Solutions Manual for
Applied Electromagnetism SECOND EDmON
•
Shen
Huang
,
solutions Manual for
Shen and Kong ' s APPLIED ELECTROMAGNETI SM Second Edition
by
Liang C. Shen and Frank S . C. Huang
rlB 8C PWS-KENT Publishing Company Bos ton
.'
PWS-KENT Publishing Company 10 P"fk PI1.U Blmon. MUIKhowcu 0 21 16
Copyright c> 1987 by PWS Publishers.
All rights reserved.
No part of this book may be reproduced,
stored in a retrieval system, or transcribed, in any form or by any means - electronic, mechanical, photocopying recording or otherwise - without the prior wri t ten permission of PWS-KENT Publishing Company. J
PWS-KENT Publishing Company is a division of Wadsworth, Inc. ISBN 0-534-07621-1
Printed in the United States of America 91 -- 10
9
8
7
6
5
4
CONTENTS
Chapter 1
Complex Vectors
1
Chapter 2
Maxwell ' s Equations
5
Chapter 3
Uniform Plane Waves
8
Chapter 4
Reflection and Transmission of Waves
12
Chapter 5
Waveguides and Resonators
16
Chapter 6
Transmission Lines
20
Chapter 7
Antennas
25
Chapter 8
Topics in Waves
30
Chapter 9
Electrostatic Fields
32
Chapter 10
Electric Force and Energy
36
Chapter 11
Solution Techniques
40
Chapter 12
Direct Currents
44
Chapter 13
Magnetostatic Fields
49
Chapter 14
Magneti c Materials and Magnetic Circuits
54
Chapter 15
Electroquasistatic Fields
56
Chapter 16
Magnetoquasistatic Fields
58
r 17
(eclo rs
Problems
"Rand
NOlicethal A · 8 * - -2pmrllhat (All). S (I) -11/ 2IReIA . S ' J - O. We should a lllO nollilhal A . B _ 0 a nd Ihal A ll] . 0 11]- 0 Thu$.lhe Iwo tlme-doma in Vf!CIOrll ate always perpeod icular 10 each OlhAr.
Their
Exa mple 1.20 r.nnJlliriAt Ihe two vec tON A _ i + iy a nd B - t t /y. IThey a re actually Ihe same vector.) We fi nd tha t A )( 8 - 0 and tha t A . B - O. Are Ihe Iwo veclOrll para llel 10 each othe r or perpenrlir: ular In eitch other?
wi
Solution :
Con ~ itl er
ins lead A x B ' a nd A • S' We fi nd tha i A x 8 ft - -t2 ;and Ihal A _ 8 ft _ 2. Thus. ( A ll) x 8 [1» - O. a nd tha i ( A II)- BIt» - 1. Furt hermore. A fl} x Blr) _ O. and Al II· 0 1'\ - 1. TllU~. Ihe time-domain veclor Is parallel to itself all the time.
Problems 11.19)
to half . B. B' . ler Ihe
1. 1 Lei a _ 8 + ;2 a nd h - - 3 + ;. Ca lculate fa) a t b, Ih) a - h, fl:) ab. lind (dl alb. Cive IhA a n ~wer in real a nd imaginary parts. 1.2 Repea l tc] and Id) in Pmb hUIl 1.1 wilh Ihc answer given in phasot form . 1.3 Find the rea l ~ rl, IhA imHginHry !Jitrl. and Ihc magnit ude of e'"'. whArA '" andt are rAaI numberll. · 1.4 lei c be a complpx number. Are the following sta temen ts alwaylllnlA? ( a) (c I c ' )isreal ( b) (I: - 1:' \ is ima,l(illury. (e) clc' has a magn it ude equal tn L 1 .5 Consider Ihp. P.
..
1. Show tha t the square rool nf (t ... la] is
1.7 Lei 0 be 11 JXlSiTive tp.al nllmber. amI 11:\ a ... 1. Showlhallhe sqUMft rool nfft + ju) apilroxinu:lldy t:4 uallo 1. (1 I iKa/2I" J· .... 8 Obtain the
(1 .20)
pha.~or
nolalion uf Ihe following time-ha rmonic fu nl:lions (if
i~
po~ilJ l e):
( a) Vltj_ Gcos!"" + .. 14\ ( b) 111 1- - 88illl",1) tc) AIII _ 3sinlwl) 2eos( ... 11 (d) t:(1j - 6 I:OS !l 20 .. 1 - .. / 21
! e) 011) - 1 -
eos (",'I
( I) Uitl - sin ("" ..... / 3)lIin Iwl + .. / 6) -domain ~' tJ have - f:!. and
1.8 Obtain GII\ in termsaf '" from Ihe following Ilha)l(jfll: [H)C - 1 and tC) C - 3 exp! ; .. / 21 ... " exp I/O.tI).
+ ;. (b) C -" exp 1/0.8 \.
-1. 10 Show Ihal. if V - r + /X and U _ g I 1}'. lhen VIIlI1!1J *- Rft!VU e""I. Find the corrpct eX!Jrt:SSion for VII) L1 II) in tftrms of r. x. ~. y. arId ",t.
1
18 1.it Lt:IA-
fd)"
)C
Ai+!ly
iand l' -2x - 4y
t
3t. FtnrtjajA ... B.(bJ"
Complex Vectors H, jr:IA. B,linu
8.
1.12 Find the angle
~Iwccn
A and 8 that life given In Problellll .11
1. t3 Show [hal for
VII I -V.cosl",11 tbl-ReIVe""l
,
al V(li- ""V 1. 14 Find a VPf:torC that is perpenrlkuluf toA -
8t + Dr - t, has no t component. lind
hall a IIIIIHrlitude equal 10 1.
1.15 "'Inri tht! vector C Ihat ill parallcl 10" - si -
8y
f
21: Ann hltll II mognitudA Aqual
to t
-1.11 Find II unit vectur II that points in the same dlrActiOIl as (Ill arrow drawn from point /\ to vaint B whAre the rectangular c.oordinates of A ,md B UN (1.0.2) and
(- l,:t 21. res pAr.lively. 1.17 Show thai thl! definition offhl! dot product v·u givtlll by (1 .10a) Is equivlilcnl to thlll givtlll by (1.10b). 'Ib simjJli£y the algAhrll. yuu moy choo!!fl thtl coordinates so tlnlt the x axill ill8100g Vand the 7.lIxis i8 pcrpondlr:u lar to Loth V and U. In other words, lat V • ux and U b~ ... c9. R
- 1.1' Prove 11.11b) using the approach SUgg8SI00 in the texl. 1.1. Shuw that the definition of tbe eros!! product V x U giVAn by (1.12) is equivalent to thnt given by {1.14}. To simplify Ibe ii1gebra, do what ill sug8ested in Problem 1.17.
- t .20 Figure "1.20 shows Ihat II vector V is along the x llXi! and a vector U IS on the x-y plane formiog a 135 - angle with the x axis. The magnitudes of those vectors are v pnd u, respeclivt:ly. Usc (1.14) to express V )I U In terml v oDd u. Now, we C8 n also say that the 8ngle betwfMln V 8nd U is 225°. UlW {1.U} and explain why the vector V x U is the same whether or not we choose the angle betW1;!cn them to he 135 - ur 225- . llint: Follow the right-ha nd rule. In determining tho direction vuctor II. your ringers lIIust alwaYR point from V 10 U in the diroction you rnellSUf'C the angle beIwt.'en them. 1.21 Pr(Mt {1.15} usillgtho approar.h suggostod in the t!tXt. _1,22 Find the phoser nota lions of tho following time-harmonic vectors: (ti) V{I} • 3 cos ((oil)! ... 4 sin (wtJ9 ... f: CUll (wi I ..-/2) (bl Etf) • [3 CUll (Wi) I 4 sin twillt ... 8[cos(wl) - sin (wIlli (e) H[I) • 0.5 cos I/o: - ",IIi 1.23 ~'jlld the phasor notatiun of the follnwinK vuc tor:
C(z,I) . ta181) £ (Z.I) where
r. is !liven in Problem 1.22(h).
- t.24 From the following complex ver.lors. fiod C (t) in IArms u( '-iLia) C - i - IY. (h) C _ ,ti - /y),undlcIC-expl - jKz)R ! /exp(,KzlY·
Problems
19
1.2' LetA - J + /9+11 + /2j!,and let B - - R - 11 Ic) A ' S, 811t1 fd) A )( S.
;2)9 I /l, FlndlaIA .,. B, lbj A - B,
, u y 13!i ~
, Flgur. Pt .20
1.21 Find A . A' and Re IA x B') fur the values uf A and H given In l'mhlAm 1.25. 1. 27 Skelch Ihe trace or Ihe tip of Ihe vector All). where (01 A _ .t
jf and where (bl A _
4R + 139. ' . 21 Ca lcu lale A . 8, given A _ i I6 l allli llll~.!I?
+ j2yand B - 2i I jy AreA(I) and 8 (11 perpendicular
e,+ b • S-+j3 (IV 2.-£ • lI+j
,./ (a.)
J;l ~.: , .. jt· I.IS' I1" .pt'
(e)
1l.·l.--2,+jZ
,i, gill. c-Z. Z-j'.4
lz.- -3"'j .J.I'II".S7~ !i'R - 2&.oBLI7'·"" lSI.! :J." L-I... 'J" Ne I tj.nJ . (~~ I", [ ej""tJ• s·~ MIt' I J t ;wtJ ~ J J
&
Ll
(.oIe.J
y.' '.J yes
f;f
(a)
I:L
Z':I-tj -
") y~s {feJ'(2ft"..,.~)
:.z, • .2t$.~j~
z. Z~ ei(~"''P'.) .
.::;.
..,,-".. 1
. zc.:2.~ei(1r#~·)._24.eJ!Ji
=['.fl/..)+.; ..••··· J :::::t. I ,+j(~)J,
!:l
(Ifjo.)ll..
!:J.
0.»1, ("j .. )i{ '" Ija.)J( &±/J(a)'I,.. :t{l+j)('Y.)J{
if
''''«I
.'
J.:! (a)~::b~j!4 lb)I-j8 (c.).d.·j3-2 Cd) s.a-j6 (It.)i,"poJli.61c. fh/hlpot.S"6Ic. l!l (o.) C(06l- R. {"fj)e;'" J • R. f,;r ",i ~e;""l fl "S(AA! .. ~) ,h, C(~)-R'[4ej·" ~;,,,,}. -I~,(..n ... el iC, CCt) !?f 13ej1f{~.J·wt + -Iej···ei,.;t;} _ 3cpJ(,..;t.~) ... 4 "'Hut+".tJ) 1!! 'i:. r#jX "'* V(t)- rCDSuri: -)(s.~£.,)6; U -8+-jy ~ U('t)- g.C()iwt - Yl;l'Ll6lt R( f:i y ~j",,}- R.{((rg-qJ .jl IX'ry lJ t j .. , 1- I r~ -XYJCoJ..,t -(9~+rf) s,,..,t B.t V(') U(~) c ra,os'we +XYS;n'",t - (gx +ry) s... kIt c•• ..,t R. [!til ej.,~ j ,.,.... " , . . . . . . . !J.! ,If, ) 1..8 =-6x,+SY
II
*
Cd) 1.11
~.e.
IAI'I-~x+qY-il'''146 , 1~1=12;-19'3il:(i; C4I5"(:
f./~
23x+211+14z
-S''' . ",,:r 00.;;;
~·O JX a
-o.S~S
.', d...:r
cD~"'(-".9 .. S)= 147. 7-
It Vct)J: -fteV.(Ol (wt"'~»J. -v.c..J .s '~(wt". ¢) .: 1ft! [ j v.we)li". e .
JIoIt
13~t: Vf<)=V.C.'(wt.;)& R.fv.ej~ej...tJ Vt-t) j wv, tj';;s jw ~
. . ?t
!:.!! lit CaX;+'IY"
CJ.A
--
C·A·-8x.,q~.o
'¥'
~
:'.-%,-,.: rt4S
R.fyejwtJ ~:t. v.ej~
-
'" ....... C'A-O, lAJJ..ert. .A=-8)(oI-QY-2
->9
-
J _ ; V.w e.jf
:. ~:T~
Ah.,A/""
- r 14S'(". C=:= 9~+8Y)
,'+,'"
,.,S Let C=";T~Y+};' eIIA-" AX c:t0 IN"."" A.S;-B;+2~ I- c. (- e 3- 2,); +(zx- S ~) y+ (S ~ 1 S ,) i -0 l
,', -Jt- 8 J-o } 2lI-")so 5~+8"so
,,~t
,'.
/E/ -
=/ ~ I
'*
2'-
i
J """ ~'-4~
x"1"l"1 .. )'(.,.4 + 1'+)-/ ...
...
...
C&~ . =C5'·-8Y·2')
,9>
1
~
8
=*~'=Trs" ,".~.r,Ts
_
A
"
=-1X+3!j-4J
-n
-/ . J7
(/. /0 .... ) (/. / 0
"
/ ~ ") =(-2X H)- - 4-})/ 14+r+16 =-(-2X+3J-4J '129 y
~)
-
U
J"'" V·U = Ilb Jim jJ. (j = A. h'T;> (..s 8
~~ e..s8:; J. /1".,1
.' V·v = I\,J 10\" ""J8 :: /. / F L~t A- = A,,, + A~ J" +
.
13
~
C
•
-
A' (If t
13,; '" C," c)"
+
B,;
.. b
-t
... ,... -t '~::I + C, ~
k[{ (3,+c,): + (13.. + c, )~. + (IlJ + c) )lJ
'" A, B," c, A,
+ A, B,
-A· -B + A--· c "" A, 8, + A, (3~ /.19
-
+ A, (, + A.
(j. Ilf )
,;'{'IS
--
-
-
V' u .,
It. ,
,,'&+,.
S/" (}
c.
+ AJ CJ
A
J
so.B'" c Ih'.,'"
~
-
.. ,4J (3)
A, C... + A) 6,," ;\) S
(I. (1)
b...t
/. 20
t
A· (1ft c: ) :: A·a + A· C )'v
Thtrt.j."t. -
x
v
-V xU- = ~ J.~,' $,',19 f' -
V ... U :: V-It s.;'~J!'·)
f
r;: V'''' J"" '" i"
2
"Co J"
""nAB" (3::: P, X of ..... 'j'" J l/ ", "'1'\.
Let A = A, x t A. , Tits}1 , -
1\
A C. ::: C,,Xt C, ,A .. C" J),
-
if ~ C= ; ( B, CJ
:41 (B)«( )
= ) [
-
C. 8) J.r
A.
,... "
-
'"
9( 8J C,
- c) 8,) t )" (
B, (~- CA)
(B,C~ -c,B.)-A, (BjC,-C)B,)]
.. 9[ AJ (B, c) -
c, 8,J-A, ( /3, C. - a.e,J]
.. ; [A, (B/ (, - ~ (/) -A... ( B.. S -aJc,)] o.+~. "f~tr ~."',(,
'if (A' [ ) :. (f3, ; .. 8.. t (A , /3 )
CA)
(I)
~
c
€-
("B-
J
=(c, ~ .. (, ; .. cd) ( .4,8, + A. L3... t
,', 73 (A '[)-C
/' 11.
i .. Br S) ( A, c, + A.. c, ... A C) )
(ft · S)
3; -j 4; .. j
; (3-j4); .,f/("jJ i
o.s-e"i AJ-;
-f.l}
3
~) IJ
)
, J.:.lj (A) ~U.).
i' 'O)We ... ; S/~&.IJt • ' ) . ~ t .. ""0"11 C.) CL:ot --)(At.,,6o\It, (,)
~ ( t ) . oX _
1:ll
cosc",t -It) -
_
.
-
...
YS;~("'" +~J.)
•
A
Co.) t,'!}-(-I'J3)Y""'J)1 (6)
-
~-
§ •
. ....
....
'"
2M +(I-;)Y+(U;) i
( c) .A.~ c. -'-
1. 2'
l'~ c 4-r-CI.j)JY-(l-j3)1
A.A'uU .. jY"(i.j2JZJ-(,i-jY+(I-jz)iJ=I+I';-7 A .§. - U +j Y.(I.j2Jl.] x{-x-(l+jZ) 9-j ~] : (-2'jf)' -
:.Rt[d M§'1--2 t -Y-i 1.17 (~)
e - x-j 9 .-. /
At.) - co• ..,t ;- .. • ;"wt y
y
CI+ jJ
...".f., ACt),
9 -(I+j) i
c I
,.~~It) \. ,wt-o "c~:t
."
''Ii (~)
.d: -4'·j 31
-
A(t) y
c
4c.osloJ'i.
A_ ;"'j2Y J Ee2;+jy .... A·i.2-1l:0 fjw.t AU) s
.:. At.e)
co;wt ; -
,.,.c, 8(-1)
Ar4.
9
2 "',Swt X- ,,;"we; ".1:: p.rp."dicu.U.,- ",t .....,y -I,.,.,e. .
2 $IAwt
Bt-r)-
4-nct.
4
~~-.---~--------............................................~
,
2.5
Solution:
Ell) - Rc!i E. e k
10.
c .... J- il::. COlI/WI - H1.1
HIII _ Rely - E. e -,b w~
_k (
35
Poy nting's T h ftO rfim
Sfl] - E x H ..
1:
1'1 ""' - 5' -k E. cos (wI .. hz) w~
~C09' I.:.lI _
""
1
(S ) .... Re (E)( ' ... 1_ I: -
,-
2
fl·;: 2
I-
11H
d
,.
(U,) ..
(wi - hz)
kz)
----,-!cos1tw t
'""
~~ k'
(Un)" - , ""')1
n
•
kZE'
..
~
ZI.,'I1
U, - - cos n
k
h J
,
Ei
)-
Problems 2.1 Let A .. 5.2 + tsyzy + ~I:: fino 'Q' )( A and 'V • A.
• 2.2 Lei ~ .. xyz: find 't" 41 and v . 'iit/l.
c
2.'
Let (I .. a,i -+ ali + oJ! lind b .. b,i + h" + bll:. Show that equations (2.91./2.11a), Knd /2.121 are true .
- 2 .• Show Ih,'11 'ii'
;It
fa + bl -V x a ... V x b andV · fa t bl-V. a + v· b.
