Electromagnetic Field Theory: A Problem Solving Approach Part 53. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Conservation of Energy 495 A simple rule for the time average of products is obtained by realizing that the real part of a complex number is equal to one half the sum of the complex number and its conjugate denoted by a superscript asterisk . The power density is then S r t E r t X H r t E r e 1 E r e x H r H r e E r x H r e2ia E r x H r E r x H r E r xH r i-2 1 20 The time average of 20 is then S i E r x H r E r x H r 2 Re E r xH r 2 Re E r x H r 21 as the complex exponential terms e 2i average to zero over a period T 2ttI d and we again realized that the first bracketed term on the right-hand side of 21 was the sum of a complex function and its conjugate. Motivated by 21 we define the complex Poynting vector as S E r x H r 22 whose real part is just the time-average power density. We can now derive a complex form of Poynting s theorem by rewriting Maxwell s equations for sinusoidal time variations as V X E r y a xH r VxH r J r a E r V E r pf r e V B r 0 and expanding the product V S V E r x H r H r V X E r - E r V x H r H r 2 jtae E r 2 - E r j r 24 which can be rewritten as V S 2 u wm - w -Pd 25 where Wm 4M H r 2 w. iel E r 2 26 A iE f -jftr 496 Electrodynamics Fields and Waves We note that wm and w are the time-average magnetic and electric energy densities and that the complex Poynting s theorem depends on their difference rather than their sum. 7-3 TRANSVERSE ELECTROMAGNETIC WAVES 7-3-1 Plane Waves Let us try to find solutions to Maxwell s equations that only depend on the z coordinate and time in linear media with permittivity e and permeability z. In regions where there are no sources so that P 0 J 0 Maxwell s equations then reduce to dE . dEx. dH dz dz dt 1 dHy dHx dE 1 i e dz dz dt 2 dEz n e 0 dz 3 o dz 4 These relations tell us that at best and Hz are constant in time and space. Because they are uncoupled in the absence of sources we take them to be zero. By separating vector components in 1 and 2 we see that Ex is coupled to H and E is coupled to Hx dEx
