )0sinB kx tw-2pl2 fpkw=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 LawE dl� =�rr�E dl� =�rr�BddtF=ThusdE yD =dEdx=Since E and B depend on x and tE Bx t� �=-� �E dl�rrBddtF-E dx� +rr( )E dE y+ �D +r rr'E dx� +rr'E y�D =rr0=( )E dE y+ + D0-E y- D =dE yDdBdtdx yDdBdx ydt- DdBdt-Monday, Nov. 28, 2005 PHYS 1444-003, Fall 2005Dr. Jaehoon Yu8•Let’s apply Maxwell’s 4th equation–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– B dl� =�rr�0 0EddtmeF=ThusdB z- D =dBdx=Since E and B depend on x and t0 0B Ex tme� �=-� �B dl�rrB dl� =�rr�From Modified Ampére’s Law0 0EddtmeFB ZD( )B dB Z- + D =dB Z- D0 0dEdtmedx zD0 0dEdx zdtme D0 0dEdtme-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 tw- =00EB=E B v=Bt�-�( )( )0sinE kx txw�- =�( )0coskE kx tw-( )( )0sinB kx ttw�- =�( )0cosB kx tw w- -( )0cosB kx tw w-Since Ex�=�Bt�-�We obtainThuskw=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 tw- =00BE=20 01ve m=0 0E B =( ) ( )812 2 2 70 01 13.00 108.85 10 4 10v m sC N m T m Ae mp- -= = = ��״��The speed of EM waves is the same as the speed of light. EM waves behaves like the light. 0 0Ete m�-�( )( )0sinB kx txw�- =�( )0coskB kx tw-( )( )0sinE kx ttw�- =�( )0cosE kx tw w- -Since Bx�=�0 0Ete m�-�We obtain( )0 0 0cosE kx te mw w-Thus0 0ke mw=0 0ve mv =0 01ve mMonday, 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
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