2.5 Show that V(
, "',1- <1>, 't""'J +
"'IV til, and that
'Q' •
(4)A ) ... A· V -+ tt>V · A .
.. 2.e ShowthatV )( 1"'''J-V x A "f"
2.7 In a source-free region. H ..
zy + yi. Does D vary with time?
2.8 What is thtl dllll'Ke density in a region where 0 .. 2xx? 2.' Find the mllgnctic field S(y,I) ouucitttttd with thi! fl lfICotric nllid E(y.t) given as follows: £ (y.I) *' R 0.3 cos(tolt + ky) where tol and It are cO II!Jh:lllts . 2. 10 Express k in terms of the mll.llutllic permeability and dlllleotric permittivity of the medium whAn the eloclromagnetic Oelds oro givoll ill Prubhlm 2.9 In Ii ~oul'Ce- free region.
P.,-
2.11 Lei £ " 8 H l ' lind D , SIItisfy Al'JUfltiOns (2 , lJ-{.!.4) with given " and Let also El • 8 t" satisfy equations (2.ll- IVI] with given 11 lind Pvl' What are the electromo8HI' and D:
"
!Y
IItltic fields due loa currAn t I, and charge P.. where I. -
1, t II and p", - PV l . . p.,? You must show that your proposed 1lOIulion SIIlillfiAJ Maxwell', equations. what is the appropriatA name for the theorem you have just proved?
36
2
Max~II '1
Equation.
_ 2. 12 (8) II is known that the vtlClor 0 is equal 10 zero lit one point Does that Imply tha, V )( 0 _ 0 at Ihal point? Give a counter-example if yo ur an!lwet is nu. (b) Does ,.: - 0 on a line alwaYlI impl)' V x E - 0 on that lin c? Give a co unler-t~lu:lrnple if the answer is no. (c) II is found thallhe ,.; field i8 zero on a surrace. Dws it rollow thot aO la, - 0 on thallurface? 2.13 Show Ihal equations 12.221:) Bnd 12.2Zd) [;lin be derived (rom equations 12.22o). 12.22b1
and the I:UlIlItltvllt i on equation 12.231. ~ 2.14 To repreMlnllime.harmonic fiekls, most physics books usc the f3ctor e ""'nstead of u..... which lIlost electrical engineering books use. For a time·harmonic ftlul functlull Ajx. y, Z, I) - o(x, y, 1,) COlI ,wI + 411. find the phallOT Ilototion Ihot corresponds to the phy,idlllll' cunvention . Whal is Ihe corresponding conversion rule hy which ph880r8 can be transformed back 10 the tl!al·limA Axpre8!lion? 2.15 Rdcl' 10 Problem 2. 14 about the nolalion 8'i00i adolJlw ill IIIOSt physics books.
Wrilft IhAlimit-harmonic Mltxwell'!J e4uotiolls using that notallon. 2.11 Whal is Ihe ronge of effective permittivity of the ionosphere al AM ltroltdcastinH frequencies" UMl lhe following dltla: N - IOU m - ~ and f - 500 kHz to 1 M ll z. 2.17 Show thai the dimension of each term of equation [2.36) ill walts per cubic meter. 2.tl IndiCHls in watls. mclers. and joules the dimensions of the following quantiti($: (a) £ . D. (bl " . 8, and fc) S.
2.18 LetE _ (i
t
;y)e II and H - tjl - It/e'". Find Sin lennsuf zand ",I and find .., S>.
2.20 Show thai S ,. Re IE x H
c""'1.
2.21 Show that S "" Ke IE P."" )( H e,....j. 2.22 Compare the energy stored in a cubic: region onl! meier un a side which has a uniform": field of 10" Vi m 10 the energy stored in a similar region with a uniform B field of 10' C. (One C _ 10 • Wb/m:). The medium is air. 2.23 Repeat Problem 2.22 for the cllSe where the medium is water Instead of IIlr. U8e ~ - AO f g and ~ • I-'tl ror wator.
VL
5
2.e v·o· v'( 2d) =2. =+ t;.. 2
Ii- =
~/."
,'i, ;:•
• .J e
v-E =-1 (o.JiiejA , ) =-j .. § B= •.J ~ e ji.l l'
-
l3(y,t)~ .t ./.
j ~
fl,( H ==
e,.,r ... t+ -4tJ) ;
.':.,'"
*5,
•• ~.f;kJ.,'J
n 1>" £. f == •. ] t
J = 0,
-
'"
Co,
.
(wt ~ ~)) ~
.£,.., ~ .
V](~=·:t6, . v~~·J,+ho, ~ 'i/v.EJ·-fl!,. VI(J./z-J,+/Eo, ,
v.B,."; v·o,-I!, v·A;-c, 17'2\-""1
Q'1Ce;- v-( ~f"Ez)= v",~.,. V.~ • -lci,- /til- -It, (6, +4;)a -Jt ~ 'O'''Jif •
v· '8t
r
'l" eN;
~~)=
Q. ( 6,+
'1 )'"
VI. N, +VII ~ :r J," fl· 8; + r () ~ 0
Q.F"
kP:".f, ~ It D, ae],.X)'" ~(5," ~) -1.'" ~ D't 0
&'
tI·~~wQ·(o;'Dz)W&V·5,"V.DL '" fV'''&l=-f...,.
.', il . £,
f'
~
s,,.... ,.c;tJ
,
H~. N, 1'#" •
8l- 8,,,E
1 .... ct
Iv:.
J~~J',.,.j",....-, fl't-fVJ f
D~. s, +4
,$.'1;,1'1 1'H...x1oJ-"; ~,~I,.()
.
t
TA/$ ...~ fAt. $~~"'e()S " 11'p"" .,I1tD,..,..,. .
4:.'r;. . . J.,.YS,'-;r+fs;,..,t.o
) , '" ( tl.) NI),
.,It
(D . tI,Dj.
bl4..t
v.. a:
&
£~D'1"'fUJ~f
4t(~ .• . ,,)
chI No. cc.) No, 0.",.
2.1l
2. ,,..
t
s:
£ s.:...) .. yI,:"'X co ." Y-AX"$ , h .... t V_E. yecS}. iUJ'~ ~ 0 crJ\. v-.••• f J'--J ='0 -.,. ) # 0 I'~"e, bJ v~F c VUJ} 4" 0 D'>\ I-"It.-.. ~ ~
Fe +n.t p/~e
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t - (E",JC #j. + lrlt Z) s.~ t.oS6Jt
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7
'orrn Plane Wavall
Prob lems
65
a
D' iii
1ft
111
!II
11 V
-It
1I
11 1< 1/
Figura 3.11 Various comet shapes drawn on silk fuund in Chinll. These figures were J.lainlcol bclwellll 246 10 177 H.C. Helow these figures arc Chincsc Humes for Ihese comets.
If mankind rUr wing made be:ure 3.111. Mod-
; vllry grf:lfIIJy in
dividual comet comel's tail? II whelher or not
his observation
le SlIn Ihal lire the sun. Other-
'resent explana-
.1 pH rl ide ... II lid
pressure of the m the sun, ilnd e ionized gllstJs
e plasma forms now of protons til 8peeu lip 10 et's plasma and
Problems 3.1 Eslimalp. the power density of electromagnfllic radiation from Ihl'! IIlln received on earth in the same frequency band as that of the VHF television channd 2 (54-60 MHz).
3.2 Consider the sun as an isotropic radialion source. Calc1 llale Ihe InIal power radiated by the sun in the television dumnlll-2 frequllncy banu (sile Problem 3.1). The distrlnce belween the sun and the earth is approximately 1.5 x 10' km. 3_3 Assume lhlll sulllr rauiatiull is i~ulJ'Opic. Estimale the total power radiated by the sun. The solar power density received on the earth is 1.4kW/m 2 • See Problem 3.2 for nthp.r nata. 3 .4 Dilrive p.Sb) from (J ..'ia), assuming that E .. Ex ~' and Ex is 8 functiull ofz uniy. 3 .5 The .'lIar a CAnlallri is approximalAly 4.33light-YAars rlistant from Earlh. A Ilght-yea.r
is il unit of length thilt is the dis!ilncc a light wave covers in one year. How distant is a r.enlilllri in kilometers? 3.8 An electromagnetic pulse is sent from an earth station to the moon. and the reflected
pulse is received 2.St! 9 later. Huw fllr is the moun from the Earth? fAn electrnmag. netic pulse consists of a wide spectrum of electromagnetic Wilves at different frAqllfmcies.)
Vul. I!.I!.I. October
3.7 Finu the 81 units uf the fulluwinx ltlllliltitills associatllu with a uniform t:!ectromIlSnetic wave: (a) "'. (b) k. (c) f. (d) T. and [e) >. .
4. April 1964. pp.
... 3 .8 A heli um-neun laser emits light at a wavelength 6.326 x 10- 7 m in air. Calculate its
1.
frequency. period. and wave number.
3
66
Uniform Plane WIVII
3.1 r-igure P3.9 ,;hnw~ iI dipolft aolAnnA II i~ \Iflry Rffllr:IIVA in receiving television ~i~ nilis willm its h!ll)!th is I!VVroXlIlIlllt:ly C4UIlI to ollc·llBl£ the signal wavelellKth.
What arc approximate antenna lenglhs for roceiving signals for the following: (al Chllnnel2 (f ~ 57 MHzjslld (0) ChtlllucllJ (f - :n:i MHz)?
Two-wire lran5miMIon lim;
· 3 .1 0 The following .'lP.t of Illoc tro magnAlir: fill Iii! slI.ti.!lfies the tirnll·hlirrnonic Ml:lxwell's
equatiuns in free 'poce: E-
E.e""'· x
"d H _ Il, e
tV
y
Expreli:I H, and k in terms of E. and ~ and
I'e
3.11 00 Ihe fieldll in InA prfIViou!I problem represent II uniform p\ttne wave? In whllt dir~iun dOt:S the wave travel? Find its velocity and determine Ihe lime-average Poynting vector ( 5 )
3.12 nUl Ftlderal Commun ications Commission ar lhe United Slates requires 3 minimum of 25 mV/ m field intensity for AM Atations ('.ovflring thfl ('.ommflfdai area of a dty. What LS the lJO .....er thmsity associated with this minimum field? What is the intensity of the minimum magnetic field II? - 3.13 Study the following E field in a source·free region ; E _ iE.t: ,h Does it salisfy Maxwell's eqllatinns? If lVI, finellhA k anrt th AH fiflld , If nn!. explain why,
' 3.14 Show Ihal in [3.13J, if 1/1& - tI>!. - r/2 lind (l - U, Ihe wlive is ri,ll ht·iumd cin:ullirly IJOla rizt!tl . . 3.11 "'ind the pnlaril'.JItion (linear. dn:ular, or ellipticlii llnd left·hllnd or riwht.iu:lnd) uf Ihe following field s:
I.) E .. Wi: + y} e-"" i)y -+ (1
( b ) E-((l i
(c ) £-112 +
iJiJA
Jl,.
iii +13 _Il!je - flor
( d) E .. (j i: +j2y)e · ....
3.1' Show Ihat, if 0 _ b anel tP_ .. 4rt. .. r / 4. the wave is elliptically polarized {Refer 10 [l.131.} Do not try luoblain an analytical expression for 'hflioclIs . ju.~1 ohtain a pair of parametric equalions similar 10 [3.14J, r.alculale E. Hlld E,. allell poillts (",I .. O. 10· . , 9O o ~ and sketch the lucUli.
n Plane Wavell televi.~ion sig-
al I
"'iI~ltm8Ih,
following: {a}
Problems
67
3,17 Shuw that an ellirlirJllly pnlariMd wave can be dl.'CompuseO 1010 Iwo circularly polarized waves, one Idt-hanum.llluu Ihl! othllr righI-handed Hint: Let
E _ (Iii + urI e
rho
whArA a and h are, in general, complex numuer.J. Then, lei E
-Ia'x
I
;a'y) e
!~, ...
lUi - itfy) e
'0,
and solve for a ' uou Il In termlt of a lind h - 3. 18 Show thai a linellrly polarized wave can lie uecornposoo inlo two ctrcularly polariZ4:d .... dves. 3. 19 A dipole antenllu is in the x'y planA and makes a 45· angle to the x axis, A receiVflT allar-had to the antenna is calibraloo lu read tllI'Retly Ihfl component of the E field Ihul i~ plirallelto IhR riirnill. What are the readings when the fields aTR those given.· in (a)-(d) of Problem 3.151
r..
1M: Maxwell's
• 3,20 An e lectromagnetic wave in vacuum has frequeru.:y wavAlpngth >,., wave number k.. and velocity v.' Whfln It flntAr.olll dielectric medium chllrllcterized by iJ<,anr!, - 4~", what tire the f.;\. k. lind v of th e wt!Ve in this m~rlilJm?
, 3 .21 Aluminum lUIS I - to. II - /I(Jo and rr - :\,54 "It" 10' mho/ m_ If on Dntennll fur UHF rf'CAplion is made of wood coaled with II lu)'er uf aluminum finn If its thickness oughl tu lie five lilnes grp.at~r than thfl $okin depth of the aluminum III that frC(illAnr.y. determine the thickness of tht:: IiluminulII IlIyp.r. 1~ ordinary aluminum foil thick enough for Ihllt purfJOSp·t Use f - I G llz Ordinary aluminulll fuil iliapproximatAly 1/1000 in thick
Ive' In what time-aVCI1!.HR sa minimum fell of II ciIY, the inltlOsity
3.22 Cukulule the allp.nlllltion ratA ;lnd skin depth of earth for Il uniform vlane wavp. of 10 M liz Take the £ollowill)( dula for Iht! fUlflh: ,.. - ,..., I _ 4,,,, and" _ 10 'mhO/in 3.23 Find the power uellsity 10 flarlh whel"9the field intensity is 1 VIm. Use the dRill in Problem 3_22 _
3. 24 Suppostl that an airplane uses a mdar to m~ure Its 8ltilud8. t.Althe frequency of the radar be 3 CHz. SUpPOSfl furlhAr thai the ground Is covcnxl with a meter of hardpaclr::ed snow. AIrplane
::::.-J
not, explain
I I
It! Circularly
I I
, I
I I I I
hand} of the h
Fillur. P3. 24
I •
I I i i I I
All'
I I I I
d (Refer to lin a pair of 10'1- 0,
to-,
."nllw
TI "jm""""';"~"""",""","" C;round (a) What is the ulrrecence betwften the apparent altitude mellsured by the ntdllr /Inri the lruealtltude1
68
3
Unifurm rhine WIVes
(bl How much attenuation in d B does the radAr !'ligna I !'Iuffflr hecause of the snow? Consider only the ;llIflnuation of the wave 11\ Iht: ~nu\\, antlncglcct the dfcct of snow on Ihl:! rent!Ctiuli at air-snow and at sn{)\'V-ground interfaces Refer 10 FigurcP3.24 . USCt. 1.5t-o and Ian 05 • 9 II( 10 ~rorhArd.packedsnowHt3GHz. _ 3 . 25 Thfl (0110\\'11111 uHta Hre Xiven for a u.ni£orlll plane wave in a dissipative medium;
(i) amplitude of E" at z· 0 is 1 VIm. (Ii) phasA of r:. ;I I 1, _ 0 is zero, (iii) k - 0.5 - i 0 .5 (11m), (iv) d irection of propagation is in I , (v) intrinsic: imptldHnce or lhe metliulU is 1 + j uhms faj find tht: phil sur expression for E. os a function ofz.
(hI Find the phllsor expression for II as a function of 7.. [c) Skfllch E~ At 7. . 0 and al z .. 2 III, liS fum:tiulls uf w(. (d) Skctch the time-domain H fields 011, _ 0 and 1,.2 m as functions of loll. a small space vehide with 100 kg of mass is iocHled in oute r space whArA the MraviiationHI field is Ilugligiblu and the fuel has been exhausted. A seo rchlight of 1 kW is turned on, with hopes that IhA veh ide will gain lIomA IIpood. I l ow mnch speed will it finally ga in if the seHn:hli"hl CII II last 41:1 hours? Hi nt: The li"ht wavc carries radiatioll pressu re, and thero Is a reaction force 00 the source of tho light.
3 .28 Consider thaI
3 .21 An ice pArtlde of radius a is r distance away frum the liun, The gravitational force
HetlUM 011 the particle is givt:n by (3.46). The ice particle's mass can be obl;lin9d 3 from its volume and its density which III A!IIIUmed to he 1 sram/cm , The iet! particle III AIM) suhjecl to radiation pressure which is givcn in (3.47), The force acting 011 till:: icc particle due to radiation pressure is approximately f!CluAI to tha romss· sectional area of the parlicift timAs Ihe radiation pressure. Show that. when the particle's rHdiwi is les~ IhHIl a critical value, the radiation force will be grealer than the gravitational force, and this critical radillll ill indflJl6ndAnt of r, the dislam:e from Iha IIUO. All a result, all particles with rllu ii siUolh:r thon this critical radius Itmu to be blown out of the sola r system , Find the 'Va lue orthls c.ri lir:al rAdh.II.
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• 92
4
Reflection a nd Transmission of Waves
- 4
,.leU,.4.23 Tn rf!Cf'i\'f~ II n c" rly po IOlri1.t.,t electromagnellc W.1V1!$, wi,,, grMing.~ may reo
place metal plates for refleclor aOI*lOO&5.
4.
Thus.
J. - y r~:8) cos (J e
(454)
Nole thalllu~ I:urr~nt flows in the y direclion Dnd Ihol no current flo\\'s in the i direction . In fact . if the ctJndllclill~ plHI~ is rf!I'IHC~1 by a grating of parallel conducting wi res orranged in the 9 direction , these wi res a1s(I serve as a fflfleclor thai is as effcctive as il solid conducting plale. Experiments have found that gra tes ~UH HffHf:ti ve when the spacing of the wi re in the grate is much smaller than the wavelenglh of the WHVtl. G " HIHfI (U ' t! UStld instaad of cond ucting plates to refl ect linearly polarized eleclromagnelic wave~ hecfllIsH they rt'lduce weight. sove motorial. and decrease resista nce to w ind . Based on these collsiciera1ions, some reflectors use wires to re place metal d ishes fortran smilling and receiving eleclrmnH8nelic waves. An f'lxHmp le of s uc h a structure is shown in Fi,Rure 4.23.
