School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

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1 School of Electical and Compute Engineeing, Conell Univesity ECE 33: Electomagnetic Fields and Waves Fall 7 Homewok 6 Due on Oct. 5, 7 by 5: PM Reading Assignments: i) Review the lectue notes. ii) Review sections 1.5, of the papeback book Electomagnetic Waves. Special Note: Gades have been instucted to take off points (as much as 5%) if pope units ae not included in you answes. You must specify the coect units with you numeical answes. Poblem 6.1: (Conductive Media) Conside a plane wave in a conductive medium given by the expession: jk z E( ) = xˆ E o e The medium has a conductivity σ and a dielectic pemittivity ε. As explained in the lectues, the wave decays because the wave looses enegy due to dissipation in the medium. The Poynting theoem is:. S(, t ) = [ We (, t ) + Wm (, t )] + J(, t ). E(, t ) t The time-aveage vesion of the Poynting theoem fo time-hamonic fields is (as in homewok 4):. S (, t ) = J(, t ). E(, t ) Using phasos this becomes: 1 1 * Re[. S( )] = Re[ J( ). E ( )] This is saying that if a wave is slowly loosing enegy, and so the Poynting vecto is a function of position, then thee must be dissipation going on. Veify the above elation fo the plane wave popagating in a conductive medium given above (without making any lossy dielectic o impefect metal appoximations). You need to evaluate the left and ight sides of the equation sepaately and show that they ae equal. Hint: Fist show (without making any appoximations) that fo any conductive medium: k ' k' ' = ω µ o σ. Poblem 6.: (Gound Penetating Rada - GPR) A conducto is consideed a good conducto fo the fequency ω of inteest if the loss tangent σ ω ε is much geate than unity (i.e. σ ω ε >> 1) and a bad conducto if σ ω ε << 1. These conditions can also be stated in tems of the dielectic elaxation time τ d = ε σ : Good conducto: ω τ d << 1 (This is also the impefect metal case discussed in the lectue notes) Bad conducto: ω τ d >> 1 (This is also the lossy dielectic case discussed in the lectue notes) 1

2 3 Solid gound has a conductivity of appoximately 5 1 S/m and a pemittivityε equal to 1 ε o. You need to design a gound penetating ada (GPR). You have at ou disposal two souces of electomagnetic adiation one at a fequency of 1 khz and the othe one at a fequency of 1 MHz. a) Figue out if gound is a good o a bad conducto at each of the two fequencies: 1 khz and 1 MHz (don t foget that ω = π f ). b) Figue out the penetation depth of electomagnetic adiation in gound at the two fequencies: 1 khz and 1 MHz. Take penetation depth to be the depth at which the time aveage powe in the electomagnetic adiation going into the gound dops to 1% of its value at the suface. c) To claify things a bit, calculate and plot (using matlab o you favoite plotting softwae) the exact penetation depth (as defined in pat (b) above), without making any good conducto o bad conducto appoximations, as a function of the fequency fom 1 KHz to 1 GHz using a log fequency scale. Indicate in you plot fequency egions fo which the gound acts like a good conducto and fo which the gound acts like a lossy dielectic (o a bad conducto ). d) If you have to use the ada to image an object at a depth of 5 m, and you need the powe eflected back to you fom the object to have dopped to no moe than 1% of its oiginal stating value, what fequency (in Hz and not in ad/s) would you choose fo the ada in ode to image the smallest possible object. Hint: It is difficult to image objects smalle than the wavelength of the adiation being used fo imaging. GPR Gound Visit this link if you want to see some nice pictues fom GPRs: Poblem 6.3: (Atmospheic plasmas) The Eath s ionosphee can be modeled as a Plasma. The gas molecules in the uppe layes of the Eath s atmosphee ae ionized by Sola and Cosmic adiation esulting in a Plasma. Suppose the density N of 1 (singly ionized) molecules in the ionosphee is 1 1/m The electon mass m is kg. The 19 electon chage e is 1.6x 1 Coulombs. You have been hied by NASA to design a wieless communication system to communicate with a deep space pobe. You need to select a fequency fo you communication system (i.e. the fequency of the electomagnetic wave with which you will communicate with the deep space pobe). What is the smallest possible fequency (in Hz NOT in ad/sec) that you can select and still be able to communicate with the deep space pobe fom somewhee on Eath without having you signals eflect back fom the ionosphee.

3 Ionosphee Ithaca, NY Eath Poblem 6.4: (Uniaxial medium and waveplates) Conside a uniaxial medium with the pemittivity tenso given by: ε = ε o z x y Waveplates ae used in optics to contol the polaization of light. In this poblem, you will use the uniaxial medium to design waveplates fo opeation with geen light (wavelength of geen light in feespace is equal to m). Fo the following pats, assume that the waveplate can be otated aound the y-axis to suitably and optimally oient the pincipal axes of the waveplate w..t. the polaization of the incident wave as desied by the application. It is citical that you undestand what is being said hee, The diagam above shows the pincipal axes of the waveplate (the y-z-x diections). All waves will be assumed to be taveling in the +y-diection in the medium. The answes to questions below ae not tivial and will test you undestanding of the mateial. a) Which axis is the exta-odinay axis of the medium? And which axes ae the odinay axes? b) Which axes ae the slow axes of the medium? And which axes ae the fast axes? L 3

