Review Notes on Maxwell's Equations

Transcription

1 ELEC344 Micowave Engineeing, Sping 2002 Handout #1 Kevin Chen Review Notes on Maxwell's Equations Review of Vecto Poducts and the Opeato The del, gad o nabla opeato is a vecto, and can be pat of a scala poduct, a vecto dot poduct o a vecto coss poduct. The poduct of this vecto with a scala φ is anothe vecto called the gadient of the scala, φ. The dot poduct of two vectos is a scala, and the dot poduct of and anothe vecto A is A, a scala called the divegence of the vecto. The coss poduct of two vectos is a vecto nomal to the plane of the two vectos, and the coss poduct of and anothe vecto A is A, called the cul of the vecto. The dot poduct is called the Laplacian, and is witten 2. See the inside coves of Poza 1 fo useful summaies of vecto opeatos (back cove), Maxwell's equations and tansmission line elationships (font cove). Review of Maxwell's Equations Maxwell's equations, hee in diffeential fom, define the obseved inteaction among time-vaying electic and magnetic fields, electic chage and the electic and magnetic chaacteistics of media. Geneal fom x E = - x H = B t D t + J D = ρ = 0 in chage-fee egions B = 0 Phaso fom (cosinusoidal excitation) xe = -jωµ H x H = jωµ E 1 Poza, Micowave Engineeing, 2nd Edition, J. Wiley,

2 Field Inteaction with Media The additional polaization in dielectic mateials esulting fom inteaction of applied electic field and the atomic o molecula stuctue esults in the elationship D = εe, whee ε may be complex ε = ε' - jε", the imaginay pat accounting fo loss. In a mateial with conductivity σ, a conduction cuent density will exist as J = σ E. In a magnetic mateial, an applied magnetic field may align magnetic dipole moments to poduce a magnetic polaization esulting in the elationship B = µ H, whee µ may be complex µ = µ' - jµ", the imaginay pat accounting fo loss. In non-isotopic media, ε and µ may be tensos (matices); additionally, some mateials may be nonlinea. The inteaction of fields with media can be summaized in the elations hee: Fee space case Geneal dielectic, conductive o magnetic D = ε o E, ε o = x F/m D = ε E J = σ E B = µ o H, µ o = 4π x 10-7 H/m B = µ H Fields at Mateial Intefaces The pincipal bounday conditions used in solving Maxwell's equation ae at dielectic o conducting boundaies. These conditions ae summaized hee: Dielectic Inteface Pefect Conducto Radiation Condition Nomal to inteface Nomal D and B continuous Nomal D = ρ, B = 0 Fields -> 0 fo -> Tangential to inteface Tangential E and H equal Tangential E = 0, H = J - 2 -

3 The Wave Equation Taking the cul of the fist equation, then substituting the second, and making use of the last two elationships plus a vecto identity, deives a Helmholtz wave equation fo E 2 E + ω 2 µε E = 0 and similaly can deive an identical one fo H 2 H + ω 2 µεh = 0 Plane Waves in a Lossless Medium A solution of the wave equation exists that has only an ) x component, and fo which the deivatives in the x and y diections ae zeo (unifom field in these diections). In this case the wave equation educes to a simple, but illustative, fom x 2 E x 2 + k 2 E x = 0 The solution to this type of equation is a wave descibed by E x (z) = E + e -jkz + E - e jkz with the popagation constant k defined as k = ω µ ε m -1. The velocity of wave popagation is v p = k ω = 1. The wavelength λ is µ ε λ = 2 π 2πv = p = k w v p f The cul of time-vaying E x gives ise to time-vaying H y of the fom 1 jkz H y (z) = H + e -jkz + H - e jkz + jkz = ( E e + E e ), whee η = µ / ε is the wave impedance. η In the fee-space case, η o = µ o /ε o = 377 ohms. H x = H z = 0-3 -

