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 695 Problems which is plotted versus kL in Fig 9 14 This result can be checked in the limit as L becomes very small kL 1 since the radiation resistance should approach that of a point dipole given in Section 9 2 5 In this short dipole limit the bracketed terms in 14 are sin kL kL lim tL i l coS kL kL 2 6 kL 2 1 15 2 kLSi kL kL so that 14 reduces to kL 2 lim R AL 23L 3 2i 2 L 2 L 80 2 A 3 A 16 Er which agrees with the results in Section 9 2 5 Note that for large dipoles kL 1 the sine integral term dominates with Si kL approaching a constant value of 7r 2 so that lim R kL 1 7kL 60 4 r 2 Er 17 A PROBLEMS Section 9 1 1 We wish to find the properties of waves propagating within a linear dielectric medium that also has an Ohmic conductivity or a What are Maxwell s equations in this medium b Defining vector and scalar potentials what gauge condition decouples these potentials c A point charge at r 0 varies sinusoidally with time as Q t Re e What is the scalar potential d Repeat a c for waves in a plasma medium with constitutive law w eE at 2 An infinite Re K 0 e k ix current sheet at z 0 a Find the vector and scalar potentials b What are the electric and magnetic fields varies as 696 Radiation c Repeat a and b if the current is uniformly distributed over a planar slab of thickness 2a jo eij 9 kXi a z a 0 J Izj a 3 A sphere of radius R has a uniform surface charge distribution oy Re o e where the time varying surface charge is due to a purely radial conduction current a Find the scalar and vector potentials inside and outside the sphere Hint rep r 2 R 2 2rR cos 0 rQp drQ rR sin 0 dO b What are the electric and magnetic fields everywhere Section 9 2 4 Find the effective lengths radiation resistances and line charge distributions for each of the following current distributions valid for IzI dl 2 on a point electric dipole with short length dl a I z Iocos az 1 1 b f z Ioe c I z Iocosh az 5 What is the time average power density total time average power and radiation resistance of a point magnetic dipole 6 A plane wave electric field Re Eo ei is incident upon a perfectly conducting spherical particle of radius R that is much smaller than the wavelength a What is the induced dipole moment Hint See Section 4 4 3 b If the small particle is instead a pure lossless dielectric with permittivity e what is the induced dipole moment c For both of these cases what is the time average scattered power 7 A plane wave magnetic field Re Ho e is incident upon a perfectly conducting particle that is much smaller than the wavelength 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 it vary with frequency 8 a For the magnetic dipole how are the magnetic field lines related to the vector potential A b What is the equation of these field lines Section 9 3 9 Two aligned dipoles if dl and i2dl are placed along the z axis a distance 2a apart The dipoles have the same length i I Problems 2ar 697 y I while the currents have equal magnitudes but phase difference 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 or end fire array e Repeat a c for 2N 1 equally spaced aligned dipoles along the z axis with incremental phase difference Xo 10 Three dipoles of equal length dl are placed along the z axis A di I1 p Y I dl li di 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 where the power density is zero or maximum i ii iii Io 12 21o 21o Il2 Is Is Io 12 2jIo 1 I 698 Radiation 11 Many closely spaced point dipoles of length dl placed along the x axis driven in phase approximate a z directed current sheet Re Ko e i of length L y a Find the far fields from this current sheet b At what angles is the power density minimum or maximum Section 9 4 12 Find the far fields and time average power density for each of the following current distributions on a long dipole a i z Io 1 2z L O z L 2 SIo 1 2z L L 2 z 0 Hint C e az Z eaz dz az a f b I z Iocos 1z L 1 L 2 z L 2 Hint e zicos pz dz e az a cos a2 pz p sin pz p2 c For these cases find the radiation resistance when kL 1 C
View Full Document
Unlocking...