HW6solutions - 54 ML i o n ECE6'O4 Homework 6 Out: Tuesday,...

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Unformatted text preview: 54 ML i o n ECE6'O4 Homework 6 Out: Tuesday, February 24, 2004 (Session 13) Due: Tuesday, March 2, 2004 (Session 15) [Students with tape delays: due according to session number] Problem numbers are from the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994). 1) 3.7c 2) 3.110 3) 3.18a 4) 3.18b 5) In the magnetoquasistatic approximation to Maxwell’s equations, the displacement current is assumed to be much less than the conduction current, so that Ampere’s Law may be written: V x H = J Assuming that this approximation holds, show that Maxwell’s equations for a spatially uniform medium with finite conductivity (5 lead to the magnetic diffusion equation, written VZH =HG§E [it Note: You may assume J = 6E and B = uH. 6) A spherical conductor of radius a is exposed to a uniform time—varying magnetic field in the z-direction: B(t) = Bosin wt 2 It may be assumed that the radius a of the sphere is very small compared to the skin depth 5, given by 8 = 1/(7tfuo) where o is the conductivity, f is the frequency, and u is the permeability, assumed to be no everywhere. Derive an expression for the time-average power dissipated in the sphere. Z Conducting sphere of radius 21 MW 7) Consider a simple parallel plate capacitor connected to a sinusoidal voltage source V(t) = V0 sin cot, as shown. |«-d—>l The radius of each of the round conducting plates is a, and the separation between the plates is d. The medium between the plates is a perfect insulator with permittivity 8, and permeability uo . Let d<<a<<9t, where 7» is the wavelength that a fieely— propagating wave at frequency co would have in this dielectric medium. (a) Using one or more of Maxwell’s equations (in integral or differential form, your choice) determine the D field in the region between the plates. Express your answer in terms of V0, 0), a, d, S, no and/or X. State all assumptions and approximations that you are making. (b) A time—varying D field is associated with a circulating magnetic field. Show that aVooor A the vector field H in the region between the plates is given by H = cos wt 4) where (i) is the unit vector in the 4) direction, as shown on the diagram above. Show the contour, surface, or volume that allows you to make this determination. State all assumptions and approximations that you are making. (WM) (WE) magnetic field energy and WE is the total electric field energy Within the capacitor. Write your answer as a function of yo, 8, a and 9». Discuss the relative magnitude of the ratio. Under what conditions should your results be regarded as valid? (c) Determine the ratio of the time-averaged energies , where WM is the total ((1) Determine the instantaneous Poynting vector within the dielectric medium. Discuss the meaning of its direction and its dependence on t and r. 3.4”, H v.3 sf... (magnetic, phat-See) -_‘_A ,_ A I 5%w.3) =V.§ 3m. butag'. =,"_.v.;ym bbflwnfifiuwfiu. ‘ ) $0 to V“? =‘ “3% , mud: add m ncbic' current: then V. vaE) 3 0= '50- [in + §%3 ‘m _ dfir'eemgnt with above, .3? :k Z 6 2. __oo3é/..é ._ _. .9 [1°44 .Cé.°:/‘...Oi_l../|Mé1.z__. ‘\ . ‘P/égsm w M_,/IM‘_‘_— , /. . E35 1 ‘ , '55 r3 -' bl?" 0" o‘-zy‘ ‘3‘ A‘ a > F, - am w» =3o 4x =J. E—‘ié—ax =iifz-r-[riaww ...‘|___.=r°‘ e I me Prob. '_3.|'1b' or Er“. 341 (J) I, 3-5 = $0 ‘3.“ . . J! t_, 50 w.=—1L:g—= L5H. I / / .3 .L L i A . ; 1k... .. .. ., = 4 a 4 I, _. Agnyigaww > “2 (3 flag; .. H. ....4/9,,7= .2 [lib/2’ 12.5.. .‘ . fl; 2 fl_,¢.§%55..43. swfow ' 3/85“ . . .. .— -— R __L ’1 2.. .. . .. 24.1% . _, J > «To 6 am 4 a 4 . _ ' .. L 5. I3 f _._.;._L__._ L...“ ._ .. _. . __ . A 4 UM, ' .3 .. _. t‘é. . I H “4-561? NW4). 5:; 2H Powhss 2 .Fm A‘Ummw ).. 325...;- ‘okw . 5? 3529.190”? is 0.6M.) VLHHQ.» ,. ‘ 3* HT“ . . ., . , , 1"“ ‘ 5”“ .I‘ ,- " are? is? :xo?Xm~%~- Fm Min/fl M». /oss :5 mes/:24? /o<f HIV/M2. if) In the magnetoquasistatic approkimation to Maxwell’s equations, the displacement current is assumed to be much less than the conduction current, so that Anipere’s Law may be written: V VXH=J . Assuming that this approxiination holds, Show that Maxwell’s equations for a spatially uniform medium with finite conductivity'o lead to the magnetic diffusion equation, written ' . an. V2H= "—— r” at > Note: You may assumeJ=oE andB=MH. . "J . i V 4 —- _ Q "FM-Si ‘4’- ' 5er; ma WW. J M.“ M" 6 5- ~ ' -V’Lk—t: (sing V'H"o) I ‘ ——-—>. 4+ VCV° V KT =' Frwimaaw‘s (m, We" aaw- Vaz- .. 14., r v 3‘: ,xfo-a'i 505M556. ,y, /.M 2, +5.5 759V. I" perk WM (5 saw cs :& :Bosmfié‘. 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This note was uploaded on 12/12/2010 for the course ECE 604 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.

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HW6solutions - 54 ML i o n ECE6'O4 Homework 6 Out: Tuesday,...

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