4 c) What is the minimum thickness L of the waveplate (answe in metes) that will let one convet any linealy polaized incident wave into a ight-hand ciculaly polaized output wave? Explain in detail you answe. How would you oient the axis of the waveplate w..t. the incident polaization diection to get a ight-hand ciculaly polaized output wave fo the minimum thickness you calculated? Hint: You answe should include a diagam that shows the oientation of the pincipal axes of the waveplate w..t. the polaization of the incident wave as desied by the application. Fo example, to answe pat (c) you should daw something like the following: diection of the incident polaization z x and say that to get ight-hand cicula polaization I will otate the waveplate so that the diection of incident linea polaization is at 45-degees to the z-axis of the waveplate (as shown in the figue) and then choose thickness L of the waveplate such that. You should use the same pocedue to answe all the pats that follow. d) What is the minimum thickness L of the waveplate (answe in metes) that will let one convet any linealy polaized incident wave into a left-hand ciculaly polaized output wave? Explain in detail you answe. How would you oient the axis of the waveplate w..t. the incident polaization diection to get a left-hand ciculaly polaized output wave fo the minimum thickness you calculated? e) What is the minimum thickness L of the waveplate (answe in metes) that will let one convet a ighthand ciculaly polaized incident wave into a left-hand ciculaly polaized output wave and vice-vesa? Explain in detail you answe. f) What is the minimum thickness L of the waveplate (answe in metes) that will let one otate the polaization diection of any linealy polaized incident wave though an abitay angle at the output? Explain in detail you answe. How would you oient the axis of the waveplate w..t. the incident polaization diection to get the desied otation angle fo the output wave fo the minimum thickness you calculated supposing the desied otation angle was 3-degees? Poblem 6.5: (Real metals: conductos o plasmas?) In this couse you wee told that the esponse of metals to an electomagnetic wave can be modeled as that of conductos with conductivity σ, and can be descibed by an effective dielectic pemittivity given by: σ ( ) ε eff ω = ε o 1 j ω ε o Notice the switch fom ε to ε o in the above expession compaed to lectue notes fo simplicity. You wee also told that electons (and atoms) in metals can be modeled as a plasma with a pemittivity given by: ( ) ωp ε ω = ε o 1 ω 4

5 And so the question aises which one of the above expessions gives the coect pemittivity of eal metals. In this poblem you will exploe this question moe caefully. It tuns out that both the above expessions can be coect depending on the fequency of the electomagnetic wave. Conside a metal in which the density of electons is N. The electon mass is m. The electon chage is e. In the pesence of an electomagnetic wave, the electons ae acceleated by the E-field. Howeve, in all eal metals as a esult of electon collisions with impuities o defects in the metal the aveage motion of the electons is not eally descibed by the simple equation in the lectue notes: d(, t ) e = E(, t ) t m but by the equation: d(, t ) 1 d(, t ) e + = E(, t ) (1) t τ c t m whee the constant τ c is the mean time between collisions, and 1 τ c is the collision fequency. The effect of collisions on electon motion has been included as a damping effect in equation (1). Collisions damp the motion of electons in the same way as fiction damps the motion of any eal pendulum. a) Assuming time-hamonic fields, convet equation (1) into phaso notation and solve it to find the value d E. of the phaso ( ) in tems of the E-field phaso ( ) b) Following the ecipe outlined in the lectue notes, find an expession fo the polaization phaso ( ) (capital P ). ε of eal metals, and show that it can be witten in the fom: σ ε ( ω) = ε o 1 j ωε o ( 1+ j ωτ c ) What is the expession fo the conductivity σ that comes out of you analysis? Those of you taking ECE315, does the expession fo conductivity seem coect? c) Using you esults fom pat (b), find an expession fo the dielectic pemittivity ( ω) d) When the fequency of electomagnetic wave is much less than the electon collision fequency (i.e. when ω τ c << 1) show that the expession fo pemittivity in (c) educes to that of a conducto. e) When the fequency of electomagnetic wave is much lage than the electon collision fequency (i.e. when ω τ c >> 1) show that the expession fo pemittivity in (c) educes to that of a plasma. Gold chaacteistics: Gold is one of the most commonly used mateials in micowave devices and cicuits, integated antennas, and in metallic mios fo optics. Below you will look at some of the featues of this mateial. 13 f) The mean time τ c between collisions in gold is aound 1 sec. So fo fequencies much much 1 smalle than 1.6 THz (note that 1. THz = 1 Hz), gold can be appoximately modeled as a conducto. Find the conductivity σ of gold assuming: P 5

6 8 N = / m 19 e = C 31 m = kg τ 13 c = 1 Sec 3 g) Using you answe fom pat (f), find the skin-depth in gold of electomagnetic waves of fequencies 1 Hz, 1 KHz, 1 MHz, and 1 GHz, and 1 THz. h) Fo fequencies much much lage than 1.6 THz, gold can be appoximately modeled as a plasma. Find the plasma fequency of gold assuming: 8 3 N = / m 19 e = C 31 m = kg 13 τ c = 1 Sec i) Figue out (fom a Google seach pehaps) what colo of light (e.g. X-ay, Ulta-Violet, Violet, Blue, Geen, Red, Infa-Red, etc) does the plasma fequency you calculated above coesponds to. 6