Problems - 4.1 ThR E ARid measureu ill .. ir just above a glass plate Is cqual to 2 Vim in magnitude lind is direct at 45 ° 3way from the boundary.:ls shown in FtgurR P4 .1. The magnitude of the F. Aeld mp3SIJrP.O jllst I.lelow ttlt: boondllry is eqoal to /I Vi m. Find the angle (} for the E fidd in the glass just below the boundary
'.
Problems
93
I
i I
i
E,
Perfect
•
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rMlOllCIQI
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/
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- •. 2 The H field in air JUS! above a perfect conductor is !!liven 0)'
H , _ 3x
!
4i amperes per meier
all .~ho",n in Fil!Ur9 N .2. Find the surface current conductor. The conductor occupies the llpar:e y < O.
J. on
the surface of the perfect
•. 3 Match IhA following dt::;t;ri~[ions with the figures shown in I'igure P4 .3. FiAirili a re lIell l' ,he interface but on opposite s tdM of Ihl! boundary.
(al mAdium 1 and medium 2 aredielcClric! wUh. , ;.. ( b) medium 1 and medium 2 arA rlielflctrics with t, <
5'1
.he
'I
t~
(e) imp"IMihle ( d) impossible ( e ) Ihere is a poEillive lIurface chl:l/'Xt! un tht: boundary between two dielectria ~
( I) medium l is II J}t!rrecl conductor
II~I
r
50
we
or
.'es
,d.
.,1
-z-.
,,1.uI.P". 3
is
~
,
or
-
E,
,
,
r~
1
• E, - u
, -
,
" i,
"
,
~E'I ~ till)
>
'""-'-
~ (III)
(II)
1'1
---~
, ~} ,
1°,
,
In,
1·1 1"'1
r
~
'P,
.::. k . ~< ~
94
4
Reflection lind 'l'r.n8mi!l1l1ion ofWav9s
.... Calr:lllale the critical
...e In the IhMfHnecJia configuration lIhown in Figure P4 6. Ihtl wltve numbers are k,. k,. anr! Find Ihe transmission angle in merl ium 3 in terms of I, ;md tilt: wave numbers. A5Sllrne all k's arc real.
"I'
'. ------>, k.
k.
--- /
I
C laS!! rod
L--_
•• 7 Solid -stale lasers (ruby or glass) arc orlen fahrkall!t.1 uf rods wiTh Thll entb bevelled aT
the BreWSTer angll!. Lt!1 ( - ~ for the rOO. Skelch the proper arrangement of the exTernal mirrors and Their angles. Indicate the twlvellt!d angle of the glass rod. What III the polarizaTion of the uutpuT of Ihe IMllt i.lt:ulIl? (Sec Figure N .7.) ••8 A parallel-polarized wavll is incident from mtttlium 1 on rhe plane boundary tw.lwtlt:!o medium 1 and medium 2. Holh lIIt!dia arc dielectrk.jJ with ~ \ - 1'1 - J; and
real permittivilitls t\ and {z. WA know that when thl! incidl;lnt angle II larger than Ihe criticK lllngle 9,. no lime-llverage power IlIlrllllldcrrcd to medium 2. Also. when the incidenl aoSIA is equal to the I-Irl!w~ter tlogle ~. thA ref1tlctcd power is 7.flro. Now Imaginl! It lIituation in which tlltl Brewster angle i~ greater than thll critical angle. A wl:lve incident III the Brl;!wster angle will nul be refiocled, lleClIUllt:! Ihe incident lingle is cqualln 9", nor will it be tranllmitted. because the incident angle Is grel:lh:f than 9,. Is thi~ situation poSllible? Why?
4.8 Consider an Illeclromagnelic wave of 1 MHz impinginH 11.1 60° on the inno~pht:!rl;!. ThilGase ill similar to Ihatshown in Figure" 13. Assume tho the plasma frL'quency of Ihe Ionosphere is 101,. '" 2r x II x Hf rad/ s. anti plot IE IGsa function oh (like in Figure 4.14). Mark thAItCI:IIt! or z in meters. Solve only for IhA OIlIt! or parallcl polarization wtth Eo - 1 Vi m.
ofWlLves
Probl ems
95 x
cashown imes Ihal
~
l
IE llvolts per IIIclcrl
Ivk..J,ulJI 2
/"',
I possible
i ll nm be .he co\'er tion, and
~-
,
--- - -~
Ire k" k~
3
•
he wave
"_u,.
P4 , 10
"tu,.
2
2
lmelenl-'> '
P4, 11
• •• 10 A perpendicularly polarized electromagnetic wave impill)(es frOIll medium 1 (char. acterized by ~) - ~ and t, - 4(~ 1 10 medium 2 (characterized by ~ J _ ~ ond fl - IDI. This situation is shown in Fi)(urll N.IU. ( 8) Wh,lI islhecriliUlI angle" ~ '1 (b) Lellhe im;iuent all,l!le lre 00°; finu k. and k. in l11rlllll uf k" - !.J.fii7.. (c) Find k" in terms of Ko. (d) In the second medium. find Ihi! tlishlllCi!:to. al which the fii!l!! slrength is l i e of
thotolz-O '
'e) Finrll R, l and the phase shifl <11')1 (Rrl. - • • 11 A uniform plane \\'a\'e in air illlpill)(t:S lIoruliIlIy 011 a dieitJctric wall. The magnilude of the tulal E field measured in fronl of the wall is shown tn Figure P4 11 ( 8) Whllt is thfl [lflrmillivity of thp. diflifICtrir. wall? AssLlml! " 1 - I'fr ( b) What is the frequt:ncy of the wa ve ' 4.12 A uniform plane WIIVt: in air impinw.'S on a lossless dielectric matt!rial at a 45 8 an.!!le, (IS sho..... n in Figure P4 12 The Iransmilled W(lve propagates In a 30" direction ..... ith resf>ect to the normal. The frequeucy is 300 MHo:.
elled al
(_) I,'ind ' . in terms o f ~...
I of the '- Wh lll
(b) Find the reflection codficitJnt Ru. ( e) Obtain the mathematical expressions for
~h!d i llm
the incident E fi e ld . the refltlcltld E fitlld, undal}'
( d) In Imth mp.dia. l'Ikfllr.h thfl l'Itonciin,!! WOVA
,
---~
potlt!rn of IE, _ rIos 0 function of z
han the 'H:n the
Medium 2
k,
ond the transmitted E field
~and
1
k,
R'
o. Now
n,llie . A
II angle Ihan {I,
5phere ency of
Figure iz.ation
• • 13 I"or t .....o iMtropic media with~ ) ,. ~, 8nd f ) ". I " lind the Bre .....ster8ngie for both tht: pc:rpt:ndicular pulari:tatiun and the parallel polarization
•. 1.
U 8 wire ilnlenna is attached l-'llrullel to the metililic surface of a vehiclt: and is insulated from the surface by (I thin layer of dielectric material with a thickness approxImately flqllalto 1 mm , would it fl'!CfliVft an AM ![snlll If _ 1 MHz)? Hint: Wire antenna interacts only ..... ith E field in the direction of the .....ire
96
4
Rc n ec::llo n . lId
n-lI n l mi ~lio n
of Wave.
- 4.15 It i,l known Ihllt the Iramlmilling anlp.nns of an PM slation lllioclited In the dirl;!Ction perpendicular 10 a metallic plate. as shown in Figure 4.15. The frequency of the signal is 94 MHz. (a) Where should II rtlCCiving antenna ~ placed to t9r.eive m
E fiAid be if the rect:iving S)'MtJ!1Il in Prohlflln 4.1!:i werAlo delecl II coming in al a n anglA lOa off lilA normal? All/lllme that 311 nlher paramelel"!l relnain tilA same.
4 . 17 What would the W iW*!
· 4 . 18 I IArive (" .!:ill. 4 . 19 For 11 parallcl-polarized uniform plane W!lVfl Impinging nn II perfect conductor at:1n An)!lc 9, find the eleclric. and magnetic: fields E ;1n(1 H for the incident and for the reflect'lft waves 4 .20 Consider a 90' "comflr rt:flcctor" shuwn in Fil!jun: P4 .20 11 is made of two conclllC:lin~ plateR placed perpflndicularly In 811Ch other. A uniform plane wllve with E - iE. cXPl/kx cos 8 t /ky sin 8J impinges on the structure a t an angle (1 Show that the tota l electric fielr! is E _ - 14E. sinfkx COlI 8) sin fky sin (1). Hint: Tht! field is the sum of four waVflS with fo ur k·vt:Clors shown in Figm8 N .20 4.21 Ulle the fnrmula given in Problem 4.20 for the totlll electric field . Find the optimum poSition of II dipole antenna placed in fronl of the 90- comt:r reflector. The' angle of the mddent wllve is 30-, The frcquAnl.)' i5100 MH:£. Express tht! position in x - y coordintltes In meters What is the "gain" of this rccelvlnSll.ntenna? GHin Is defined as follow!!: GHin - 20 lOS"
l~
I
(dB)
when! E. is the 1-; field at the Rntenna poIIiti!)n and f'.. Is the field strength of thl! incident wave,
y
!"i.yr. '4.20 Top villw of a 90 comer 1'8neclor aoll Ihe rour lI _veeloTI, G
,>
...
,
Problems
97
lion the
Med iu m I
Medium 2
(.,
The the
,
Figur. P4.22
the
ltml 'ield still ~I
a
)Iers
2.0
2 .U
II
an
o
-the
u
,
• two v.ilh
2.0
thot i
..0
Ih~
-y ined
f Ihe
o
,
>
"urn :Ie of
==~
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--- -
-
,
,
,
)
t t85
0.67
o
.J:<:
>,
lef re-
>,
---1ll~ > ,
(b' Fll'!reP4.22
,
)
98
4
Rcneclion a nd Transmission ofwlilves
4.22 Malch tilt: following descri ptions In tht: stand ing ..... lIve palterns "howl! in Figure 1'4.z:l . The int:idelll wove in medi um 1 has an tuu phtudA Aquui to 1 VIm. Nole: There are thret: patterns Ihllt du not fit anv of the folJow inMutlscriplionll. Cross out the~fI patterns. (i) Plot of E, ....... ,wilh mooium 1 hf>lnM Hir. medi um 2 hUlling (l .. 4fDHud #12 .. 1£0. t\u rnu:al indrlem.:c.
(iiI Plot or 11-;, ..... ,with medium 1 being churaCleri7ffli by (, .. 4(0 Rnd p, and mt:dium 2 MinK Hi r. Normal incidence. IIIL) Plot of I 1::, _II ' with mmliulU I being chlll1:lc lcrized hy I, .. "1 0 and 1',
Po~.
and medium 2 being /lIT. The incicfllnce auglo Is grARler than the !"Tillettl anglo. (iv) Piol of E,IIII&II. incidence anglA is tll.juoilo the Brewster a ngle. (vi r iot of I Euotell ' iucidencfI anMh: is equal to thu Brews'Ar aU8lc. tl is greater than r~. (vi) Plot of I Ey1uIall. Meuium 1 is lIir 8ml modium 2 18 j.Jcrfoct conduc tor. {vii) Plot of ) II n",.1 I (Ih). MerllulII 1 is ai r a nd meuium 2 is perfet:lconductnr. 4.23 Cunsider the i.:ItSC of normal iucidellrA of a uniform plane W6\'C on 11 perftKt conductnr IHI ~hown in Figure 4. 15. 11 Cilil ue seen in [4.47) that an ollcilluli1l8 currAnt 18 IIU.lUCed on the surface of the conductor. Tlwfefore, the fullowing expre~~ion may be w ritte n for the vtliudly of II c hargc 011 the con d Ul.:!u l": v. tdqEncos(wl- h)
The abovtl aquahOIl is exactly the ~ltmo as f:f)u llliull (3.39). Con \IIlUO to wnrk alullg this line lind 1-l1"O\'0 thattha timo-averaSfI raJiation prells ure on the pflrfel.:! conductor III twu limes that given in (3.45).
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bllf'
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"
Problems
15
Using t5.45~
WI!
QuIllin
- iJA • - fle Vx A - f l - ' - . - - e ..
:al
,.
127
•,
p
:bJ
Example 5.8
Id
Cak:ulallt the IQlal time-averaije electromagnetic power Iransm iUed IIlon)l a coa ~ial line when the fields ilft: ~ivcn by (54ft).
~t
Solution:
The time-average Poynting vector (5 ) is I!IVtlll by
loJ ( 5 ) _ -I Re[E )(
2
Ibl
WI -
IV'P e
I Rc -
2
''''P)(~
I
V ·e'.... _ 1._' V_ ' 'IP
2'1p~
Therefore P _ (r V~Il))lJn{blaJ.
Ia
,v. h.
'c.
Problems - 5.1 Show thaI the complete fields of tilt: TE wave In II para lltd-plalc w3veguinf! aTe ,IIiven as follows E,. - £.sin k.x e ,.....
h. ;ul
- kE.
., ;k.E. .... II, _ - - cos k.x e .,
H. _ --'-sin k.x e ....
'c. ice ilh
00"5
lid lid
h.
E.-O: E.- O: HJ .. O
•••
Find the r.omphlle fields of the TM wove In ;) paral14!['plate waveguide
- 5 . 3 What is the lowp.sl frt:qut:ocy of an eieclrulIIllHnclic wave thaI can be p ropagated in the '1'1': mode in the carth·inno91lIu:n: wavegUide? Mod~1 the latter as two perfectly
r:onducting parallel
plate~ ~eparatcd
by AO km.
- s •• Find rhfl9udllcc·cha rge den9Hy p. on the upPflr lind the lower pllllps of Ii parallel. plate waveguide for lal the TE. mode. and Ib] the TM", mod A_
me
- 5.5 Finrl the IIIl1thcmatieal pxpre~ions for a Tt-:M wllve in a parallel.platt! waveguide thaI propagatl!l'I in the t dir€'r:lion I~e Figure 5. 11. Skelt;h the parallel.plate waveguide. and indicate Ihe rllTt!ctions of E. II . and J•. -5 •• A rnicrostrip line has the dimeMions f,I - 0 Hi em and w _ 0.71 em. and the permittIvity uf the sullslralp. LS ( _ 2!i t ". I' - ,... II 0 Eslimlllp. the tUllc·averag€' powerlhat is Iran .~m ltted by the line whtm l E 1_ 10' VIm
• 128
5
Wavftg u lclflfl a nd ResontllOrs
' 5 .7 The hrf'akdown vo]toge of Ihe dieler.lric .!luiJsll1lt~ used in Ihe Slriplinf! clAscribed in rrubll:!lTI 5 6 is 2)< 10' VIm. Ust: .. safety foctor of 1050 !hatlEI is ltlsS than 2 )( 10' Vi m eVf!rywhere in the linc. Find Ihe maximum tune-ilverage power Ihat the strilJlinl:! can transmit Neglp.cllhe ohmic loss 5.8 With the fields in a rcctangular wavJ>glllde, lind the surface current I. nn the top Iy - bl of the w;J\'egtlinll. If we v.anlio CU i a slot along 7., where should Ihe sial becul in order In mInim]',,\:! the disturbance it will r.aWle? Assume that only the TE,. mnde eXIsts ill the \\il\'cguide 5.9 Shnw that, If tht: .... avclcngth of an f'lectromll.,ljnetic wave In an unhnunrlP.rl medium chllrilclcrizcd by " and f is KreHIt!T than 20, then this Wa\"A cannot propagate in Ihe rectangular wiivtl.i\uide (shown in Figurf! 5.8) wllh the dielectric inside IhA wave8U1de al50 Ch(lr
4 HI melp.T
U "fl
:;~,
,..",."11.10
Frc~h"lrdUCI
ml!tflfS
5.10 An AM rodio in an aUfomohile cannot rl..'Ccive any signal when the cor is inside a tunncl Why'f [.Ill m ilSlIUIIH: Ihollhe tunnel i8 thA LIncoln Tunnel. which was huilt ill unA nnder tlte Hudson River in New YurkoFigure PS 10 showA a cross sa;lIun of the Lincoln Tunnel· 5.11 "'incl Ihe fn:qucncy ranges for TE.. mooe optlr'l!lon for 'hMA rAClall,ljular waveguides listed in Table I. 5. 12 DfI..!Iign an air·filled rectangular waveguide to be U8ed for Inmsmission of electromagnetic power al 2.'15 CHz. This frequenr.y Ahnult.l he al the middle of 'hI! opP.rating frf!
"G. t:. &tnOslrom. Turn"ll'lI.. New York: Holt. RinclwM.t WlnJllnn. 1963. p. 242.
ttnlltors
129
Prohlem"
7iUetJ III :! w
10'
hat the I
I
the hJV becu!
J(
FI9""·",·16
I
I
moo,
,I o
)
,
nedium e in tht> 5.15 The cleclromagnAtlc fiehJs ossociated with the TE ,o mode pmpaglltinx iii the t direction Me givt!lI by (5.23). Find Ibt! t!lcctromagnelir: frelch ossociated with the TEIO Illude propagating in the f direction, wilh maximum eiectrir: field equal 10 E,. 5.16 ConsinAr a reclunguinr wavegllinttllhuwlI in Figure 1·~ . 111. Fur the region z < O. the medium is nir ann for z > 0 tho modium II ciulrdclorizoo. by ,~ lind $/oz. A TE lo mode with maximum Efield OCluIIlto En implngos on thA boundary from the iftft. Thfl resull is that somA power is reflected ann lIome is transmittod . Alisume tilal tho reilocted wave is tliso TE ID . with maximum E field aquallo E I, and the transmitted wa\1! is TEIQ mode with mllxiJlLum E-field ACl1l81 tu Ez' Find the ratio E,lEn in It:rlJl S of 0, W. fa. f'u.' l' alii.! $/oz· 5.17 Tilt! corncr refleclor sluLiied in Problem 4.2U rCi.juires Ihe solution E
inside a \ huilt ill In orlhe
- i:4E. sin{k)tcM8Il1l11ikysinOj
Show thai althou.IIh thc coordinates IIrt! different this solutlOlI is in fact the resonator mooe thai wc studied in Sedion 5.2. Placing condut:ling plates al )t - fI And y - b to furm a cavity ff'.sonulor as shown In ProlJlem 4.20, what are the restrictions on the incinpnl anKI!:! 8?
eleclrop!'l
5.18 (a) Find (hI' real-time expression of tht: fiddsof the Tt';,... moot: in Ihe reclangular cavity shown in Figure 5.9. ( b) Find Ihe tolal slun:d eleclrlc enArgy III the cavity as a funclion uf lime. Find the corrf'..spundinglotal storM lIlaHnclie energy Ie) Show that enl'flI)' is stored alternMlngiy in !:!Ieclric and In magnetic fields. thai the maxUlium stored eleclrie IInt:f!~y is equal tn Ihe mllximum stored magnetic ellt!tgy. and 'hal Ihl! total siored elpclrolllll.llIH:tie energy In the cllvily is a constant inLit!jJt:ndenl of limA. Note thol these proPftrties IIrt: similar to those of the luw-frequency I.e rl!lfOnant circuits.
fill (he O. The
5.19 Find the lowelll resonant frf!C]Uflncy of the TE,., modI! in an air-filled rRClanHular OIvity III!:!Qsuring 2 )( 3 )( 5 crn l . Note thlll tht!re ore three different choices for designating thA Z axis ond thatthA.'If! result in three dlffenmt TEIOI modes.
