MIT OpenCourseWare http ocw mit edu 6 013 ESD 013J Electromagnetics and Applications Fall 2005 Please use the following citation format Markus Zahn 6 013 ESD 013J Electromagnetics and Applications Fall 2005 Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution Noncommercial Share Alike Note Please use the actual date you accessed this material in your citation For more information about citing these materials or our Terms of Use visit http ocw mit edu terms 6 013 Electromagnetics and Applications Fall 2005 Lecture 19 Electromagnetic Radiation Prof Markus Zahn November 29 2005 I Retarded Potentials A Maxwell s Equations E B t J f H D t 0 B f D H D E B B Vector Identities 0 A 0 A A 2 A C Potentials 0 B A B A B E E 0 t t A E t E 1 A H Jf A Jf t t t 2 A A A 2 A J f 2 t t 2A 1 1 1 2 2 J f c2 2 A A 2 c t c t Setting the gauge Lorentz gauge 1 A 2 c t 2A 1 2 A 2 2 J f c t f f 1 2 2 2 2 E A 2 t c t 1 D Solutions to the Wave Equations 1 Solution in Time Domain In spherical coordinates 1 1 1 2 2 2 2 r 2 sin 2 2 r r r r sin r sin 2 Consider a point charge Q t at r 0 Then 2 1 r2 r 0 r t 1 2 r2 r r r r2 Then 2 1 2 r 1 2 0 for r 0 r 0 r2 c2 t2 c2 t2 This is a wave equation in r r t 2 r r r t f t c 0 r f t c No sources for r 0 so only waves emanating radially outward from point charge at r 0 f t Q t Q t f t r 4 4 r Q t rc r t 4 r lim r t r 0 2 Solution in Frequency Domain j t j t r t Re r e Q t Re Qe d2 2 r r 2 r r 0 dr2 c r r A1 e jkr 0 A2 e jkr k No r traveling wave A1 Q Q lim r A1 r 4 r 4 Q r Q r e jkr r t Re ej t c 4 r 4 r r 0 Spherical Wave 2 c Vector Potential A 1 r2 r r 1 j A 2 2 c t c 0 0 A 1 1 1 r2 A r A sin A r sin r sin r2 r 0 j Q jkr r2 A r 2 j e c 4 r d 2 j r2 A r r Ar Q re jkr dr 4 j 2 r A r Q dr re jkr 4 j Q e jkr jkr 1 4 k 2 j Q 1 jkr e jkr 4 k 2 r2 j Q 1 jkr e jkr 2 4 r2 A r jQ 1 jkr e jkr 4 r2 3 Solutions for Volume Charge and Current Distributions Image by MIT OpenCourseWare r f t QP c r t dV 4 rQP all charge r 2 J f t QP 1 A 2 c A 2 2 Jf A dV 4 rQP c t all current 3 II Radiation From a Point Electric Dipole From Electromagnetic Field Theory A Problem Solving Approach by Markus Zahn 1987 Used with permission j t iz i t Re Ie dl I at z j 2 p Qdli z dipole moment Q A Vector Potential Az r t Re A z r ej t 0 dl r 2 j t QP c Ie Az r t Re dz 4 rQP dl 2 I j t r c Re e 4 r Idl e jkr k 4 r c j t Az r t Re A z r e A z r i z i r cos i sin A Az i z Az i r cos i sin A r A z cos A A z sin 4 B Scalar Potential jc2 jc2 1 2 1 A r A r sin A r2 r r sin Idl cos d r 2 e jkr sin sin jkr j e 4 r2 dr r r sin r j Idl 1 jkr e jkr cos 4 r2 Qdl 1 jkr e jkr cos 4 r2 C Magnetic Field 0 0 1 1 1 A A H A sin i r r sin 0 i 1 A r r sin r 0 i A r r A rA r r 0 1 1 A r H A i rA r r 2 Idlk 1 1 H i sin e jkr 4 jkr jkr 2 with A 0 D Electric Field 1 1 1 1 H E H sin ir rH i j j r sin r r 2 Idlk 1 1 ir 2 cos 4 jkr 2 jkr 3 1 1 1 i sin e jkr jkr jkr 2 jkr 3 E Far Field Limit kr 1 0 2 E Idlk lim E H sin e jkr E 0 kr 1 jkr 4 Same relationship as plane waves 5 III Power Density 1 H S Re E 2 E E 0 2 cos i r 1 1 1 1 1 sin i e jkr jkr 2 jkr 3 jkr jkr 2 jkr 3 E 0 1 1 H i sin e jkr jkr jkr 2 E 0 2 1 1 1 1 ir i S Re sin 2 cos 2 jkr 2 jkr 3 jkr jkr 2 i sin 1 1 1 2 jkr jkr jkr 3 1 1 jkr jkr 2 i i i r E 0 2 S sin Re 2 1 1 1 1 2 cos i jkr 3 jkr 4 jkr 4 jkr 5 imag cancel imag 1 1 1 1 1 1 sin i r jkr 2 jkr 3 jkr 3 jkr 4 jkr 4 jkr 5 real cancel imaginary cancel 2 E 0 sin Idlk i E r 0 2 k 2 r 2 4 2 2 IV Total Time Average Radiated Power A Power P 2 d Sr r2 sin d 0 E 0 2 2 k 2 0 2 d sin3 d 0 0 E 0 2 2 d sin3 2 k 2 0 1 E 0 2 2 cos sin 2 2 k 3 0 4 3 4 2 E0 3 k 2 2 k 4 2 4 Idl 2 3 k 2 4 16 2 k 2 Idl 12 6 2 imag From Electromagnetic Field Theory A Problem Solving Approach by Markus Zahn 1987 Used with permission Antenna Gain G S r P 4 r2 E 0 2 sin2 3 k 2 4 r2 2 k 2 r2 4 E 0 2 3 sin2 2 B Radiation Resistance R 2 k 2 Idl 1 2 I R 12 2 2 2 2 dl R kdl 4 6 …
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