Solutions Manual For CONTINUUM ELECTROMECHANICS James R. Melcher £12; / / / Copyright© 1982 by The Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the author. ACKNOWLEDGMENTS Although the author must take responsibility for the solutions given here, each student taking the subjects CONTINUUM ELECTROMECHANICS I and II has made a contribution to this Solutions Manual. Some that have made substantial contributions beyond the "call of duty" of doing homework are: Kent R. Davey, Richard S. Withers, Peter W. Dietz, James F. Hoburg, Alan J, Grodzinsky, Kenneth S. Sachar, Thomas B. Jones, Markus Zahn, Edmund B. Devitt, Joseph M. Crowley, Robert J. Turnbull, Jose Ignacio Perez Arriaga, Jeffrey H. Lang, Richard M. Ehrlich, Mohammed N. Islam, and Paul Basore. Eleanor J. Nicholson helped the author with the typing. Introduction to Continuum Electromechanics ( 1.1 CONTINUUM ELECTROMECHANICS Used as a Text Much of Chap. 2 is a summary of relevant background material and care should be taken not to become mired down in the preliminaries. The discussion of electromagnetic quasistatics in the first part of Chap. 2 is a "dry" starting point and will mean more as later examples are worked out. After a brief reading of Secs. 2.1-2.12, the subject can begin with Chap. 3. Then, before taking on Secs. 3.7 and 3.8, Secs. 2.13 and 2.14 respectively should be studied. Similarly, before starting Chap. 4, it is appropriate to take up Secs. 2.15-2.17, and when needed, Sec. 2.18. The material of Chap. 2 is intended to be a reference in all of the chapters that follow. Chapters 4-6 evolve by first exploiting complex amplitude representa tions, then Fourier amplitudes, and by the end of Chap. 5, Fourier trans forms. The quasi-one-dimensional models of Chap. 4 and method of character istics of Chap. 5 also represent developing viewpoints for describing con tinuum systems. In the first semester, the author has found it possible to provide a taste of the "full-blown" continuum electromechanics problems by covering just enough the fluid mechanics in Chap. 7 to make it possible to cover interesting and practical examples from Chap. 8. This is done by first covering Secs. 7.1-7.9 and then Secs. 8.1-8.4 and 8.9-8.13. The second semester, is begun with a return to Chap. 7, now bringing in the effects of fluid viscosity (and through the homework, of solid elasticity), As with Chap. 2, Chap. 7 is designed to be materials collected for reference in one chapter but best taught in conjunction with chapters where the material is used. Thus, after Secs. 7.13-7.18 are covered, the electromechanics theme is continued with Secs. 8.6, 8.7 and 8.16. 1.2 Coverage in the second semester has depended more on the interests of the .class. But, if the material in Sec. 9. 5 on compressible flows is covered, the relevant sections of Chap. 7 are then brought in. Similarly, in Chap. 10, where low Reynolds number flows are considered, the material from Sec. 7.20 is best brought in. With the intent of making the material more likely to "stick", the author has found it good pedagogy to provide a staged and multiple exposure to new concepts. For example, the Fourier transform description of spatial transients is first brought in at the end of Chap. 5 (in the first semester) and then expanded to describe space-time dynamics in Chap. 11 (at the end of the second semester). Similarly, the method of characteristics for "first order" systems is introduced in Chap. 5, and then expanded in Chap. 11 to wave-like dynamics. The magnetic diffusion (linear) boundary layers of Chap. 6 appear in the first semester and provide background for the viscous diffusion (nonlinear) boundary layers of Chap. 9, taken up in the second semester. This Solutions Manual gives some hint of the vast variety of physical situations that can be described by combinations of results sununarized throughout the text. Thus, it is that even though the author tends to discourage a dependence on the text in lower level subjects (the first step in establishing confidence in field theory often comes from memorizing Maxwell's equations), here emphasis is placed on deriving results and making them a ready reference. Quizzes, like the homework, should encourage reference to the text. 2 Electrodynamic Laws, Approximations and Relations r------r-/-- - ----- ~ ill{] j } / 2.1 Prob. 2.3.1 a) In the free space region between the plates, J =P=M=O and V Maxwell's equations, normalized in accordance with Eqs. 2.3.4b are ~H q )(. E. -~ (1) C.