Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves Electromagnetic These notes Waves may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide 1 Lecture Outline Maxwell s Equations Predict Waves Derivation of the Wave Equation Solution to the Wave Equation Electromagnetic Waves Slide 2 1
Maxwell s Equations Predict Waves Electromagnetic Waves Slide 3 Recall Maxwell s Equations in Source Free Media In source free media, we have J 0 and 0. Maxwell s equations in the frequency domain become v Curl Equations E jb H jd Divergence Equations D 0 B 0 Constitutive Relations D E B H Electromagnetic Waves Slide 4 2
The Curl Equations Predict Waves After substituting the constitutive relations into the curl equations, we get E jh H j E A time harmonic magnetic field will induce a time harmonic electric field circulating about the magnetic field. A time harmonic circulating electric field will induce a time harmonic magnetic field along the axis of circulation. A time harmonic electric field will induce a time harmonic magnetic field circulating about the electric field. A time harmonic circulating magnetic field will induce a timeharmonic electric field along the axis of circulation. An H induces an E. That E induces another H. That new H induces another E. That E induces yet another H. And so on. Electromagnetic Waves Slide 5 How Waves Propagate Start with an oscillating electric field. Electromagnetic Waves Slide 6 3
How Waves Propagate This induces a circulating magnetic field. H j E Electromagnetic Waves Slide 7 How Waves Propagate Now let s examine the magnetic field on axis. Electromagnetic Waves Slide 8 4
How Waves Propagate This induces a circulating electric field. E jh Electromagnetic Waves Slide 9 How Waves Propagate Now let s examine the electric field on axis. Electromagnetic Waves Slide 10 5
How Waves Propagate This induces a circulating magnetic field. H j E Electromagnetic Waves Slide 11 How Waves Propagate and so on Electromagnetic Waves Slide 12 6
Derivation of the Wave Equation Electromagnetic Waves Slide 13 Wave Equation in Linear Media (1 of 2) Since the curl equations predict propagation, it makes sense that we derive the wave equation by combining the curl equations. E j H H 1 j Solve for H 1 E H j E 1 1 E j E j Electromagnetic Waves Slide 14 7
Wave Equation in Linear Media (2 of 2) The last equation is simplified to arrive at our final equation for waves in linear media. 1 E 2 This equation is not very useful for performing derivations. It is typically used in numerical computations. Note: We cannot simplify this further because the permeability is a function of position and cannot be brought outside of the curl operation. 1 2 E E E Electromagnetic Waves Slide 15 Wave Equation in LHI Media (1 of 2) In linear, homogeneous, and isotropic media two important simplifications can be made. First, in isotropic media the permeability and permittivity reduce to scalar quantities. 1 2 E E Second, in homogeneous media is a constant and can be brought to the outside of the curl operation and then brought to the right hand side of the equation. 2 E E Electromagnetic Waves Slide 16 8
Wave Equation in LHI Media (2 of 2) We now apply the vector identity 2 2 E E E E E E E E EE 0 A A A In LHI media, the divergence equation can be written in terms of E. D 0 E0 E0 E 0 Electromagnetic Waves Slide 17 Wave Number k and Propagation Constant We can define the term k as either This provides a way to write the wave equation more simply as Ek E 0 or or k 2 E E 0 Electromagnetic Waves Slide 18 9
Solution to the Wave Equation Electromagnetic Waves Slide 19 Components Decouple in LHI Media We can expand our wave equation in Cartesian coordinates. Ek E 0 Eaˆ Eaˆ Eaˆ k Eaˆ Eaˆ Eaˆ 0 x x y y z z x x y y z z Eaˆ Eaˆ Eaˆ keaˆ keaˆ keaˆ 0 x x y y z z x x y y z z E k E aˆ E k E aˆ E k E aˆ 0 x x x y y y z z z We see that the different field components have decoupled from each other. All three equations have the same numerical form so they all have the same solution. Therefore, we only need the solution to one of them. Ek E 0 Ex k Ex Ey k Ey 0 0 E k E 0 z z Electromagnetic Waves Slide 20 10
General Solution to Scalar Wave Equation Our final wave equation for LHI media is Ek E 0 This could be handed off to a mathematician to obtain the following general solution. jk r jk r E r E e E e 0 0 forward wave backward wave Electromagnetic Waves Slide 21 General Solution to Vector Wave Equation Given the solution to the scalar wave equation, we can immediately write solutions for all three field components. jkr jkr Exr Exe Exe jkr jkr Eyr Eye Eye jkr jkr E r E e E e z z z We can assemble these three equations into a single vector equation. E r E ˆ ˆ ˆ x r ax Ey r ay Ez r az jk r jk r Ee 0 Ee 0 forward wave backward wave Electromagnetic Waves Slide 22 11
General Expression for a Plane Wave The solution to the wave equation gave us two plane waves. From the forward wave, we can extract a general expression for plane waves. jk r Er Pe Frequency domain E rt, Pcos tkr Time domain We define the various parameters as r xaˆ ˆ ˆ x yay zaz position k wave vector E total electric field intensity 2 f angular frequency P polarization vector t time Electromagnetic Waves Slide 23 Magnetic Field Component Given that the electric field component of a plane wave is written as jk r E r Pe The magnetic field component is derived by substituting this solution into Faraday s law. E j H jk r Pe j H H kpe 1 jk r Electromagnetic Waves Slide 24 12
Solution in Terms of the Propagation Constant The wave equation and it solution in terms of is r r E E 0 Er E0e E0e forward wave The general expression for a plane wave is r E r Pe Frequency domain backward wave The magnetic field component is 1 H Pe r j The wave vector and propagation constant are related through jk Electromagnetic Waves Slide 25 Visualization of an EM Wave (1 of 2) We tend to draw and think of electromagnetic waves this way Electromagnetic Waves Slide 26 13
Visualization of an EM Wave (2 of 2) However, this is a more realistic visualization. It is important to remember that plane waves are also of infinite extent in all directions. Electromagnetic Waves Slide 27 14