MIT OpenCourseWare http://ocw.mit.edu Electromagnetic Field Theory: A Problem Solving Approach For any use or distribution of this textbook, please cite as follows: Markus Zahn, Electromagnetic Field Theory: A Problem Solving Approach. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Problems 695which is plotted versus kL in Fig. 9-14. This result can bechecked in the limit as L becomes very small (kL << 1) since theradiation resistance should approach that of a point dipolegiven in Section 9-2-5. In this short dipole limit the bracketedterms in (14) aresin kL (kL)2---l---kL 6lim (kL)2 (15))tL• i coS kL 12kLSi(kL) -(kL)"so that (14) reduces tolim R (kL)2 23L-2= 8 0L(2 L 2(16)AL'< 2i 3 3\A A Erwhich agrees with the results in Section 9-2-5. Note that forlarge dipoles (kL >>1), the sine integral term dominates withSi(kL) approaching a constant value of 7r/2 so thatlim R -7kL=60 •-r 2 (17)kL>1 4 Er APROBLEMSSection 9-11. We wish to find the properties of waves propagatingwithin a linear dielectric medium that also has an Ohmicconductivity or.(a) What are Maxwell's equations in this medium?(b) Defining vector and scalar potentials, what gaugecondition decouples these potentials?(c) A point charge at r = 0 varies sinusoidally with time asQ(t) = Re (( e'"). What is the scalar potential?(d) Repeat (a)-(c) for waves in a plasma medium withconstitutive law--= w eEat2. An infinite current sheet at z = 0 varies asRe [K0 e ('-k"-)ix].(a) Find the vector and scalar potentials.(b) What are the electric and magnetic fields?696 Radiation(c) Repeat (a) and (b) if the current is uniformly dis-tributed over a planar slab of thickness 2a:jo eij(9-kXi, , -a<z<aJ 0, Izj >a3. A sphere of radius R has a uniform surface charge dis-tribution oy= Re (&o e"•') where the time varying surfacecharge is due to a purely radial conduction current.(a) Find the scalar and vector potentials, inside and outsidethe sphere. (Hint: rep=r2+R2-2rR cos 0; rQp drQ=rR sin 0 dO.)(b) What are the electric and magnetic fields everywhere?Section 9.24. Find the effective lengths, radiation resistances and linecharge distributions for each of the following current dis-tributions valid for I zI <dl/2 on a point electric dipole withshort length dl:(a) I(z) = Io cos az(b) f(z) = Io e-*11(c) I(z)= Io cosh az5. What is the time-average power density, total time-averagepower, and radiation resistance of a point magnetic dipole?6. A plane wave electric field Re (Eo ei') is incident upon aperfectly conducting spherical particle of radius R that ismuch smaller than the wavelength.(a) What is the induced dipole moment? (Hint: SeeSection 4-4-3.)(b) If the small particle is, instead, a pure lossless dielectricwith permittivity e, what is the induced dipole moment?(c) For both of these cases, what is the time-average scat-tered power?7. A plane wave magnetic field Re (Ho e••) is incident upon aperfectly conducting particle that is much smaller than thewavelength.(a) What is the induced magnetic dipole moment?(Hint: See Section 5-7-2ii and 5-5-1.)(b) What. are the re-radiated electric and magnetic fields?(c) What is the time-average scattered power? How does itvary with frequency?8. (a) For the magnetic dipole, how are the magnetic fieldlines related to the vector potential A?(b) What is the equation of these field lines?Section 9.39. Two aligned dipoles if dl and i2 dl are placed along the zaxis a distance 2a apart. The dipoles have the same lengthi · IProblems 697ywhile the currents have equal magnitudes but phasedifference X.(a) What are the far electric and magnetic fields?(b) What is the time-average power density?(c) At what angles is the power density zero or maximum?(d) For 2a = A/2, what values of X give a broadside orend-fire array?(e) Repeat (a)-(c) for 2N+ 1 equally spaced aligned dipolesalong the z axis with incremental phase difference Xo.10. Three dipoles of equal length dl are placed along the zaxis.(a) Find the far electric and magnetic fields.(b) What is the time average power density?(c) For each of the following cases find the angles wherethe power density is zero or maximum.(i) =Io,12= 21o(ii) 1= I ,Il2= -21o(iii) Is = -Is = Io, 12 = 2jIo2ar'II1A diI dl'li diýp Y698y(a) Find the far fields from this current sheet.(b) At what angles is the power density minimum ormaximum?Section 9.412. Find the far fields and time-average power density foreach of the following current distributions on a long dipole:(a) i(z) Io( 1 -2z/L), O<z<L/2SIo(1+2z/L), -L/2<z<0Hint:C e azZ eaz dz = -(az - 1)f a(b) I(z)= Iocos 1z/L, -L/2<z <L/2Hint:zi az (a cos pz + p sin pz)e cos pz dz = e (a2+ p2)(c) For these cases find the radiation resistance whenkL << 1.Radiation11. Many closely spaced point dipoles of length dl placedalong the x axis driven in phase approximate a z-directedcurrent sheet Re (Ko e'"'i) of length L._C_ _ __
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