1School of Electrical and Computer Engineering, Cornell University ECE 303: Electromagnetic Fields and Waves Fall 2007 Homework 6 Due on Oct. 05, 2007 by 5:00 PM Reading Assignments: i) Review the lecture notes. ii) Review sections 1.5, 3.3-3.6 of the paperback book Electromagnetic Waves. Special Note: Graders have been instructed to take off points (as much as 50%) if proper units are not included in your answers. You must specify the correct units with your numerical answers. Problem 6.1: (Conductive Media) Consider a plane wave in a conductive medium given by the expression: ()zjkoeExrE−=ˆrr The medium has a conductivity σ and a dielectric permittivity ε. As explained in the lectures, the wave decays because the wave looses energy due to dissipation in the medium. The Poynting theorem is: () () ()[]()()trEtrJtrWtrWttrSme,.,,,,.rrrrrrrr++∂∂=∇− The time-average version of the Poynting theorem for time-harmonic fields is (as in homework 4): () ()()trEtrJtrS ,.,,.rrrrrr=∇− Using phasors this becomes: ()[]() ()[]rErJrSrrrrrr*.Re21.Re21=∇− This is saying that if a wave is slowly loosing energy, and so the Poynting vector is a function of position, then there must be dissipation going on. Verify the above relation for the plane wave propagating in a conductive medium given above (without making any “lossy dielectric” or “imperfect metal” approximations). You need to evaluate the left and right sides of the equation separately and show that they are equal. Hint: First show (without making any approximations) that for any conductive medium: σµωokk ='''2 . Problem 6.2: (Ground Penetrating Radar - GPR) A conductor is considered a good conductor for the frequency ω of interest if the loss tangent εωσ is much greater than unity (i.e. 1>>εωσ) and a bad conductor if 1<<εωσ. These conditions can also be stated in terms of the dielectric relaxation time σετ=d: Good conductor: 1<<dτω (This is also the “imperfect metal” case discussed in the lecture notes) Bad conductor: 1>>dτω (This is also the “lossy dielectric” case discussed in the lecture notes)2Solid ground has a conductivity of approximately 3105−× S/m and a permittivityε equal to oε10 . You need to design a ground penetrating radar (GPR). You have at our disposal two sources of electromagnetic radiation – one at a frequency of 10 kHz and the other one at a frequency of 100 MHz. a) Figure out if ground is a good or a bad conductor at each of the two frequencies: 10 kHz and 100 MHz (don’t forget that fπω2= ). b) Figure out the penetration depth of electromagnetic radiation in ground at the two frequencies: 10 kHz and 100 MHz. Take penetration depth to be the depth at which the time average power in the electromagnetic radiation going into the ground drops to 1% of its value at the surface. c) To clarify things a bit, calculate and plot (using matlab or your favorite plotting software) the exact penetration depth (as defined in part (b) above), without making any “good conductor” or “bad conductor” approximations, as a function of the frequency from 1 KHz to 1 GHz using a log frequency scale. Indicate in your plot frequency regions for which the ground acts like a “good conductor” and for which the ground acts like a “lossy dielectric” (or a “bad conductor”). d) If you have to use the radar to image an object at a depth of 50 m, and you need the power reflected back to you from the object to have dropped to no more than 1% of its original starting value, what frequency (in Hz and not in rad/s) would you choose for the radar in order to image the smallest possible object. Hint: It is difficult to image objects smaller than the wavelength of the radiation being used for imaging. Visit this link if you want to see some nice pictures from GPRs: http://www.geomodel.com/ Problem 6.3: (Atmospheric plasmas) The Earth’s ionosphere can be modeled as a Plasma. The gas molecules in the upper layers of the Earth’s atmosphere are ionized by Solar and Cosmic radiation resulting in a Plasma. Suppose the density N of (singly ionized) molecules in the ionosphere is 1210 1/m3. The electron mass m is 31101.9−× kg. The electron charge e is 19106.1−x Coulombs. You have been hired by NASA to design a wireless communication system to communicate with a deep space probe. You need to select a frequency for your communication system (i.e. the frequency of the electromagnetic wave with which you will communicate with the deep space probe). What is the smallest possible frequency (in Hz NOT in rad/sec) that you can select and still be able to communicate with the deep space probe from somewhere on Earth without having your signals reflect back from the ionosphere. GroundGPR3 Problem 6.4: (Uniaxial medium and waveplates) Consider a uniaxial medium with the permittivity tensor given by: Waveplates are used in optics to control the polarization of light. In this problem, you will use the uniaxial medium to design waveplates for operation with green light (wavelength of green light in free-space is equal to 6105.0−×m). For the following parts, assume that the waveplate can be rotated around the y-axis to suitably and optimally orient the principal axes of the waveplate w.r.t. the polarization of the incident wave as desired by the application. It is critical that you understand what is being said here, The diagram above shows the principal axes of the waveplate (the y-z-x directions). All waves will be assumed to be traveling in the +y-direction in the medium. The answers to questions below are not trivial and will test your understanding of the material. a) Which axis is the extra-ordinary axis of the medium? And which axes are the ordinary axes? b) Which axes are the slow axes of the medium? And which axes are the fast axes? Ly z x ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=900040004oεε EarthIthaca, NY Ionosphere4 c) What is the minimum thickness L of the waveplate (answer in meters) that will let one convert any linearly polarized incident wave into a right-hand circularly polarized output wave? Explain in detail your answer. How would you orient the axis of the waveplate w.r.t. the incident polarization direction to get a right-hand circularly polarized output wave for the minimum thickness you calculated? Hint: Your answer should include a diagram that shows the orientation of the principal axes of the