.. cguides
transmit
1.20 ~:IACtrumagnelic waVAS in air with wavAlflnKlhs mnging from 1 1010 mm are called millimeter WHVI!S. Millimeter wavl!S may be gulden by dielectric slabs. r.onsidl!r a diAIACtric slab with" - Ilk, and I: _ f .. all !Inuwn in Figure 5.12. What should ils Ihickness he in orul!r that only Ihe TE. mude may be exdted fot' frequencies up 10 aoo Gil ...?
-5.21 IJM direct substitulinn Inlo Maxwell's Aquatiuns to show thai the rll!lds given by (5.481 ore solutioos uf Moxwell's ACluations in cylindrical coonJinalcs
5
130
W.veKuidcs and Ruonaton
5.22 Use th e formula!:! of divergence and curl in t;ylin drical coordin .. rell 10 provf;l thaI V. V x A _ OforanyveclnrA 5.23 Find the fP.Ctan)!uiar coordin a tes of a p(lint P where the cyli ndrica l coordinules are p _ I.~ _ 30o.andz _2.
5.24 " lnd IhA cylindrical coord inate.-; of 8 point Q whe re the rectangula r coonJinales are 1(, y. and z. 5.25 Show thaI the di fferential voillme in the l.'ylindrical coordlnatM is p up dIP dz. 5.21 To con ve rt It veclor expressed in cylindrk.al cOlnpullents inlo the same In tP.Ctan,!luil:lt oomrxmlwUI. or vice versa, il ill conven ie nt 10 prepare II ta hle fur dot products Ut:lwccn unil veclors in these coordinate s)'!!If'ms. For example. i . p _ (:0,0; 6, all IIho wn in the fullo wing la ble_ Complete the tab le . Dot Products aetween Cylindrical and
Rec tangular Unit Vectors
!
p i
•
,
cos .j
i'
•
0
5.27 Use the aLovtl table to find the rocto ngulor componpnls of the fullowing vector luca led at II - Z, Ib - 30°, anu z _ 3: A _ lIp+ 4 ~
3f
5,28 What is the maximum time-average power a coaxial line ca n transmit w ithout call.'ling brtlokdown? Assnme that the coaxial line is a ir-filled and Ihal the breakdown E of the air is 2 x lit Vi m Use a Aafety fac tOr of 10 50 tha i the ma ximum E fie ld a nywhere in the line does not 8xct!f:!d2 x 10' Vi m Thfl rlimenllion of the line is 2(1 - 0.411 cm a nd 2h - 1.14:t c m. Neglect ohmic: loss, - 5.21 Cuns idc r t he coaxial li ne shown in Figure P5,2A. HaH uf the line (z < 0) is filled wi th ai r, and half uf it (z > 0) is filled with a IUllterial charar.IArized by tl and ~I ' The 81ectrulllagnclic wave inc:ident frum tho loft hi" the folluwiu8 field s: - V, c - ,u... E' - pp
• V,
H , _ q. -
e
i ....
''''
The fields uf Ihe reflected wave may Iut tlx lJ rcsscd as follows:
E' _
V'
p~el""
p
atonl
Problems
. thai
fa) Write down the fields of the transmitled wave In
131 1
'>'
0 What wave number k
!lhould tw. used? f b) Finu V~ and the amplitude of the transmitted fields in terms of V.. ",. and.,. by
'5
arc
matching the boundary conditions at 7. _ 0 C'..ompare your r&sult with the reflection lind tnmsmission cot!fficitmls Obtoilloo in Chapler 4 ror waves reflected from dielectric boundaries.
I
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v~
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18
tIIr1
w.e-
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s:_
p
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w
x.- COS 30"z; {J/ z; ~. Sin Jo". Vt ; S·'+ 1':: (7.'~'1' ; ¢ .,.;.".'(~/'lt ) ; !-..!
~
Hi
d,,= df·(Pdl» ·. i epa!,d?d' • •
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cos~
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19
180
6
Tron,milll ion Lines
We can find thft rtlnecled wave by carrying out an analysis similar to the one for II transmission line with a capacitor. The result is as follows: V
_ VL
V, _ V.(e 2.0(.
TVL -
~I
[6.58)
Figure 6.39h shows the voltage VIz) on the line during the lime period T < ! < zT
~
8.1
0. 1 0. 1 0.1
Problems 1 . 1. Wh", t81he voltage in the striplinc discussed in Example 6.1 whtlnlhe timtl-llverage power uein,l! transmitted is 10 kW?
e.l. ConllldfIT IhA coaxial line discussed in Example 6.3 . Calculate the maximum time-average power thai may lre tra nsmitted in the IinA. IlI1e Ihe hreakdown E - 2 )( 10' Vim and a safety faclor of 10. ' .3. Two coaxia l lines have equa l characteristic impedances: 50 fl Bolh ort! air-fillt:!ll. The firAl Itnll hall a power c.apadty of I MW. and Ihe ser:ond line's capacity Is I leW. Find the ratios a,loz and b,/b a. Consider only the breokdowll vollilKe. 11.4 . 115f116. Ibland the boundary condition (4.31 to obtllin the surfllce-currellt density J. on the lower plate of the parallel-plate waveguide Then calculale Ihe lolal current on the lowAr platA. Compare the curren! with the cif!finilion of Ilzlxiven by (6.3bl·
e.s.
"'Ind the surface-current density J. on the inner conductor of a coaxial line. Then calculate the total current on it. CompBre the total curren! with I lzl defined for the coaxial line.
11.11. 1\ trunsmission line is short-circuited (ZL"
01
(_) FinclthA AxpMMion.( fo r j V(zl jand j l1zljll8 II function of kz.
0.1
o. ~
0.2
..11.1:
. .2!
z.. II ncl V, .
( b) SkctchjVlzlltlndjI(zll
(e) Find VSWR nn the line. II.?
Rcpctlt Problem 6 6 for a transmission line with on open circuit at the
• •• • Ref}t!lit Pruuitlln 6.6 for II. transmission line with a mll!chAC! loarllZ/ -
IUlltlIZ~
- 001.
2..1.
+ /1.0 Calculate la) the VSWR. fbi the posit ion of the voltage minimum. a nti IcJ thtl pt!rcentll..IIe of lhe incident power thaI is reHeeted by the load Sketch IVlzljas a funCllo n of z/A .
-• •8. 1\ tr<'lll8miAAion IlIlA is terminated with a normll1i7.ed load of n.R
• •10. Solve the problem discussed in Example 6_6 by usin8 the Smith chart. Find the
po!Iition of A Ahunt susceptance that can tune the line to have the value fin mhos) of the shunt sUSCt!ptam;t! .
II
perfect malch
DelennllH~
•. t 1. For an 0Jlfln-drr.uilArl 50 U transmission linft of lftnglh p. Ihft input ImpNtAnce Al lhe
other end is ;33 0_ Find the lcngth 2 (in A)_ II. t 2. Repell.t PrulJlem 6.11 when the line is short-circuited lit ona find • • 13. For Ihe first WAvegUIde in Tahle t of Section 5.2. design an iris Ihat will give a it_57 IiJulittanct: til f - 8 CHi:.
..21
·n Lines
he one (6.58)
Problems
181
e .14. From the Smith chart. find r L for the following ZI,oo; fall + jZ. fbI "". fel O. And (dJ 0.55 iO.38. ~ e . 15. UI6 the Smith chart 10 find Zl.oI from the following f L (01 06 e 'll•. fbJ - 0.3. Cllld
(clO
period
e. 1e. Fora load impedance of 0.4 ;0.5. find the location of the first voltage minimum and Ihe first voltage maximum allhA load end
.e.17.
Prom Ihe Smilh cha rt . find Ihe ad miHances for tilt: fo llowi ng Iml>f!dancAlI: lal 0.3 - iO.6 amI (bl 5 + i3.
z... -
z. _
8 . t8. A traMmiMlon line is terminated with a normalized impedance Z.... _ Z + ;2. as shown in Figure 6.19a. The incident V. - 1.0. linn Ihe characteriSlic Impedance of
Ihe line is 1.0. Show Ihal V .... - 1-... - 1.62.1 vloll- 1.55.lvf - O.21!:l>.I1 - 0.76. V",,,, _ t.", _ 0.36,11(011 - 0.55. and I 11 - 0.219>.) 1_ 1.45. Ivenlge ximum '.: - 2 x
--filled. ,1 kw_ ,"sily I. ::urrenl 5.3bl. '. Then for Ihe
e . Ut. A shunt IIdmillance of Vr... - -i 1.57 is added to Ihe transmission line that ill terminated by o lood ZI.'. from Ihe load, so !hallhe lin e is perfectly motched. Let V. _ 1.0 and Zo _ 1.0 nnd show Ihal V .... - 2.lm.1 VIOII - 2.00. I ..., - 1.86.1 11011- 0.7 1. and I.", _ 0.49.
e .20. In Example6.B. find anolher set of solutIons of ~I and iJ (In cenlimcters). e .21 . For the IIOlulion found In Example 6.8. how much lotallime·average power ca n be fed 10 Ihe array without causing breakdown in Ihfl difllftClrtc? Use the value 181.000 V/cm as Ihe breakdown sirength of Ihe dleJcclric. use 6 saft!ty faClor 10. and lei a - 2 mm. Hint: consider the standing waVe on Ihe stub tuners as well as on the transmission lines. - 1 .2 2. For Ihe circuit llhown in Figure 6.25a. lei 2'. - SO II. Ht _ 70 n. It, _ 50 n. e _ 2 m, v10' m/s..I1 - 10 's. and V. - 1. Plotthe vol!age and current at :t _ e/2 ItU function of
lime.
e .2J. Ca lculate Ihe percentage of energy generaloo by the puilltl Jlenerator that is abllOrhect by the load in the circuit of Prohlem 6.22. 8.24. "or a four-digil code system. design a D-A converter sim ilar to Ihal discussed in Section 6.5 using Ihe transmi.ssion Iine8hown In "~18ure6 27a SpecIfy Ihe value of R.
the location of the sampler. and the lime that a 1I1HIlpie shuuld he 18klln. L -
•lale tal , of Ihe
!Od the match_
e at the
z..
e .25. III plotlin,l! Figure 6.32. it ill implicitly as/lumAd that Hr > 7" and Ihal H, > $0 that both r t Bnd f , ore positive numbers. Skelch a similar diaMram for Iht! clllle in which Rt - 0.52. Hnd R, - 0.52'.. 1.21. Draw Ihe voltage and the current refleclion dingrnms for Ihe trulUlinillSiun line which is sliort-circuiled as shown in Figure Pfi.26. Piol V and I (i5 fun clions of time at z - t /2.
~o---===---, v. :
a j1_57
z.
T
•
·1
F 6
182
~ ,-
J~
Tra nsmission I.I nllll
-I
~1.u,." . 27
--1
_ 1.27. Draw th~ vohoge ond thecurtenl reflection dj~KllIm5 for the transmission liRAlhal is perfectly matched. all shown in Figure P6.27. Plot V and I lUI fum.: tioll!J of time at 7._ 912.
• •28. Refer to Figure 6.31, and It!! R, _ 22'.0 and H~ - 057'\1' Draw the voltllMtl reflection dia,l!ram for 0 <. t <. 6T, and plot Val Z - 3i/4 tur 0 <; I <. 6T
~
• •21. Refer In Figure 6.31, and let R. - 2 Z. and Al - O.S1... Draw the current reflection dill)lr8IU for 0 <. I < 6'1', and plntt al z - 3f / 4 fOf 0 .... I .... 61' , .'o. Refer In Figure 6.388, and obtain an exprflAAion for I . Sketch I II} and liz) versus 1. fur the time period 'j' < I < 21'. The SktllCh should be simtlar to Figure 6.3Sb and Figure 11.31k:. 8 .31 . Derive (G 58)
7.1
Vector ;
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Problems
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to Puerto Rico. This mdio lele~r:oJle system is ca lled the Very Long Baselin" Array. The angular resolution will be of the order of 10 • radian, which is smaller than the angle spanned by a dime located ill New York City when it is viewed from Los Angeles. In ordinary arrays, individual antenna clements arc connected by transmission lines. For the very long baseline array. such physicHI inlerconnection of antennas is DOt practical. IIISif!Hd, Ihe signal received by each 811huUlH in Ihe Hrray is recorded on magnetic tapes which arc later transported to a central facility where the tapes arc replayed simullfwl7 ously. The key to such processing is a ve l'Y Af!I~urHte lime standard for all rer:orch!ti dalH. AI present, lime synchronization of the recordings is provided by hydrogen masers which arc accurrate within 20 nanoseconds·.
Problems - 7 . 1 Find Ihl! roclangular coordi nales of a poinl P where the spherical coonJinales are (r-l.8-60°,._300). - 7.2 TI14~ reclonsular coorrlinates of a point Q are (1. 2.
4). Find ils llpherical
coordinates. 7 .;) SllOw liIal V . V )( A - 0 in spherical coordinales for any vcctor A
o·
-- 7.4 Show that the differential spherical surface element is tlqUllllo ds Hill l: Rerer 10 Figure P7.4.
that
r
sin fJ dB dr;.
!lleur. P 7.4
find
.49) j -
. 11 ..aH
7.5 To convert a voclorin spherical coordinates 10 the same in rcclangular coord inates. it
is conven ient 10 prepart:
Ii
table fur dut prtxiucls belween unil vAC:lors in Ihese
'K. I Kellermann and A. K. Thompson. -rhe ..... ry lonll baseline.nay;" Science, Vol. 229, No. 4709. luly 1985. pp. 123- 130.
7
222
A ntenn••
coordinate systems. Pnr example. i . t ... sin 8 cos lb. as indir.atM in the following tahle. f'.omplete this table.
, t
•
smScos,;
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i
- 1,8 I JM! thA I"ble prepared in the preceding prnhlp.m 10 express the fo llowing vector A located 01 It ... 1, 8 ... _ 60°, tP ... 45°} in rt:Ctongulor coordinates: _
" ... 12t + 58 - ~ 7.7 Show thlll the distance function It ... r'l llia\ appears In 17.7) and (7.8) can be I!xpresscd in spherlc
Ir _ r ' ll_ r'+ r'l
2ft'COS )'
cos ')' ... COlI 8 COlI W + sin 8 sin 8' cos('; ... .
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l'
'1
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tPl and If'.
8'. ,,') are
spherical coordinates of r and r ', respectively.
7.' A vertical receiving dipole antenna at P Is 15 km away from
\I
capacitor.plaut
antenna thai is abo placed vertica lly, 8uhown in Figure P7.8 The receivins antenna mellSUres an E field equal 10 10 mV/m. What Is Ihe vahut of E tlutlllle same receiving antenna will detect Ht a height 3 Icm ahovfl P? What must the orientation of the rACflivinX dipole be to obtain a maximum reading? (A maximum readinx is obtained if Ihe dipole is p3rallei1o Ihtl E field .)
-------------H]= _-
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7.11 The power losl on a cylindrical conductor thllt is!).z lunx and that carries I amperes ut current Is given by P_ _ YIl t H,!).z
where P_ Is the loss due to finite conductivity of the wi rt!. It. is the surface rftllililanct!l(iven by 1/ (od,2TO). and d,ll! the skin dtlpth. The efficiency of the ante nna is given by 'I, -
Power radiated Power radiated + P_
AMumflthat a short antenna of length 0.7. hal an efficitlncy of ten percent. Is the efficiency improved by increasing the ienXlh 10 2 0.% while maintaining the same current and, if 110, by how much? Assume thai the anlfmna is still II short antenna after ils lenxth is increased to 2 6.7~
Antenn ... 2'
223
Problems
following
y
,
•
Power fl ivifler
18 Vf!Clor t\
r
-I
PhillltJ
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90- [kIll)
'.8) can be
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lcifor-plale ngsntenna e receiving tion of the is obtained
7.tO. Consirier the IIntenna system consisting of two short dipole/! arranged perJ.l8ndicularly to each other In space, aSllhown In Figure P7 .10. ThtlllC dipoles are driven by the same amOllnt of IlOwer from a common source, Ilowever. the current on Ihe i:-uriented dipole has a 90 v phase with rflSI>fK:t 10 thlll on the y-orientcd dipole because of a phase shirtHt ill~tlrted in the transmission line that leads to the former . Find thtl total radia ted electric field on Ihe 7. axis. Verify Ihllt Ihis IInlenno system radiales a circularly polatizArl wllve ill t}u::! i direction. Is the wave left·hand or right-hand cin;ulllrly polarized?
'.11.
Find the expression of thl! total nuJilltoo electric field on the x axis that is due to the antenna lI)'slem discussed in the preceding problem . What is its polarization on the x axis?
7. t2. A certain application rp.(luire~ that II fitlld strength of 1 V1m be maintained al a poinl 1 km from all antenna located In free space. What power mtl.~1 be fed 10 Ihe IIntenua if il is (a) an Isotropic an tenna. (hi It !lhurt diJ.lOle. and (e) a half-wave dipole? Neglcct ohmic loss. An Isotropic antenna radilltes an equal amOllnt of power in all dirtlClions. 7. t3. The curtfmt atlhe ctmterof on antenna Is 100 A ; what ill the E fieldl km away from il on the horizontal [0 _ goQ) planA al 10 MHz ir the IIntenna is (a) a dipole with hi _ h, - 0.5 m, (hi a eapllcitor-plule antenna with uZ - 1 m . and IcJ a half-wavfI clipole?