~) VX\-\ = ,6'~ -;)t - (1) V·E = 0 (,1) V·\-\ : o For fields of the form given, these reduce to just two equations. ~'Ex = -~'.i ~ ~ ~t -t8~Ex ~'i = ) i. 'c)t Here, the characteristic time is taken as 1/c...> so that time dependences exp j cJ t take the form (1) > 't i. For the time-rate expansion, the dependent variables are expanded in ;:3 = "".,M'€ .X " = L00 A 11 E El(~ (§) ,<'.3 X l'l:O ) so that Eqs. 5 and 6 become -alf l f ~ EK" f./']:: H,n ;3~ J (2) [to o,l. l'I • 0 flt: ti ~1 H~n toE.,.n[3 ~ 1-3"] =- j{Jl (19) .. Equating like powers of ;.S results in a hierarchy of expressions -;;, t ... i \.\ CU) d :a.,."" '::. - tr\ ~ ~'11" d. ~ (12) =. - L. >e(n-1) Boundary c~ditions on the upper and lower plates are satisfied identically. (No tangential E and no normal Bat the surface of a perfect conductor.) At z=-j , z=O where there is also a perfectly conducting plate, Ex=O. At Ampere's law requires that i/w=H (boundary condition, 2.10.21). (Because w)") s, the - y magnetic field intensity outside the region between the plates is negligible compared to that inside.) With the characteristic magnetic field taken as I /w, 0 where i (t) = i.Ct) I , it f.o. llows that the normalized boundary conditions are 0 la E,.( o) = o \-\ (-1)-: 1 > (13) ~ 2.2 Prob. 2.3.l (cont) .. The zero order Eq. 12 requires that = ;>H10 o (1]) ~:a and reflects the nature of the magnetic field distribution in the static limit /3-+ o. The boundary condition on H , Eq. 13, evaluates the integration y constant. A H~o 1 The electric field induced through Faraday's law follows by using this result in the zero order statement of Eq. 11. Because what is on the right is independent of z, it can be integrated to give ... -a = E~o r (15) Here, the integration constant is zero because of the boundary condition on E , Eq. 13. These zero order fields are now used to find the first order fields X The n=l version of Eq. 12 with the right hand side evaluated using Eq. 15 can be integrated. Because the zero order fields already satisfy the boundary conditions, it is clear that all higher order terms must vanish at the appropriate boundary, E at z=O and H at z=l. Thus, the integration constant is evaluated xn yn and (,16) This expression is inserted into Eq. 11 with n=l, integrated and the constant evaluated to give 3 E• =.-,1 I- ( -I i! -2 ) (17) ><I d'~ 3 If the process is repeated, it follows that l• -\p = 4I ( cI ; ~ o\, - (18) E =- l) i .!.. ( ..!_ ~ S' - .1 ~ 3 + £ c.gi) )(.'2 d 4- 30 3 ~ so that, with the coefficients defined by Eqs. 15-19, solutions to order are 4 = f3 iA"' I\ A 'Z. E" C + t>'I + C (.3 ; H~ = \-\~ + \..\~, {3 + H~t. /.$ ,c.o X.2. 0 (£9) 2.3 Prob. 2.3.l(cont.) Note that the surface charge on the lower electrode, as well as the surface current density there, are related to the fields between the electrodes by (~) The respective quantities on the upper electrode are the negatives of these quantities. (Gauss'law and Ampere's law). With E1s, 7 used to recover the time /3 dependence, what have been found to second order in are the normalized fields E~=~l1-\(ic1.-1),8+~(~ c4 ~c1.+i)f311~ t = '1 (~) - 0 H = [ 1- ~(t1.-1)'5 + ~(ic~ - c~+ ~)l-3z1 c:.,k..t =\ -{~ (~) 1 The dimensioned forms follow by identifying E :_,P' w) Io (24) 0 w ~ e) Now, consider the exact solutions. Eqs. 7 substituted into Eqs. 5 and 6 t• give ~ +!-3 Hl4 o d-l_L ... ti ( 25) E i = J "~ " fl° di (~) Solutions that satisfy these expressions as well as Eqs. 13 are ... He;)-= ~(~ l)/~ ~ ( 27) E~ =-i ~(~i)/ ~$ (~ ti' These can b~ expanded to second ord er i. n ~/2 as follows. I '2. I 'Z. ~ 1-J:,->~ +4~(3 l -\··· (~) I - ~ f3 +- ~'fl.,_.\--· · · - ( I - -h-/9 / +{/l')(1- H~. ~l)+ (-1~+ ~!13')')