4 In this case E and H ae othogonal to each othe and to the diection of popagation z ), so these waves, having only tansvese components, ae temed tansvese electomagnetic (TEM) waves. Plane Waves in a Lossy Medium The same deivation applies in the case of lossy media (eithe σ 0 o ε = ε' - jε", o both). The popagation constant becomes complex, signifying a decay tem as well as the popagation tem of the popagation constant. A case of paticula inteest is that of a good conducto, defined as σ >> ωε. In this case a wave popagating into the mateial can only penetate a depth called the skin depth, which is defined as δ s = 2 ωµσ. The skin depth vaies as 1/ f, and is of the ode of 10-6 m at f = 10 GHz fo most conductive metals such as aluminum, coppe, gold and silve. The solutions fo geneal plane waves not necessaily popagating along a single axis have the same fom as the simple examples given above. It will be found that each geometic situation will esult in chaacteistic foms of the solutions fo waves that fom a family of othogonal modes defined by the bounday conditions. The plane waves discussed above had thei electic field vecto pointing (polaized) in a fixed diection, and this is temed linea polaization. If we supepose two plane waves, one polaized in the x ) diection and the othe in the y ) diection, taveling in the same diection, the linea combination of the two can fom linea polaization at an angle o, if thee is also a time (phase) diffeence, can fom cicula polaization in which the polaization vecto otates as the wave popagates. Similaly, two ciculaly polaized waves can be combined to fom a linealy polaized wave. Abitay field pattens can be expessed as a sum of othogonal modes, which ae the solutions of Maxwell's equations that fit the geomety of a paticula class of components such as waveguide. We don't have to solve Maxwell's equations fo evey new poblem, we just esolve the bounday conditions to detemine the mode coefficients necessay to meet the bounday equiements. A Note on the Histoy of Maxwell's Equations Maxwell's heoic contibution was the successful combination of the laws of electostatics and magnetostatics so that they descibed accuately the phenomena of time-vaying electomagnetic fields. The esulting mathematical desciptions of the inteaction of electicity and magnetism, when combined, esulted in an easily ecognized - 4 -

5 wave equation pedicting the existence of electomagnetic waves whose velocity of popagation in a vacuum would be v p = 1 µ o ε o, whee µ o = 4 x 10-7 H/m and ε o = 8.85 x F/m. If you substitute these quantities into the equation fo wave velocity, you find as Maxwell did that the pedicted velocity is v = 9 x = 3 x 10 8 m/s. As he knew this was aguably close to the known velocity of light, this confimed fo Maxwell his ealie conjectue that light was electomagnetic in natue, and at the same time pedicted electomagnetic adiation at othe wavelengths was mathematically possible. Excellent eviews of field and wave electomagnetics is found in Cheng 2 and Inan 3. These books contains a clea eview of electostatics, magnetostatics and how they undelie the waves that ae the solutions of Maxwell's equations. Once a discipline is well-established, texts pesent it in a logical athe than histoical ode, so few moden texts give a pictue of what was known when duing the life of Maxwell himself. If you eve have a time fo eview and eflection, look fo the wondeful Dove book, The Scientific Papes of James Clek Maxwell.. Although the fou equations ae called Maxwell's equations, his genius was the postulation of the existence of the displacement cuent, which then esulted, in combination with the othes, in a wave equation pedicting electomagnetic waves that popagated at c = 1/ µ o /ε o, the speed of light. Fom this he concluded coectly that light was electomagnetic in natue (he sumised this fom the esults of Faaday's expeiments in polaization of light by magnetic inteaction with dielectics). Many solutions of inteest ae in chage-fee egions, but note that popagation though chaged plasma 4 epesents a significant aea of study. We have the luxuy of knowing how histoy woked out, and the study of electomagnetic fields and waves now takes fo ganted many of the geat unknowns esolved by Maxwell's illustious pedecessos who ae memoialized today in the units of electomagnetic quantities. The physical desciption of the inteaction of electicity, magnetism and matte began with the discovey of the phenomena and the mathematically stated laws of electostatics. The scientific heoes of this peiod include names like Coloumb, Gauss, Faaday, 2 Cheng, D. K., Field and Wave Electomagnetics, Addison Wesley, Inan, U. and Inan, A, Engineeing Electomagnetics, Addison Wesley, Poza, Micowave Engineeing, 2nd Edition, J. Wiley, 1998, pg

6 Ampee, Voltea and Ohm. Late, discovey of electodynamic phenomena bought about the development of the moe extensive set of elationships assembled by Maxwell, which themselves wee polished futhe by the mathematicians who developed vecto analysis as a shothand to descibe the equations of xe electic and magnetic fields