I amperes
1e surface Ie antenna
:ent ls the the same
rt
antenna
7.14. Show that if the radialion fil!ld pattern IIhown in Figure 7.4 for the infinitesimal dlpoll! or the CHllltcitur-plate antenna Is plolted in )(-7. plane in linear scale Ihtl vuttern is exactly formed by two circlfl.'l. 7.15. I,'jnrl the clirp.clivity of la) an isotropic antenna. lb) a C
- 3 ml and Ihilt
7.17. ( :nnslrler a uniform linesr array of two half·wave dlpolplI that are 1.5 wilvelenJojlhs apart. The currents on Ihe&fllwo dipolesllrtl ill ",lUIlit'. Sketch Ihe radiation pattern in the horl7.ontal [8 - 9O "llJlalle. Show clearly the number of lohel!;n this pattern. Also. estimuttl the beam width of each of IhA major 10UeIi. The beam width is the angle between Iwo rlirp.ctions in which the radiation intensity is one-half [ - 3 d B) the maximum valutl of the beam
Chapter 7
224
Anie nD II
7.18. "'Igure 7.21(bl shows the array factor of a h~'O-element array separated by 20}... Find the beam width (ill terms of the ulIKh: between t\\'O adjacent nullA) of thill. array fa ctor near 1ft- 90- and 0- 30 e • Use the approximate furmula Kivell by (7 .49). (bl Find the exact value starting from (7.45). 7.11. Find the d ircctivity of the two-wire transmission line shown in Figure 7.27 with rarliation fiAlrlll. giwln by [7.42). (a)
7.20. Find the field pattern of a two-element array with d - )' / 4 and", - O. Sketch the field paltern on the x-y plane. 7.2t . Find the field pallem of a four-element array wllh d _ ,\/4 and 1/--0. Sketch the field ptillern un the x·y plane. (a) Use 17.37) to oblltinlhe fie ld . ptiltern forlllu lH. !:tnll fh) USII
the result obtained in the preceding problem and In Figure 7.16 ond the pattcmmultipliclltion technique. 7 .22. Write a computer program to plot field pallerns of a len-element phased ilrray with d - '\/4 I!OJ varyins phases. 7.23. A uniform linear array consists of 6 short d ipoles. Tho spacing botwoon ad jacent
AhtmAntAill. Al4, Itllllhnwn in f'iguffl 1'7.23.
la) What should the phase shift'" be. in order to point the maximum radiation in the 41 90 0 (thai ill.:91 direction? (b) Suppose that the E·field due to the first cle ment (the dipole at far left) is givcn as follows : 1000 e-iit sin 8
,
I f.:.1
of the entire array at point AIO.lOOO,O}, point R{IOOO,O,O}, point 1)( - 1000,0,01. separately. All positions arB given in rectangular coordina tes in meters. Use the phase shift found in {a}.
Calculate
qo, - 1000.0), and point
Ie) Sketch Ihe field pl:ltlt!rn uf tht! Iltfl:lY in the x·y plHIlt!. Irll Skell:h the fie ld pl:ltlern of the array intht! x-~ pla ne.
y
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'lbpi£1 in Waves
Figure 8.17 shows 0 typical arrangement of H liquid-crystAl display.* II is operated in the so'(;HIlHd "dislortion-of-aligncd-phases" or DAP muue.
Figure 8.1701 shows the normal stal e ur Ilut crystal before activation. The light enlf'ring the crystal is polarized and then Irllnsmi llHd through the crystal with no aitentlilill in polarization. The second polaroid absorhs Hlllh61ighl, and no light IS IransmiU~1. In its activated Siale, the crystal changes the
polarization of the transmitted light . which propagates through the second p(lt.lruitl and becomes visible.
Problems 8.1 The d£!rivalion of 18.5) only considers the electric field Why is Ihe magnetic fielti nf'gif!f:IP.d? Hint: Cum pare the magnitude of R, wilh I!H, nt!lll' Ihlt !:Ipllt!n:, or the stort,.'
8.2
Wh~'
is the rising or setting sun rl!rI?
8.3 The smuke emiUoo from engines of boa's cnn lain .. fine purticles. Against a dark bilckground 'he smnkt;! Innh bill!! Imt lI)juinsl a bright oor.kground it looks yellow \Vhy? 8 ." Explain the :lpPp:lranr.A of shuTts of sunlight through bre:tksin a clOlld·r.nvAreri sky. 8 .5 Shuw that
I,-f'
/
erdx_.;i
1.1 A
"d,
I-IInt:
I; -
f'
e' dx .
f'
e '" dy
Then. transform x.y r.oo rdinates inlo cylindrit,;al (:oordin,lles 10 perform the exact mtAgr.tllull .
e.e
Show that
I, -f.·
~ //
I.' "0h.
r.
Ih
AXPi _p'.'( i+q.'(IJ.'(- ;ex{4:')
d
"
8 . 10 A
Hint:
Q'. QJ' +_ 4p-
pl,r ... qx ... _ px _ 2p
(
ThUll. uS(: the result obtained in prohlem 8.5 after IrHllsforllling the inlegrillion varlahlp. from x In px - tj/2p. 8.7 Assume that on earth a microwavl! Imam of 10 GHz is rudiuleu by a 20·meter· diamPll!r rlisk antenna uillltlu til the moon Estimate the size of Ihe microwave heam on tht: moon OSee R W Curlier ,lilt! C. Mare. "I.iqulo CoI"\·",,,1 Oispla)'s, . IU::t; Spectrum. Nu\'emt.t:r 1972. p. 25.
'I 'I
Problems
:. in Waveli
253
lisplay. * It AP mrxlB.
The light ·he crystal Ithe light. langes the he second
!.
wnetic field erc. or Ihe rgy de nsity
insl a dark lkll Yl!lIow. '\ered sky.
a.a
1
the exact
J\ [)f!rson leaving his home by train mails a leiter home every utl y. Suppose thaI the troin lrovt:ls ZOO miles per day and thai th~ ma ll moves at a speed of ZOO mile' per da y. Ilow frequently do his leiters arrive home? Try 10 solvA thill rmh1em by simple relilJOning. nol by substituting numhers;n some formula
a .g On II foggyday.lhe driver of an automobile SIOPpt:J al a rtlilwllYcrosl~i n g hecause he heard u whillt le from II moving train . 'rhe sound of Ihe w histle came from hi' lelt. A few seconds/aler he heard Ihe ceho. ond Ihe lJilch of the fiNllllollnd was lower Ihan Ihut of the echo. AlISlImA Ihat thA echo was due 10 rencetion from 0 neurl,y mountain close 10 Ihe track. If yo u were the driver. wuuld yn u r. rf)llll the track- that Is. could you tAli whAlhAf the Irain was approaching Of leaving you? (See Fi)(Uft! PH.9.1 8.10 A Dopp/er radar sends a signolot 8.800 GHz. Hnu IhA TACAivAr displays a frequency lIptlctnlm of fAlurnM ~ignals as shown in Figure P8.l0. Whot ClUJ yo u 9Ily aboutlhe speed of the ItlTKeqsj?
ntcgration 10-melerhPam
.iI\·f!
AmflHturie of rhe rclurn ....,J sil'nuJ
-
10kHz
15 kH z
8.800 r. Hz
bkHJ.
£requen
,"!9ur.
"'.10
254
8
Topic. i n WIlVH
n.
Au.o r l'lIon I
I
Randomly poolatiW
l'g"1
Passing "x , ~
1~ ;-----i!I7--I-----1I-
, Cfobserverl
• ..i,wr. ".15
B.1 t Fur th.: FM-C\\' 1lopplf>r rilJltf discussed In Sectinn 84. assulIlt: Ihat f.. the upper frequf'nr.y of Ihe l'IuJnr. is 8.8 G lb'! Suppose the f dunr is to measure targp.1 speeds rftn)(lUlI: from 0 to 3 "-lar.1I and IlJ~ l lini,;C5 from I km In 10 km. Fmu the system's opproxima lf> frfl11uenq hdnuwidth and ,hA lim p. inltH'VlI1 the system must be ahle 10
rf'lIOlvp..
z - 0 ann z - tI inciden t from IhA left will become II
8 . 12 If d _ >.s/4. all shuwn ill Figure 8 16a, ann if rAflecliulIs III intcrfnc~ ~,
art! ne,llli,!!iblc.
linearly rnlarized
WilVC
circu larly polarizHd \\ilVl:. as discussed in Ih A Il!xl. Whal is the polarization of the pxitm.!l WIIVC if the rcncclion~
t. _ 2 11 t, IIno. ;!.JIk .. Find thc minimum Ihlc.kne.\l.ll uf a (IUllrll. 4uiHlcr·wave plall! for a h,llht having>. - 6500 A,
8 . 15 In !llgnff> Pe.1S Ihl:! Polaroid film III A I.~ oriented ltut;h Ihat il passes lighl rm1arizal
In
the x llirL'clion and IIh.'iorh.~ li,llhl IJOJilrizcd in Ih~ 5' dlrp.(:l\on T he film al B passes r-polarb:f'i(! light Hnd .. 1)Surbs x-pobri7RfI lI ~ht A nllldurn l) pola rized light 8Oun:e. silch as a n.:lshlighl. shed~ hgh! from tilt: lefl illong z Clln an observer HI C sec Ihe lighl? Explain 8 . 18 Cunsidcr Ihe arrangempn! ~hU\\1J in Figure pe 16 Thi'l figurl! dLffel'S from Figure P8.15 only in the 11Iat;t:Ulcnt of;) Ihlrd PolMoid IiIIII at D bclwCC!IJ .... and B. The
IIxisof thc IhirO film l.!1 ~5 ' frum eithcr Ihe If o r Ihl! r aXL9, Nu\\, can Ihc ubserver al c.: AAH rhl' Ii,llhl? (If ~OU do not h(1l1l!vf> in yuur alls\\'cr. do an ex periment with th rfle '1<111".::1 uf polarized slingla!l.M'!9l1ud stte for roursclfl ah~urptjuJi
•
~r,.
,,,, ,,,' , <":Iobservcrl
9.1 Ele
CHAPTER 8
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288
F. lflCtmll talic Fieldi
Problems e . t Considt:r tht: dipole arrangement shown in FiHUI'C !I2a Ull Q _ 1.6 at:
X
10 - 19 C. find 4>
(a' x - O.l .y- O.l .Z-01 (b) x_l , y _ l,z _ l U~ the exact ronnula 19_ 14) first . TIltm use Ihe approximate formula the accuracy of the laller. The medium is air.
[!l. 1 7~
and find
' .2 Thret: point charges are lor.ated 011 tht: x axis with q, - q al ,< - 0, 4 , _ 2Q at x _ 1. and q, _ 3q 1'11)( - 2. Find the positionls) on rhl! x axis wlltm: 4> is equal to zero. 11.3 Four point charges are located on Iht: corners of a reclanglf!, fill shown in Figure P9.3 Find Ihe pll1np~'I on which the potential is eqnal to ztlro. Skt:lch these planes y
• q
"I
---tlin 201
y
I 'm I
" , q
-Q
- 2q
•>
I"'e"r. PI,I ' .4 Two ('IOint CltHr'MtlS arc scparated by 0 meteNl III air. as shown in Figme P9.4.
( a) "'ind the poltmlial funclion (x, y. zl. ( b) Calculate 4> at x _ 1000. Y _ lOOa. Z _ 0 (c) Show Ihat, for distances much gre"lpr IhlUl potential is approximately given by
.-
-q
U
Imay from these charges, the
1
-01 ..~ ;-;,---;'---"",", Ix' + 'I + z' f~
e.s
whAfAlx! + y' + ZI)11I » o. UM Ihill approximate formula to r.alr:tltll le 1]()O
t .1 For Iher.harge distribution given in Prohlem 9.2. cah.:ulate E~ at (a) x - - 1.th) x - 0.5. Ic) x _ 2.5. and (d) x - 3. 1 .7 Apply,.; - - V to 19.14) to find the": fiflkl\O(:aleJ ulthe origin and produced hy two chuf)lfls + q and q locatAd allO. 0.11) ElIld (0. o. h). respectively. 1I1IlihoWll in Figure 9.23. where h - 1 Clll. Show that E _ !tq/2 .. ~hf).
e .B Solve Ute same prohlf!ffi all in 9.7. Lut usc (9.19c).
e.1
SkAlch the dirt:Ction of the fo: fialrllocaled ul tile center of a square shoVl.'Il in Figure P'J.9. The E field 18 producetl by four charges al four comerll uf the square Ot\BC. Ihe56 fo ur charges corry q, q. q. anrl {j coulombs. respectively ,
Field.
Problem!!
287
y
-----.-q , ,/1
C
Q
. Find 4>
I
8
,
'x/
2h
/
I
,..,.".. H .•
I
/'
I
// ',I
-Q
A
I- 2h---l
•
g.10 A line chnrgc 2h mllter.!! long ill located along tnc 2: axis II.lI:ihuwn in Figure 9.6a. The chargA ctAnsity is p, coulombs pllr meter.
(a) Calculate thl! electric fif!lct at
p -
O.lh, 4J - 0, and z _ 0 Wling thA exacf formula
IY,21). (b) Calculate fhe ell!ctri(.: fillld ol lhf! IIRme poin t using fhe 8:iSUllllJliun thilt the li ne is
infinilA ly Inng. (c) Find Ihe I}f!rr.f!nfagc error of the value outllined in /bl iI~ r.ompa red w ilh thl! I!Xl:lct value •. 11 "or the same line-chaf)!1! dll.!ICribed in Vrohlem 910, (e) Cakulatl! the elf!r.lrir. field al p - 20h, tP _ O,:t _ 0 U.!!1Og thf! Axact formula fb) 1>0 the same usinlil thl! assumption Ihat lhe line is a point chaf}(e I:Itlhf! origin. (c) Find the pAN:f!nlage error of va lue obtil in et.l in /b}.
g.12 A plan(' charlile of p. coulomb:. ~r SflIJaM mPler 15 located on Ihe )( _ 0 planA, and
another plilne of - p. coulombs per SQuare mt:lter I.!! localAri on Ihe )( _ 1 plane. Find the lotal electric-field in the Mgion Ill) x '". I. /b) 1 ~ JC :.. O. ond I(.:I}( ..: 0
:es. the
'.13 COllsidt:lr the pmhlpm discussed in Example 9.11. Axsumlng thai everyth ing is the same except for the far.:1 Ihl:lt the tnlal charge on the conductinH .!!hllllls nnw equal to l.ero. r:alr.ulalp. Yo everywhere. and skl!tdl E, Vef"5I1A r Aimllar to the sketch shown in Figure !U5. g.14 A dlill")(e dislrihllfinn of the following form is set III) in Illr I~phe rical coordinates):
1. 100o,
ketch
JC _
II.
0.5,
( a) Fill£! Ihp D fiAld forO fb) Find O foru·, ,.' U.
(c) Finri
bytwu Figure
Figure
OARC.
J)
forb
<
<
r< 0
r
8.15 A charge distributiun uf the followIng fnrm i~ SPI up in air p,
1U ' . e
r:nlllnmh..~
per cubic meter
IJ!p Gil 11M' law 10 find the £ field cvel)wherl!. Hint: Tn finrl thA Intal charge in a Gaussian :.urfilce. you mllst rln thp integration becau$C Ihe charge is not unifonnly fh~lrlhllted Ilowever. symmetry still exists \\ Ilh rf!lI(lfM:t In ~ and 1/ '.11 Electric chargl!:' aredistnLuled IInHonnl), in the region 0.1 .... JC <. +0 1 with density fl. :/m' Elsewhere. Ihe density is ~u"llo zero. FlRrilhA "; field everywhere.
10·'
9
2BB Plot E,
\tArtlUS)t,
Find the potential difference V.
V. for
Electrostatic Field, iI
point x with
fp.specllo
the origin.
1 . 17 Fmd the [)(llp.nlia ! rliffp.rp.nce V ... - V. for two points A and B IOClitOO 81 r _ 0 and r _ 1 in the E field obtainoo in Problem 9.1S.
1 . 18 Tilt! solution for the electric fielrl of an oscillaling Hcrttion dipole with anxular frequency w is given in 17.1 4 ) a&"follows: E-
t: '~'!'1.,IftT1 I f~12cos8 .,. ell +.~ V~/hl.l.z 7 /I\fl
/f{f
+
.,~
IIl1t)
,Isinsl
Derive the solution (9.20) for a Sialic di ~le by !JtIUin!l W - O. Nnlir:elhal k _ lind 1~z - npliH - , ... p.
wf,utl,n
,UI In the electric field E - 3x + 4y - 5i. find V" - V. if A islocatAl'l aliI. 1,2) and B isa! till! ori~ln. Dop.~ the difference depend on the p3th of !he integration? 9.20 (:onsider the spherical-shell problem s hown in Fixure 9.1 4. Find thfl pntenlial ( r) at (") r t:: (hI h .:; r < c; (r.) b :> r :> 0, lind (d) r - o. Assume '" - 0 at in finity. Plot .pfrl versus r, • •21 Rcpt!" t the preceding problem for thl! r.Ase In which the total charge on the
conducting shell is equal to zero whill;l all otil!;!r cuntiili unll remain unchanged . You may v.ant to use the re.'Iul! ohtained in Problem 9, 13 • •22 Consider the coaxial line shown in figure P9.22. The inner conductor is a solid conrluctlng cylinder with a radius equalla U.l m. The outer conductor has an inner nuJiulI equal to 0.4 m and an ouler radius equal to 0.5 III. Tht! morlium bet"''fIen the Inner and the outer conductor is air. The inner conductor carries a net charge of - 3fo CJm Rnnlhe outer conductor carries ollet charj(e of - 1S10 CJrn. The sym· bol to used hero represt:uts 8 constant Aquai 10 8.854)( 10 - 1l . (II) Pinn E. In Ihe region 0.1 m
<
p
< 0.4 III.
