Table Of ContentSolutions Manual For
CONTINUUM ELECTROMECHANICS
James R. Melcher
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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')')
