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## by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus

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**Fundamentals of Electromagneticsfor Teaching and Learning:A**Two-Week Intensive Course for Faculty inElectrical-, Electronics-, Communication-, and Computer- Related Engineering Departments in Engineering Colleges in India by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, India**Program for Hyderabad Area and Andhra Pradesh**FacultySponsored by IEEE Hyderabad Section, IETE Hyderabad Center, and Vasavi College of EngineeringIETE Conference Hall, Osmania University CampusHyderabad, Andhra PradeshJune 3 – June 11, 2009Workshop for Master Trainer Faculty Sponsored byIUCEE (Indo-US Coalition for Engineering Education)Infosys Campus, Mysore, KarnatakaJune 22 – July 3, 2009**Module 4**Wave Propagation in Free Space 4.1 Uniform Plane Waves in Time Domain 4.2 Sinusoidally Time-Varying Uniform Plane Waves 4.3 Polarization 4.4 Poynting Vector and Energy Storage**Instructional Objectives**23. Write the expression for a traveling wave function for a set of specified characteristics of the wave 24. Obtain the electric and magnetic fields due to an infinite plane current sheet of an arbitrarily time-varying uniform current density, at a location away from it as a function of time, and at an instant of time as a function of distance, in free space 25. Find the parameters, frequency, wavelength, direction of propagation of the wave, and the associated magnetic (or electric) field, for a specified sinusoidal uniform plane wave electric (or magnetic) field in free space 26. Write expressions for the electric and magnetic fields of a uniform plane wave propagating away from an infinite plane sheet of a specified sinusoidal current density, in free space**Instructional Objectives (Continued)**27. Obtain the expressions for the fields due to an array of infinite plane sheets of specified spacings and sinusoidal current densities, in free space 28. Write the expressions for the fields of a uniform plane wave in free space, having a specified set of characteristics, including polarization 29. Express linear polarization and circular polarization as superpositions of clockwise and counterclockwise circular polarizations 30. Find the power flow and the electric and magnetic stored energies associated with electric and magnetic fields**4.1 Uniform Plane Wavesin Time Domain(EEE, Sec. 3.4; FEME,**Secs. 4.1, 4.2, 4.4, 4.5)**Infinite Plane Current Sheet Source**Example:**For a current distribution having only an x-component of**current density that varies only with z,**4-8**The only relevant equations are: Thus,**4-9**In the free space on either side of the sheet, Jx = 0 Combining, we get Wave Equation**Where**velocity of light represents a traveling wave propagating in the +z-direction. represents a traveling wave propagating in the –z-direction.**4-14**E4.1: Examples of Traveling Waves**Thus, the general solution is**For the particular case of the infinite plane current sheet in the z = 0 plane, there can only be a (+) wave for z > 0 and a (-) wave for z < 0. Therefore,**Applying Faraday’s law in integral form**to the rectangular closed path abcda in the limit that the sides bc and da0,**Therefore,**Now, applying Ampere’s circuital law in integral form to the rectangular closed path efgha in the limit that the sides fg and he0,**Thus, the solution is**Uniform plane waves propagating away from the sheet to either side with velocity vp= c.**In practice, there are no uniform plane waves. However,many**practical situations can be studied based on uniformplane waves. For example, at large distances from physicalantennas and ground, the waves can be approximated asuniform plane waves.**4-22**x z y z = 0**E4.2**x z < 0 z > 0 z y z z = 0**Review Questions**4.1. Outline the procedure for obtaining from the two Maxwell’s equations the particular differential equations for the special case of J = Jx(z, t)ax. 4.2. State the wave equation for the case of E = Ex(z, t)ax. Describe the procedure for its solution. 4.3. What is a uniform plane wave? Why is the study of uniform plane waves important? 4.4. Discuss by means of an example how a function f(t – z/vp) represents a traveling wave propagating in the positive z-direction with velocity vp. 4.5. Discuss by means of an example how a function g(t + z/vp) represents a traveling wave propagating in the negative z-direction with velocity vp.**Review Questions (Continued)**4.6. What is the significance of the intrinsic impedance of free space? What is its value? 4.7. Summarize the procedure for obtaining the solution for the electromagnetic field due to the infinite plane sheet of uniform time-varying current density. 4.8. State and discuss the solution for the electromagnetic field due to the infinite plane sheet of current density Js(t) = – Js(t)ax for z = 0.**Problem S4.1. Writing expressions for traveling wave**functions for specified time and distance variations**Problem S4.2. Plotting field variations for a specified**infinite plane-sheet current source**Problem S4.3. Source and more field variations from a given**field variation of a uniform plane wave**4.2 Sinusoidally Time-Varying Uniform Plane Waves (EEE, Sec.**3.5; FEME,Secs. 4.1, 4.2, 4.4, 4.5)**4-36**The solution for the electromagnetic field is**4-38**Parameters and Properties**4-42**E4.3 Then Direction of propagation is –z.**4-43**E4.4 Array of Two Infinite Plane Current Sheets**4-45**For both sheets, No radiation to one side of the array. “Endfire” radiation pattern.**Review Questions**4.9. Why is it important to give special consideration for sinusoidal functions of time and hence sinusoidal waves? 4.10. Discuss the quantities ω, β, and vp associated with sinusoidally time-varying uniform plane waves. 4.11. Define wavelength. What is the relationship among wavelength, frequency, and phase velocity? What is the wavelength in free space for a frequency of 15 MHz? 4.12. How is the direction of propagation of a uniform plane wave related to the directions of its fields? 4.13. What is the direction of the magnetic field of a uniform plane wave having its electric field in the positive z- direction and propagating in the positive x-direction?**Review Questions (Continued)**4.14. Discuss the principle of antenna array, with the aid of an example. 4.15. What should be the spacing and the relative phase angle of the current densities for an array of two infinite, plane, parallel current sheets of uniform densities, equal in amplitude, to confine the radiation to the region between the two sheets?**Problem S4.4. Finding parameters and the electric field for**a specified sinusoidal uniform plane wave magnetic field