(bl Find £~ in the reMian 0.4 m < p < 0.5 m . Ic) lo'ind E.. in the rcgionp > U.5 III. (til Fil1ti ~ at p - 0.2 m, knowing that'" _ 0 at p - 1 III. (e) Skotch £, 05 functio n uf p foro
Ftgur. P.i . 22
"OItath:: Fields \lith n:spect 10
:falr-Oand
289
Pro blems
'.23 Modellhe dome of II Van tit: Grllarf )ltmerator as a conducting sphere. The dome is charged to hold the maximum amount of electric charge Q", before the ai r SU t · rounding the dome breaks down. Ulle the following data : radius or the dome - 0.11 m ,
with angular
I, Zland B ill at
)Iential '11 rJ at inity Plot o$(r) harge on the changed . You :tor is 8 solid tor has an In· hum between
; a net chargc 1m. The lIym·
I p.
breakdown E of air _ 3
II
10' VIm.
(8) Calculate the maximum Q", accumulated on the dome just before the breakdown . (h) Calculllt" th" 1IOlta~e of til" dome in reference to the potential at infinity just before breakdown occu rs. rcl When the dum" is c1l1irjoloo with the U1uxiUluUl charge Q",. a person uses a conduc ting rod to discharge the electricity. Assume that the discharge takes 0.01 seconds to complete, how IllronglllthA dlAChatging current Inn thl! avtlrage)?
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This electrostatic adhesive surface is widely used in desk -top calculator-driven curve tracers. A typical voltage used to charge the embedded conductors is 300 volts. and typit:1I1 spacings between them are approxi· rrmtely 2 mm "
Problems 10. 1 A point charge of 4 cuulom bs is located at the uri)Cin (0.0,01, ami (I. !lecaml point cha rSA of lJ' cuulombs Is a t II , 0, OJ. A ~Ula ll test-charge il placed at 13. O. 0). and it is ruumJ 11'101 the 101111 force 011 Ihtl test charge ill ~uallu .:era. Find Q' in IfOrms of (j. 10. 2 Two irienticill small balls are altllcht:d 10 weightless IIITin!!!! 15 em long. Each hall carries 10 • C of c.harge, lind cach has a ma.'IS of I )! The), achieve an Aquilibrium state lIndp.r the influence of electrostatic force and gravilalinnai fun;t:. as shown In Figure PlO.2. Find the angle cr. Hint: n is small
10.3 Consider a long \intl·charge with fI1 - 1U -~ C/m . Find thl! fon:tl acting on 11 dust parllde carrying 10' C, I m away from the liM CharM!!. 10.4 A line charge wUh p, _ 10 6 C/m isl(lf:aled in tlil" ot )( - 1, Y - 0 A plane charge with P• .. 10 • e l m is located al x - O. A positive point chArge of 10 I C is at (1{z, 0, 0) in rt!chUlMulaf coordinates. WI,al is the 'olal force actinM on .his point charge? 10,5 Cha'"8e is uniformlydilii,ributed in the spherical volume r.$ 0 with p~ andp. - 0 forr ~' u.
-
2 x to 'C/m'
tal u~ Gauss· law 10 find E for r .:. n (b) Find Ihe fon;tl acting on .1 1 1'~t churgtl uf 10 I. C al r - 11/7.. (e) Is the force ob'aiMd in [hila be changed if 'he dntrgc distribution extendslu r20 instead of heinM limited to r ~., o? 10.1 In a .'IPOO sorting machlnp-, undesirable seens IIftl deposited with an tlh:ctrostalie r.hargt! while they p
1111(.1
0 R r.OT!IOn. E:1t.-cltomogM.tism ISan FnlllcitOO W. II. Freeman lind Co.
I [ner~y
• ca lcuhedded pproxl-
Problems
333 VlvQhsl
""
,
5
[milllsecondsl
At room temperature f:lU "CI and standllrd Atmosphere, whfll should be the S17.e of the cnrona wire if b - 3 cm. Vo - 10 kV. and the fOughM" factur uf the wire is equal to 0.8? (RefAr to Figure lOA..I What should the loweS! vollage on A Van de Grad! generator be in order to htiVA It product! corona on its surface? Assume that ,.;< _ 4 x 10" V1m and Ihlll the radius of thA melal sphere ill equal to O.H m. Refer 10 Figure 10.1. If Ihe voltag9 applied to the parallel plall!llsihe sawtooth signal IIhnwn in figure Plo.g. find the loclls of tht! electron on tlll~ fluorescent !K:reen located at x - 20cm .
for the cathode-roy luhe shown in f igure to.8. whlllllhnuid Ihe voltagM V~and V, be in order 10 make the electron beam trace 11 circular palh on the x:reen at 60 TflVnlulions per second ? AssulOt! Ihat the vcl1iclliand the horizon tal deflection plates are identic.a!. ~n Aleclron is IIcceleraTed by a difference In poten tial of 1 kV between Ihe anode and Ihe r.alhode. It e nlers the pnrallel -plate region with Ihls kinetic energy. Its velocity makes a 5" angle wilh thA plane of thtl fllu-allel plate lit thA entrance end, as shown in Figure P10.11.
j
tu r-
(a) Find V,. v..' und v"" al I _ O. (b) OLtain Iwo equati uns for the coordinatAII of Ihe eltlCtron (II. z) as [unclions of I. NOlc thut x - Oand z _ 0111 1_ O. (e) find the posil ion of the electron lit the exil end uf IhM parallel jJllltA.
oslatic r T he ·oltll.lje 10
- 3cm_1
the
be V, region Figure plille.
T-- F======~ ~ 10m -+-~"'"'
1
100 V
__ --,.,,.-_> -1.0'
L--F====~'V m._gll x 10 " kll q.- 1.60xl0 " e
,
......r. "10.11
10
334
E lectr ic force Bnti Energy
10. 12 Conyider the ink-jel printer shown in FigurA to.l0. Define q~
_ c:hargt! on the ink drop
/lid -
mass of the drop
V. _ deflection-plate voltage d - deflt:CIion-plale spacinH _ velocity of the ink drop al entry 10 the deflection plaIA 9dp - deflection-plale length ~ _ distance from the deflection-platA Antry 10 the print plant! v~
Show Ihl:lllile vertical dillpla!:flmeni of the Ink rlrop is Kiven by
L1~ V. RdpI [ "p - -'t1dp ) '. - --'mdUVoj 2 10. 1' Find the capacitance orlhe IIpherical capacitor 8hown in Figure 10.13 by HllinH (10.42J and (lO.50). Start from
Q £ ___, ' 4 ... u
for b > r > o
and show that your rellull agrees with (10 47).
10. 14 Find the cllp8citance of thA cylindrical capilc!lor shuwn in Figure 10.14 by using (lO.42) and (10.50). Shnl from
E -~ jJ for h :;:.p> u 2lffP
and show thai your result agre4!S with (10.49). 10. 15 Consider the parallel.plote capacl10r shown in Figure 10.12. What is the maximum capacitance on~ can obtain by Ilsingmica as the Insulalor1 L~t the area of the plate be 10 em' and the voltage rating of the capacitor be 2 kV, with 0 safety factor of 10 Use Table 10.1 for the value of ( for mica. 10. 10 Consider the cylindrical capacitor shown in figure 10.14. What is the maximum capacitance one can obta in by lIsingoil es the insulator? Tak~ 0 - 1 cm, h - 2 cm and the volta)!!! rating _ 2 kV, with a safety factornf 5. UlW Table 10. 1 for the value of t for oil. 10,17 A pmallel·plale capacitor Is filled with two dielectric: mB!t:!riais In a configurBtion shown in figure PIO.17. The lulal area ofthA plBte is A (a) Find the cBpacitance Gin lerms of A. d, f" Bnd (I ' (bl Suppose Ihal the poslllvA plale carries Q coulombs of charge. and find QI and Q1 in lerms of Q, whArA QI and Q, are charges un the left· and on the right.hand sides of the plate. respectively. Neglect fringing fields. 10. 18 Consider the capacitor shown in Figure PIO.17. Letl , - 3••. (, - 5fp. d - 0.6 mm. and A _ 20 cml. The potential between thA plat~ is 300 V. Find the 10lal slored eleclric energy In Ihis capacitor.
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fora: Bnd Enerxy
Problems
335
10.1' Find the capacitanr::e per unil length of Ii c.;ultxla] capacitor with two layers of insulating materials. asshuwn in FigllTe 10 15c. ExpressCIl1 in tennsof n. h. c. ',. and
'"
10.20 Find the capacitance C of a parallel-plale capacitor witla two lavers of insulating materials. al'! .~hown in Figure P10.20. Expre.u C In tt>rms of .A {the orea of the plalel, d,. dz. 'I' and ( J.
13 hy using r 10.42J
10.21 Refer 10 the capacitor shown in Figure PlO.:W. Let (, - 3fo, f , _ 5(0' d, _ 0.3 mm. d l _ 0.3 mm. anu A - 20 r::m z. The vohage across the capacitor is 300 V. Finrlthe total stored eltXtric energy in this clll-lticHur. 10.22 Derive {lO.tI:ll.
ra HlI4 hy using
10.23 A pltrallel·plate capacitor ctll"l'ics ... Q un onR plaIA lind - Q on the other plate. The orca of ellch plale is 1\ and the separation between the platfll'l is S. The medium is
air. (0) Finu the tolal stnred energy UII; In this capaci tur ill lerms nfQ, 1\, Sand (0. (b) What is the c1cctrosllltic fUTCe acting nn the platos? Is it attructive or repulsive·'
Hint: finn the change in VB with fCSpt!t;1 tu S.
is the maximum "ea of the plate be factor of 10. Use 15 the maximum cm,h-2cmand . the value of f for
configuration C in s Q coulombs of uges on the lefting fields
1 II
clt~citam:e
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368 r.onnl1l::lOr
Conductor
Cuu,ju"lor
., %
~
+, _ 0 Case II
Solutiun Thch niques
"''lure
"'1.1
Cue JIJ
Problems or
11 . 1 Consider the thrAll LlOunuary-volue problem!! !lhowllm Figure P11 I The Milltion UlSf'l I is <1>,. lind the solution /'If Case II is "'1< In Case III , IhA r.hllr,!jI!.!H/,ltnd Q2 l:HC the sumll charges Ihal appellr in cases I and II, and thAy IIfJVt!ur in exactly correspondinll positionH. ExprllSll , in terms of $, and "':.
11.2 Consider the IhTRA buunuary,villuc prohJAm.~ shown ill Figure P112. ThA Milltion uf CaSI! I is "',. and Ihc solution of C:ase IT is "'I' In Case Ill, IhA .. harges Q, lind Q2 orc Ihe SUint: chargcs thaI appear in C8St:S I and II, anI'! !hAy IIppt!I:Ir in exact ly corresponding positions. Nole Iht! differences in the boundtlrY conditions for Ihe three ClilIeS. Can.J hfl Axprt:SSed in terms of "'1and 4>21 U so. obtain the exprp..Mioli Ir not. explain why_ 11 .'
The radius IIf the innl;!r conductor of a r.oaxill.llint! is 0 and that of the ouler conductor ill o. The potential of the illllerconductor is Vand that of the outer conductor is zero. There is no volume chltf)!t: density between band u. Slur! from the Laplar.e f!quation to ohlain the potential in the coaxial line.
11 .4 Two concenlric conduclingspheres have rarlii ultnd b. respcctivelyth;;. 01· The ouler IIphere ill CIt zero potential. anrlthe inner sphere Is main lainp.{1 at V volts. There is no space charge helwAen the conductors. Start 'rum Ihe Loploce cquallon til ohtain the potential f rl for b ;>0 r ). o. CunJuClnr
\
•
Ca.II@I
_ 0
Conductor
ConJuctor
4> _l 00V
C3s.>11
Cas.' III
l'e(;hniq ues
P robl ems
369
1.1
;ululion of Ql 3re the !sponning
:olution of Q1 3re the ~sponding
~s.
Can 4>a l::Iin why.
:onductur or is zero. t!(jualion
fhe uutA r lere is no 'utllin the
InSlilarillg8 up ,~ -
0
11.5 In Figure P11.5 a {;unuucling conp. i.~::It a potential Va. and a small gap separates its verlex from II conducting plane. The axis of the cone is per~enrliclJlllr to the conducting plane, which is m involve (/ only. is independent of rand 41 when spherir:al coorninMes are used . Find the pulen tial {O) in the region 81 .:!O 8 ~ 90°. Hint: f (l/sin 0) dO ~ In (I!HI 012J. Find Ihe surface charge density on Ihe cone.
'*'
11 .6 The uppflr pl
'*'.
11 .7 Model a de vilcuum · tllhfl rectifier as two parilllel platfls with a Spilce dlllrge in Iw.tween, as shown in Fi.l!ure 11.3 . Let Ihe separtltion be 1 cm. Find the voltilge needed lu prmlllr.fl I Alml current.
11 .8 Find the surface chnrge distribution on the vertiC31 and the horizont::ll conducting Willis for lhe case rlisr.l1ssed in Example 11.8. Plol p. for 7. '> 0 and x _ y _ U. Lei Q _ 10 ~ C,andQ - b _ l. 11.0 Find the inHl)o!es uf a point charge netlr n corner uf a conductor similar to the one shown in Vigure 11 .11 excejJllhlll ¢.O _ 45°. 11 . 10 Find the c!ectru~tlltic forcfl Ih::ll acls on the point ch
to.
0, d) and is due lu
11 . 11 Calculate the capacitcull;e IJflr mp.tflr of a 12-inch (0.3048 mJ-diamfltflf steel pipe locllled 6 ft II.A.1 m) above and pOflllJel tu the grollnrl. 11 . 1.2 Example 11.10 slote~ llml the milximtlm electric field on lhe surf::lr.e of the conduct. in)o! cylindAr is located ot the point ll1mrest the ground. Show the villidily of this st3tement by jJlulling Ollt E. nn Ihe surf3ce as
11
370
Cundll~tor
I'"
Solution Techniques
01
Flgur. Pltt,tll
11. 15 Equation 1I1.42J )lives the polenti:!1 dUll In II lJUint charge in the prf'_wnce uf II ~ rounded
wnducling sphere Equatiun jll.ol41 gives the pOtfmlia! ,Iue Iu II point chorge in the presence of an isol.tt'd sphere carrying no opt elia!')!!:. From these resultli., find IhA polfmlial JUt: to Q poinl charsI' q. (/ IIlCIt:I'S from an isolated r.oncluclin)o! sphere tAm y ing 0. net charge of fill_ 11.16 A lint'! r.har}!!: PI is insidc 1\ conrh lColill,l( tUllnu l of radiuli. 0, a.~ shuwn in Figure PII .t6 Nutice that the liM char,llt: is b meters off r.An IAr. Firu.1lhc potential fuor:tinn in tht:
tunnel. Ilinl: This i~
I)
compiemAn ra ry pl'ul,lcm of the one !lhnwl\ in Figure 11.1 2.
11.17 Calculate Ihe forCA [lpr mettlf aCling on the linA char,!!!! in lht: lunllclshown in FigurE' PI Llfi.
11.18 A point (;harMt: Q is inside a J;pherit:a.! cuvit y 01 a connur.tor. Ii~ ~ilown in Figure Pll.16. The radius ofthftclivily is 0 and Ihe cav ity ill rllltKI with air. (a) Jfind the VOlential" in thA cllvity wilen b. O. [h) Find Ihe surface chuge dellsity of the cavity Willi when b - o.
(c) Find the potenlilli 4> in tho cavity when h - ul'J. . (d) Finn the sur£ace charge denf;ily or the cav ity wall when b 011..
11.te Calculate the elP.r.trolltlitit; force acting on the VOinl c harge in thp cavil)' shown in Figure PI1.HI. 11 .20 Sketch the E linp.ll due to 0 point chargp nfMr the intc rfac(' of Iwo dielectric media The ,llilualiull is similar 10 thA nne lIhuwn in figure 11 . 17. except that ( 7 - 0 ·5f 1· 11 .21 A rectangular conduc ting trough of widlh u and height h j,ll maintained at zero pnlRntial. as s hown in Figllrf! 1'11 .21. The potenlial nn Ibft IU[J plate. which covf!l'!Ilhe trough. is known to ue 4>{K. bl - ZOO lIin(2u/ul volts Find the (lOlential 4> in the trough. There is no volume chargfl in the trough. 11.22 Three sides of a recll:lll)(utllr conducting pilH:! 111'1: grounded . whllfl the fuurlh side is maintflined allOO V. OS shown tn FiKure Pll .22. Find Ihfl pohmliul in the pipe. ThArA i9 no volume charge in the pil)C.
'" '0./"
- --"
•
Pie"'.
"11 .22
~I----I /oo\-> •
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Problems
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11.23 The lJoumJulj' potentials of II rRr.langular conducti ng pipe arc shown in Figure PI L23 Find the potential in the pipe. There i,l no vol ume chaTgfl in Ihe plpi'! .
1ere
11 .24 CUlilliuer the boundary valu!! probl/lm lihnwn tn Ftgtlre Pli 24 The upper Qnd the lower conduc ting plates are mainlQintld at :':I:I\J ~lentilll. Thl! IllatA III the left is IIHllnlf1inflfl fit 100 V. Two gilps insulate the side plBte from the grounu . Tht:re is no
1.16.
volume charge in the reKion and Il~prollche~ ZAro HlI x apprnaches infinity. (a) 11M rhe method of separation of variables lu uutu ill two oNli na ry differential
fintl
'he
tl4UUlioll1l.
(b) Solve Ihe differential equations. (The rUllet ion involving y must be Il sine (uncti on.]
(e) Mateh the boundary condit ions. and find thft fiMI Mllnion 11 .25
1
in
:Jia
A sphftrical capacitor is filled with a dlelectrie material of (1 in half or the s pace and with ano ther moleriul of f l in the remaining IpIlCIt, I'll shown In Figure Pll .2S.
{a! Find thtl pollmtial function (rJ in the region o
011
till: inne r conducto r and Ihlt
em
'he 'he flis
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Figur. P11 . 25
" b
CApACitAnce
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Solution:
387
The resistivity read hy the !IOnrlJ! willlWl infhLf'nr:P.rl fly mooillm 2. Thus, IhA Mmde Will nol feMcJ p , to \1'11\. IIlthouxh it is IOCillt:u cOlinll), in medium 1. To find the cxpt,.'Clcd read ing. we must first calculate Ihe potential detected at /I We solved Ihe polf'n liaillroh]pm In F.xampif! 12.S. In the presen t Clist!. \\e Imve"l _ 0.1,)( _ U, Y _ O. " 16 in. x 12.5-1i HlOl m/ in _ 0406 m, Z f h - 32 )( 25<1/ 100 m - 0813 m. and
I.
l' _
1[° 1 + 0.01) _ 0.8181 01
0.01
'l'hPfPfoff!. according to r12.2211},
-,
I [-I- + O.81~ -;--'--;" - - -I 4. x 0.1 0.406 0.B13 4-w
134.11
Substituting the above value 101('1)1 2 26), WA ohtain 2 . 5~
p, _ 411' x 16 x -
100
1 x - x 34 .7 _ 14 I n·m 4 ..
Problems 12.1 A p(lf
The lul(ll area or the plote is A The dielectric constan t and the conductivity nf nne materia 1are. , an n q h rf!RpoctiVf~ ly , ThCllle uf Ihe Olht!r lIIult!rilllllre ~ 2 unu 0'2' Finu Ibe f+(llIlvu!tml dn:uil fur this parallel pl(1tc. and express the circuit paramete rs in terms of A . d, ' " 0" ' 2' and fJ l
12. 11 '1 and
12.2 A pilr'dlld vlule is filled with t .....o materinls In Q configuration s hown in Figure "" 2.2. Find its equivalent circuil, and exprp..M IhA r:ircllit paramfllfu's in terms of A tbe .. reu nf thf! 1l1a1f!. and d,. <12, t ,. t 2' II,. 1llU11l2, which ore defined in the figure .
, dary
h.811 exact IIff'nl
12.'
A cooxialline haslwo laye r.'l nf in.'llllatinn . FigurA Pt2.3 shuws the Malmetry. Find (. ) Ihe IJUlt:ntial 4> , fo r a < p ... b ( b) the potential 4>, fnr h ..: p '" c ( c ) the resistuJ\<:e uf a section of such a line'! mete rs long
~ltI .. · .
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dlf·
boun· t reo ~ 1i ve
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12
388
Direct Currenls
y I, .• ,
-~-T I •
,
Fleur. "12 .•
I I p'. I I
,
0
PArfocr conductor
y /
,
,,
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I r 2{lcm
I
Perfect conductur
fI-O.Glmho/ m
,
.,
12.4 A lIphericctl conuuctor of radius a is inside a spherical conducting shel l of radius c. Two malerlals arc used 10 fill the space between these cnnnll!':l n n!. The dielectric con51an~ and the conductivities of Ihelle malerial.\! a rt! (,. 0'1' f l . It:_ respectively. FiMura P1 2.4 shows the configuration. Find the equiva lent circuli of Ihis system, and express Ihe ci rc uit pa ramelp.fs in fArms of Q, b, c, t o. fl'
12.S Two oil wells ilrc 1 km apart. The resislllncc between IWO steel pipes in these wells I, measured li t lAIIi. What is Ihe conductivity of the ~roUIIJ I1Imr Iht:lltl well.\l? Use the
fulluwinH d(l\(l : the length of both pipes - 1 km. and the diameter of bolh pipes - 10
em. 12.' A currenl e l ~lrod e is ncar a pcrf~t1y conducting plate that Is bent to form a 90corner, as 5hnwn in FigurA Pt 2.6. ThA output from the electrode is I UIJl~r~~, und the material fillinJij th~ spact: has a conductivity cqual to a. Find the potential function
"'( x,
y,
1.1.
12.7 A currellt electroUe is nt:ar a perfectly conducting plate that is bent to form a 60· corner, as shown in Figure PI 2.7. The electrode prodllcftS 10 A nf r.urrAnl. anrl lhe mate rial filling the region defined lJy 0 .~ ••~ tiO° ili wuter wit h conductivity equa l to 0.01 mho/ m. find the potential at point B shown in the figure
u .s
A point electrode puis oull amptlrtlS of current above a conducting piane, as shown in Figure P1 2.8.
(_) Find "'Ix, y, zl for z > O. (b) Find Ihe current density 1.lx, yJ al the surface of the cnndllr:lor. (e) SlcAtch the paths of Ihe current flow.
4 Pmbl ems
389
fC I
I 3m
I
I i
0a.1 SlIIl
'%\"\\\'@. '{
{~·;\. .:,fu"\'\~\~
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FJgure P12.10
12.8 For thtl Coltse shown in "'igure 12.9, find InA pfHGAnlagA of Inti curren t emitted from
Ihc elect rode crosses the boundary ond cnters iUlllcdium
"
f, _ eo
ius c. celric IVAly.
and
t:urrent, u show" in FI,I(ure P12. 10.
(al Calculate Ihe potential at point B. (b) Calculate the polentillial point C. 12.11 A wAII.logging rAsislivHy lnol similar 10 Ihe one shown in Figu re 12. 1215 near a buumlury between two beu!!, as shown ill FiKure PIl.11. Tht! boundary is making 060" angle with the well . Find the ilpJXlrcnt resistivity measured by this tooJ al the po'!it!on "hown.
,p.lhA
12.12 Rflfflf to Examplfl 12.6. Ohtain Pe (the apparent resislivity mtl8surtKi by the tool) as o function of tool position for Zo" j 160 in. to Zo" 160 in .. where Zo is the position of the center of the tool (the midpoint between electrodes A And UJ relative to lile uuulluliry. Clilcullilc Po for Ilt ICllst 21 PoiUls. IInu!)lut Pit V(1J'~US lon.
. - 10
12.13 RCPCOf Problem 12.12 fortne situation shown in Figure PI2. tt .
t.
ells is
• 90' d Ihe clion ill
SO"
d the nallo
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P. - 1(11)l1-1n
12
390
Direct Curre:nll
12.14 A point electrode is located at (0. YI' nJ. lind 8 perfuctiy conducting sphere of ra· dius 0 is located at (-t, 0,0) as showl! ill Figure P12.14. The electrode gives I amperes of cUfmnl. The cunductivity of the medium Is (7. I"ind the potentilll oil on the y axis. Hinl: usc (11 .44). 12.15 Consider a woIl-logging resistivity 1001 simillif to the one shown in f'igure 12.10. Lei the spacing bel'oWtlll the current eloctrode A and the potential electrode B be 6 m. The tool measures the conductivity oflhe earth furmation as it trawls in a well. Assume thai the well passes near II mineral deposit modeled by a perfectl y aliidueling lIphertl. 8S shown in Figure PI 2.t5. Find the Apparent resistivity measured by the tool as a function of y. Use the follOWing data : 11 • 0.0 1 mhnlm for Ih. grou nd ; the radius o f the mineral deposit. 50 m; And the dillhmctl between the Ctmttlf of the sphere and the well • 70 m. Plot 0. _ _ versus y for - 70 < y < 70. Hint: use the result ohlAined in the proceding problem.
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13
422
MIl8 n tllOILIa l ic Field,
Oecause magnetic fi e ld is present in the wflxial li ne. we know that magnetic energy is stored there. The mngnclic field is giVtm hy (13.7): -
1-1. _
I
b>p>o
Z/rp
{
o
e lStlwhere
SubsWu ling the ithove ex pression in (13.34). we obtain
Utt .. ;
2
JJ
l
lo
0
dq,
1"p dp 4,..I:P
l _
g
,1011 %In 4".
(~) U
T his result is the sto red magnetic e nergy pe r unit length o f rhe cooxinl line. Conseq mll1t ly. we Cun calcula te lht! inductance per un it length IIf tht: line from [ IJAl j: L ..
~ In (~) 2'11"
{lJ.4Y}
II
Th is induc ta nce pe r un it length also appea rs in Ih H tra ns mission · line repreStlrl tHtion of the coaxia 1 lim~ in (6.191 of Chn pie r 6
Problems 13.1 "'inri the IIIll,l!nctic fi eld " althA cflIlttlr of a square loop carrying It current I. The side uf the square loop i.~ b meltlrs long_ 13.2 A r:in:ulllr loop tha t has ratii ull u amI thai carries a curren t I IJrOOuces the some UllIgnetic·field strpngth III its center as thot al thA cAnler or u square loop that hall lIine band Ihat cll rries the some curre nt I. Find tht! ratio of b 10 0 , 13.3 Consider a la rge con cl uct i n ~ plate of thickMAA II locutt:d ilt
in Figure PI3.3. U nifo rm current o f d e nsity ulll'egions
I is flo\\ing in
d/2 .s y . d/Z. all shown Ihe f: ni rection . Find H in
13.4 The eH rth's magnetic fie ld al the norlh mognetic pole IA approxilllHtcly 062 G 11 C _ 10 • Wh/ m:j Allsume thot this magne tic fill id ht vrwuccd by a loop of CIIftll nt fl owing II long tht! t:q ua tor. £Slimal/! the IIIH)!ui tude of this Climmi. T he rutl ius o f the mirth is ap proxi mAte ly 11,500 kill, y
d
l-~·-/ .
___. Y-
Ftlur.
~
U ,:I
l,
,. i.
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11 11 Itt
13.5 An Infinitely long tubulul' (,:undut:lur uf inner radius 0 and outer radius b carrit!9 II direct curTent of I amperes. as shown in Figure P13.5. Find the H Rflld at fI: where (a) p :;, a, (bl a .so p .so b. IlIllI (\:1 b ~ p. 13.8 An infillltely long tuhular conductor has outer radius b and inner radius a offset by a distance c from the axis of the Olllflr cylinder. as shown in Figure P13.6. This eccentrir: tuhular conductor carries a direct current of 1 !impArl',&, Find the " field at point A shown ill the lisurfl. Hint: Consider the tube to be a superposition or IWOllOlid r:ylinders that have radii b and a amI thlll !;1m), IIniform current density 1 in opposite diret:tiolls. 13.7 An infinitely long wire is bent to form II 90- \:ornflr, al shown in Figure P13.7. A di rect (':UlTt:llt I nnW!! in tllfl wire. At point A find 11'10 H field due 10 this current. Follow the steps given below. (a) Usc tilt: Biot-Stwllrt law 10 express the " field at A due 10 It typical segment of wire dyon the wire axis. Express tilt! field in rectAngular coordinates (b) InteNl""dtt! the rp.!!uit ohlained in (al to find the H field dUt! to the semi-infinite wire OC. Note. to faci litate inteHration, lilt y - fltan fI, 110 thai dy _ a SCC Z 8 d6. ( c) Finrllhe H RAid Al A due 10 thc ~mi-infini l e wire 80. (d) Add the results olJtuined in fh l and (c] to yield the lotal field at A due to the current in the wire DOC. 13.8 Follow a similar procedure 10 the one rlltllCrihed tn Problem 13.7 to find the H Relrl at ~inll\' . 8.'1 .~hnwn in Figure P13.7. 13.8 Consider a circular loop \:lIrrying a currenl I counterclockwise. as shown in Fi8urt! 13.11. Plolthe mAgnetic field 11, on the z axis for - 0/2 -::: :t -::: u/z. Find the value Z. in lerms of a. such thllt. ;rl:.d ..: 1.., then H. ill uniform within 10% of the valut! uf H.!:It Ihe (".f!ntf'r of the loop_
13
424
Magne tos' . I\(; Fields
,
PltU,. "13. 10 Helmholtz t'OI!t.
,
13. 10 '1'0 ImprovA the uniformity of Ihc magnetic field along Ihc Axil! of II circullif loop (see Prublem 13.9). onc may use two iden lir:alioopil S8!)l:1rl:ll~ by tl distollcC cqu31 !alheiT radII, as shown in Figure P13.10. Such II pair of current-carrying 1()('1pt; i.. r.illlild Helmholtz coils. Find H, as 3. funClion of z on Ihe axis of the Helmhult:e coils. Piot H. for 0 < Z <: o. Find, in terml! of c, the valut:!:t,. such thot. within the range Izl < 7. H. il'luniform within lO"k of the magnetic field at the middle of IhA two coil" CompiHe your res ult with thai obtained in ProhlAm 13.9 for Ii sin)!]e loop.
13.11 A 5tjuArA conductor loop 20 IUcters long on each side carries a (Hreet cu rMol1 u shown in Figure P13.11 . laJ CaJc:uilltf! Ihe magnetic fit:ld B al (b.O.O). Express the magnetic Reid in terms of 4 integrals, where each represents the contribution from thtl currellt all each side of the squaNl. UIWt the Biol-Sllvilrl law. Do nOI try to integrnte those Integtlll9. (b) Assume thnt b is much greater than o. Now, INllluulll thc inlcgtnis approximately to ohtain an approxinllilll value of B at (b,O,O). y
c
T "
. I
lb. u. uJ
,
Flgut.P13.11 A
B
13. 12 A surface chAtge of P, C/m1 is uniformly distributod on a t9Cord disk. The inner radius uf thtl disk is a and the outer radius Is b. Thfl rACord disk is turllinlol at II constant angular velocity w radII in the duckwiStl diruction. Find the mngnetic Reid at the r..anter of the disk dUll 10 the surface charge on the turning disk. Ignore the prestlllctl of the metal post on the turntable.
< Fields
Prublflm s
425
13.t3 The tlHrlh's magnetic field at the equtllor illllpproximlltcly B _ to late the cyclotnm freqllfmcy of the electron in the ionosphere.
4
Wb/m7, Calcu-
1:1.14 lIecause natural uranium contains a slight Illlluuni of Uranium 234, llit! electromagnetic: Isotope separator I:tlil also yield U. If tho radiuli of the circular path for I)lU particles (st!8 Figure 13.14) is cquollu 10 m, where should one place collectorI for U~ U and 2:HU partidell? Expreu spacings In melers. 13.1e Rcrcr lu Figure 13. 17. The maglletic fittlri ill r::hanged from 5 x 10 ~ In 10· J Wblm 1, All other parameters remain unchanged. Find the following: (see Their
J
liliAn 01H .
".h II, l~Dre
(0) tbe position of the electron 1I1lhe exilltide of tho magnet ic·field fegion (h) tho exit anNie (the lingle between tho truitlctury lind tho K axis after the elecl ron hA~ passed through tilt: magnetic field) t3. tO Consider an electron having initial kinetic onorgy III. ~2 IIml enlering B region uf uniform magnetic field. as uelJicted in Figure P13.16. This ~ilulttiC)n il> similar to that SIIUWII in Figure 13 .17, except thot tlJtl!:!leclron In Ihe prescnt COstl is inclinmi AI an a anglc with n:specl to Ihe x axis. (a) Shnw Ihal v. and v, of tbe electron after il enters the UUI,I(nelir: field are given
t I AS
by
v, of cach int&-
'01>
.roxi·
V.
v. -
cosj"',r , uj v,sin(\o.'
wheT!:! 101< - Cl,.B./m~ And I - 0 corrcsporuJ:! to the mnmenllhe electron !:!lItel'lllhA magnetic field (b) Fint.1lhe cooroinares x and zofthc !:!1t!Clron i!.llime t NOle that x _ 0 and z _ 0 al ,- 0
(e) Finn Ihe poinl where Ihe electron Ip.awt' Ihe magnetic field. Alisump. v, ... 2 x 10" mis, a _ ao. "', - 8.77 )( 10" rad/s. and d _ " cm (d) Find the ongle beh-\cen the x axis and the Iraj ....'Clory uf the e l ~lron after il hall left the mAgnplic field . Sketch 1111:: entire Iri!.j9CIOry, and compare il with the one shown in Fi)!ure 13. 17.
, J----j ------,
.l::lecrron
..,,,
• • • • , , •" • • ,
Figur. P13.1'
,,
t
)
• • • ~--~ • • I x
)(
K
K
X
1<
I
--------'
no. :tt Ii clic 0 ..
t3. t7 'I\\-,., flarallel wires arc carrying 100 A of current in 0pPQ:!lte diroclions. On each wire £ind Ihe force pAr unit length due tu the mllgnetlc field produced by the other wire. Is the force repulsive or attractho-e? Assume that thf!linf!S IITf! 1.5 m apart.
13
426
Mll8nelollltic: Fifle.
13.18 " wo idtmticul circular loops of radii n ATflllepartltoo by a distance d, where d «(l. Qne of tho coils carries I amperes current clockwise. and the other carriM I amperes coun terclockwise. Find the fort:e betW8fln Ihft8f! coils. Hint: Becaux these coils IlrC close togethor. you can appmxlmate lhe magnetic field that Is ODe coil and is produced by the current 011 the othor as III .. Izl'f2Trl). the field due to an Inn nileiy lonx win:. Let 0 .. 1 m and d .. 0.05 m. How much current is needed
or
I'
tu produce a force of 9.8 N?
13.1' A circular loop of radius 0.5 m 1:IIld l00 lurns is excited by II 2 A direct curreni. This loop is placed in the Earth's magnetic fiflld, which b IIpproximatcly equal 10 5 )( 10 - ~ WbfmJ pointing nnrth . How do you orient this loop to produce II max;· mum torque? What is the value of this torque? Find thAt oriAnttllion uf the loop in which it t:Xpt!ricnccs no torque. 13.20 'fhc »quare cond ucting loo p AIICD ~hown in FiJ\urc P13.20 cn rries 2 A of direct current. I~ach sine of the iuutJ is 0.1 III long. The loop 15 pillced in II. uniform mag. netic fitlld B. Find the force on each side of IhA lootJ II.nd the torquo on the entim luop if:
(II) H - ~ 0.2 Wblm 2 (bJ B · t 0.2 Wblm 2
, D
C
T
-U.lm
f .. 2 A
•
-.
Flguf. P13.
2.
8
13.21 An Infinitely lung conductor of radius n cllrries Figure P13 .21 .
II.
uiroclion current I as shown in
(A) Finel the H field in the region 0
Itatic f'jelds
111M d « o. ler curies I ~t: Because halls alone field due to It is nettd9d
}Ct current.
lly cqutll to lell II maxithe loop in
Problem.
427
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t
y
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T
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1m
I"
!-- -1m-...j
" of direct form magthe culil'9
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Figur. P13.22
FlgvrePt3.2t
13.23 The mo.gnt:tic field in a cOIlxiallille is giVfln by
1'. - fOllP ~
forO.lm
Tho mcdium i8 Air. What is the totiil stored magnotic ellcrMY pltr unit length ill the line" Give tho numerical value and indictlte its unll .
hownin
13.24 (a) CHku lale lhe stored U1tlgnelir: energy per uuille nglh of Ihe parallel·plate mnductors shown in Figure 13.5. (hJ If the parallcl plate is u~ed as a capacitur to IIlore elect ric CllerNY, finrlthe volt· axe Vo for which the stored electric energy is cqutll to thfl ilored magnetic en· ergy found ill (a). Let , - l A, W _ 10 cm, And (] - 1 c m. ExpteSll Vo in volts. The medium is air. 13.25 CalculAlfi the inductance pllr unillfl ngth of the coaxia l line shown in Figure 13.3a.
0.5
e duelO Jalue of
13.21 Calcuillte the inductance per unit length of the parallel·plate conductors shown in Figure 13.S.
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452
14
Mil3uellc M. leriaJ. and Masnetic Circuit,
From 8'01, we find the correspondin)! I-t0) (Figure 14.1 7b): /l ilt _
2~OO
Ai m
The iterative method calls for substituting the above value Inll) (14.23) to obtain the "first-order ' approximation of B, which IS denoted lUi BI": /1111_ 11000 - 24(0)( 0 1211.u.. _ 0.176 Wblm1 0.005
The corresponding I Jl I' mllY be read fmm Fi"ure 14 . 17h; H I' I _ 2100 Ai m
WA ubta in the "second-order" approximation of B by substitutin8 the above H " va lue for H in (14.23):
=_2~'~OOo::;x~O~.~12~I~"' '' - O.167Wu ' / III• B'" _ cll~OOO 0.005
This procedure (".an IJe repealed to find the n _th itljl'lltivc result of B I~I. When a rli8illli computer is available, the mlignc ti7.
be written 10 cllrry out thA iti!rativc proc:eduft!. which rACluirt!9 very lillie computer time l!lei! Problem 14.71. Tht! problem ill hand can also 00 9011U!. by a gmphicaJ method. Nutc that l14.221 or. equivalently, {14.23} is an equtttion of II AlraiMh! line on tht! B-H plane. As shown in Figure 14.17b this lint! intersects tht! B oxls al 0.251 Wblm1 and tht! H axis at R264 Aim. II 81110 intersecls thlt nonlinea r magnetizotion corve lit B _ 0.19 Wb/m1 • This Ttlsuit a)!rees fairly \\..,11 with the result obtained by tht! iterative methtxl.
Problems 14.1 RefAr to the magnetization CUrvA shown in "'igure 14.3. The malt!riol is a nnnlint!8r nH:dium bec-.8wt: J' depends 011 the magnitude of If. For mognelostalic fidds. u is equal 10 the slope of the line joining the ori!jill to lhe tHo BI poin t on the mllKlletizaHon curve. In this WilY, Figure 14.3b is obtolned frum Figure 14.30. Now. if the matltrial is placed in Ii time-harmonic field . thA Afft!ctive J' will btl different from thA Il for the magnetostatic fi e lds. Consider a field H - 110 + H, cos (101' + t/lJ. where H. is the nias magnelolltatic field and H, is tho amplitude of the timt!-hormonle eompont!llt of the lotal field. Lei H, « 1I. ; IhAn the cffecllve Vt:rmcobility of a malt!riol is thA slope of the tanKtmt of thA ffittMncUzatlon t; urvc at f-I •. Skt!teh the effective I' Vt!I'SUS H. for the curve !lhuwn In Figure 14 .3a. Compare it with the magnlltoslatic JJ "hown io FigurA 14.3b. and IIhow that the JJ's in these two Cl:ISt:S are t!Qualto eilch other al " ,. 14.2 Point out the differt!nccs between the foliowillK pairs of IArms: (o) diamagnetic paromagnAtic,(bl remanen(;e vs retenlivity. and leI cDflrdvc force vs. cUllrdvily.
\IS.
14.3 What ore approximate valuAS of the retentivity and thA (;UIlrcivity of Iht! ferrite shown in Fiwure 14.91
ic Circuita
Problems
453
14.4 Consider Iht! r.arbon steel. alnico V. ond cunico malerials listed in Table 14.2. Which has the highest IJf:!nnanen l magnellc.Reld strength? Which has Ihe most difficu lty in losing its permanent magnetism once it i~ magnetlzed '~ ohtain the
14.5 A permanent ma~net of fUJIUS 1.5 cm and thickness 0 3 cm is put in It magnetic fie ld Ihal is parallel tn the disk. as in the siluatioll depicted in Figure 14 .7. The torque on Ihe disk is cquallu 1.2 )( 10 ' N m. and the magnetic field is equal 10 10- 1 Whim'
Whal is the remanence orthe permanent magnet? 14.8 To writc "onc" in the memory corf! X' YJ shown In Figure 14..11, how should the
current pulM'.'I he sen t along the wires? S~ify the pola rity of these puLses. 14. 7 Consider the magnetic·core melllory sketchflnln I"'gure 14.11 and the corresponding hysteresis curve for the cores shown in FiHurc 101.9. Now suppose that. because of
,hove H ilt
malfunt:liun in the CiN':lIitry, a positive pulse of amplitude I, which alone is capable of producing the switchinH ml:lHlletic fielri strAn8th HI' is scnt down the line YI lind Ihlll slmulianRolIs\y an identical pulse is scnt down lhaline x •. Assume that all cores arc initially ill thtl 'zero~ sta tR, which corresponds to havinK the magnelir. flux cir· culation point ing either tOWtlro thtl uppflr Iflh or the lower left (usinH Ihtl risht. hUlld fule). What afe the states of aU of thu l.:ures after Ihlts", pu ls8S have passed through?
, a di~itul standard ~ram may :::omputer
14.8 Compare tbe hysteresis lonp.~ of two ferritcs shown III Fiijufe P H .6. The curve labeled til is "thinner" than lilul lullelecl 1/2. Which fcrrite core requires leM switchin,ll current? Which ferrite has a better ability 1o Withstand magnetic Interier· pnccs'l
hat (14.22) As shown H axis at g Wb/m 1 • n
14.$ Consider the lOal!Cnetic r:irr:uil shown in Figure PH.g. The material is steel. and Figure 14.1 7 shows its magnetization curve. The flux density in the air gap is 0.5 Wb/m'. Find the curren t ' needed 10 produce this flux . 14,10 The magnetic circuil shown in FiHure P14.10 i~ made of a materia l with Jl Find the flux dellIllues HI and B:. and indicate thdr directlOn~.
-
600,"".
•
on linear ~lds. JJ. is on the . Now. if enl from I. where .. rmonlc lity of 1:1 -etch the .... itb the ases are letic vs. vily. !
ferrite
"'
"'
lOOOr\1rns
J. f O.S cm
>
fI
r. P1 • •• "1." ferril
Hy~teresilloope
ror two
~
P,
r,,
100lums
.. ------
T i Dii ,
P,
/'
I
I
0,
I
...I ___ ='.l'
r.
p.
B,
Ipngth~.
P,P,-lIcm P, P, - PIP, _ 10 em
P,",", - Ucm
/',/',-9cm
cross' N!CtioMlllrella: ":/'. - 12 em'
all other branches _ 9 em'
454
14
Magnetic Mllterillis and Magnetic Circuit,
12cm
14.1 t To produce a magnetic flux of 0.5 Wh/ m1 in I h~ air gap of Ihf! ma)(nt:licci rcuit sho..... n in 1-"gIlfA PI4.11 . what s hOllln be lhtl mag nitude of the current in the coil? Take.u • 200.uo. The cross-seclionaillrt:a of all brandlfls is t:!(juollo 4 emt, 14. 12 Write II cornpuh:r program 10 f'.H rry uut the Iteralinn proceuufc outlined in Example 14.5. First. approximate lin: nonlinear CUr\lft in Fi)!uro H .17b by a polynomial of fiflh order. Than carry out the Iteration five timos to obtAin the rourth-order approximatiull for B.
14. 13 Find the approximate value of Ii in tin: magnetic r:ircuit ~hown in Figure 14.178 for excitatiUIl currenl J _ 15 A instead of 10 A. All othor conditions givtlll remain unchanged. Ca rry out the iteration a liufrident number of timt:1I 10 obtain an ar.cu· racy to thfllhiro digit.
15.
QU;
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:. 3OoI" ~;.S'tC";'+ ;~!.]lCO.IIS"+ ~.:ttCD. 28 ~
10 20 30 40 60 70
eo 90
100
= .!t.)(3.Ii'1S,qo-~
:. I - 3 .'3'TS-"/O-'/(3N)~f.""~·") .. 8.32. A
H-O I11AGE 7A.OD.7A,D.DDD,17A~ODDDD.Dt6A FOR 1-1 TO 5 J=l-I B=(1000-.121*H)/3978.9 H=(25815.47-(129360.27-(354706.29-(447912.425-214034.26*B)*B>*B)#.B ) tB PRINT USING 20;" ORDER-",J," B-",B," (Wb/m**2) Hz .. ,H," (AIm)" NEXT I END
ORDER- o ORDER"" 1 ORDER- 2 ORDER- 3
ORDER- 4
S- .251 B= .179 B= .188 s- .187 8= .187
(Wb/m**Z) (Wb/m**Z) (Wb/m**2) (~lb/m**2)
(Wb/m**2)
HHH= HH=
Z37~.6
2090.4 2127.8 2123.0 2123.6
ftE.. ~ "'. " ().DDS + 1-1.. )( D.III-/S_IOO -ISDO ... B e/·)= ISDDIt,IT rl-IO·"/o. ct1S - (). 377 .,..
st,'= (ISCO-C./2/'t1..3000)KIITII./O·Yl1,t,OS = /j ( j ) ~(!S'"o "4./21 'It
_
(AIm) (AIm) (AIm) (AIm) (AIm)
C!~DD. 0./1.1 N'l/IC,u.
c.",.,s
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B - D. 2 ~ H. (1$0()-
55
!t!-- 0.1>11$")/4../21 -
S820
EIcc:troquasistaUc Field.
15
468 d.
Poynting', theorem fOf quasi-
1f18ur. 15•• Italic flflkiL
CirCtJilS
v I,
p - -
Ii ria . IE x H I - Ii do . ((V
-1IJ, rlV" . II"") x H) - 1IJ, - -.ff£ dV V . (
-
f1
rlV,,· I" x I..Hl
.." x HI
do· IIlV x II
Ignoring the dispIHct!lUt!nl current iJDli)t. we find N
p- -
,
#do. "1- SV.[-1l.. do.1)- ~ V.I.
where I. represents the currenl nowing into the volume through the surface An. whose surface normal is pointing outwArd. Thus, we have established thai the circuit theory's concept of power input is va lid only when the displacement current is negligible.
Problems 15.1 Considllr lilt! short-circuited parallcl plate shown in FigurA PIS.I, wilh wiuth w. length t and separation o. Find the zeroth-, fil'llh secondo, and third-order electric lind magnfllic fie lds. Show tll&llhe sum or the quaslslatic solution is equal 10 Ihe ruil wave solution as presented in Chapler 6 for II IIhort-circuit Il'I:Insmission line. Assume that the current at 1. _ 0 is 1fl) _ I. t:os(loIl) and thot a ll fields are funcllons of I and z only. 15.2 Calculate the tolal 7.f!roth-ortlt:r stored e lectric energy In the parallel-platA rAgion shown in Fi)!ure 15.1. What is the zeroth-ordAr IIIOfflci ma)!netit: enef1lY in the some region ? 15.3 Calculate the total first-order slored magMlic eneTtO' in the parallel-plate region shown in t-·igurfl 15.1. What is tilt: first-order stored electric energy in Ihl! same volume? Ot!note the maximum 10la1 fil':lll-nml!r lItored ma)!IIt:lic energy in the
Um.
c Fields
Problema
469
•
--
, !
,
F======~"
1-.1 -- .- -
,
11"1.",. P1S.1 A PIU'llllel plate whh ,horl. r.ll'f':ull CUfrenT , - '. CUll 1",11 at z _ O.
region nnd V~ the maximum lotal zeroth·order stored electric energy in tht: sume vol1lme. Show Ihat
V'I' H"_V,lk.r
Vt!~
J
15.4 Calculate the 1011:\1 second-order Siored eleclric encflO' in tha parallel.plate rCl!lion
shown In Figure 15.1. Comparc il with the 7,ernth-order stored electric Anergy. If\? is O.V, 10111!1. whllt clln you say aboul the relolivtl mll,ljl1ltlldA~ of the zeroth-ordel' !>Iored fllecl rlc energy. the firsl-onler slorM mllsnetlc energy. and the second ·order slored electric enATS}'? 15.5 Find Ihe total zeroth·onltlr slured alfl(;lric lind magnetic energies in II parallel plale with 1:1 !>hort-circuil current I - I, cos(OOI'llit ;t - 0 [rAfer 10 Problem 15.1).
.rface ished
, the
t5.a Find the higher-order stored flleclric and magnetic energies in the parallel.plale reMion shown in Figure P151 up to the thim omltr. If P _ O.IA. compare Ihtl relativA magniludcs of thtlse stored energies Use the 10lal U!roth-ordar stored magnetic energy fOllnd in Problem 15.5 for comparison. 15.7 A coaxial line 'melers lanK is filled WIth II mlllerial char.1cteri;ted by f I:Ind 11. The radii of the innAr lind the outer conductors artl U I:Ind b. rp_'1 pectively. The voltage between Ihe coaxial conductors is V I r.0I11"," . !'Ind the zerolh-orderel~troqllasjstatic field £101, IhA r.urrentl lOl, the charge Qt-I, ano thtl fint -order currenll'lI. Express thtls~ in terms or tht: pl:lra melers VI ' fl, n, ", t, 6, and lol l
th w. eclrlC e lull slime lod z
15.8 Two cunr.flnt ric spherical e lcctrodtls or radii fl and h, respcct ively, urA filled with a material choracltlrized by f and 6 The voltage iJt!tween Ihe e lectrodes is V, cos IOOI'J Find Ihe 7.erOlh·order clectrut.jul:lslstatir. fie ld ,.; ..., the currellt I t-1,lhe charge Q IOI. ilno the first-ord er current/ III Express IheSt! in Iflrms or V" a. b. t. 11. tlud wi.
_'8 i un
t5.a Show Ihatlhc time nt:eded 10 r.harge a Van de Craurr Me neralor shown in FigUl"t! 9.241:1 with radius It to a maximuill VOIiH8f1 of V .... by applying a dm'1!inl!l r.llrrent I is equal to bl. RV... .I1. Calculate Ihe chal'gin)! tillle I if H _ I m, V, .... _ 10~ V lind I - lO 'A.
"'me Iglon oma I
the
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Solution:
16.
MagnllloqualiltaUc Field.
from 116.721. Wt! find 2
.0:
4. X 10
1 )(
,.' ...1 -
2 X to} x 2 x 101 X 1.0
... )( 0.5
-
64
X
101 N
- tl.5 1Ilt!lric tons
Problems 18. 1 A 911lull circular loop of 5 mm radiull ill placed I m IIWd)' from
il
50-Hz power line
The voltage induced on this loop is measured al 0 fj mir:rnvolt WhallS the currt:nt on IIII~ power linc? 16.2 Assume thai the Cllrrent on the infinitely long line shown in FiS1ITfl 16.1 IS the Iria ngull1r pu]:re shown in Figure PI6.2. "'I ncllhe Induced vohil~t: un the rectangula r loop. Usc the fo llowing rl::llll : U - Z em. b _ 4 em. and d _ 1 em
18.3 Consider the network shuwn in Figure P16,3. The magneTic flux is increa.~ing al II nllil of 0.5 Wb/s in Ihc direction pnlnling into the IHlf)tlr. Find Ihe readings of thc voltmeters shown.
1S.4 Find the readings of the vullmtltt:rs shown in Figure VI 6A . Thll mAgnet ic increlillilll( al a ratc of 0.5 Wb/s In the directlnn poinlinl( into the paper.
, , PI.",. "16.2
nux is
= d.
Problem I
511
~----~ kQ - - - ....... I p;!
PI
I
-+ 0' \ (000\
I I I
/
, lO
\ 00 I \ 0 In
I
"J'>,/
I
\
oe
\
00
he I"
././ ./
\
./
"-
llO
"-
"-,,p.
HO
P3
Flgur. P11.5
I, he U5.5 Four resistors form
R circuit 8S shown In I,"gure P16.5. The total magnetir. flux linking thA circuit is increasing at R rale of U.5 Wbls. In the direction pointing out of thtl papor.
;,
(a) Find the dirt:(;lion and magnttmle of the induced current in the circuit. 11>1 Find the readings orlhe \IO]lmclcrs VI and Vl .
18.& Two resistors are conlloctoo by wires to form
8
circuil as shown In Figure
P16.6a. The magm:tic flux linking the circuit vorios with lime. Figure P16.6b shows the time variation of the 1l1ltKilCtic flW(, The positi", value of the flux curresponds to the flux dirt:clOO into the pHper. The mognitude of the flux is for a lingle turn of 8 circuit loop Ihal flncircltlll the magnetic flux. (a) Plut the current/tl) vtlrSUlJ time. Bo sure to mRrk tht: scale orthe curNlnt.
(b) Piol the voltage V(I) vcrsus time. Mark the scale.
r----------, 1111 I
I
,t{WulJcrs'
I
I
v,
HOO
"0
0.'
,
VIII
I I IL- _ _ _ _ _ _ _
I
_ _ -l
{-I
"I Figur. Pte.'
512
16
Magoe luqullli.tat ic Field.
18.7
What illlhe EMF induced on a propeller hlade 111111 is 1.& m long and is rotating al 10,000 rlmin in the earlh's magnetic field (0.5 x 10 I Wb/ m' f'
18.'
Find the volta)!!: induced in the rectangular loop shown in Figure 16.1 If it 15 rotating "wu! thc axis parallel tn Ihe z Itxis located at x _ d .,. ~_ AAAlIrnfl thai the angular frequency of the rotation is", and Ihallhe infinitely 100M wire carries a direct currenl of I IiIllj)efCS Show that IhA Induced EMF is not a pure sinusoidal voltase. It is approximately !'inusoidal when d ,. 0
18.8
A magnt:!tic c ore is made of a mllleriAl whuse hysteresis loop III "hown in Figure P16.9. Noto that thill hysteresis curve is not a MsqU8r8 loop.~ To read the content of the core, two pubes arc applied 10 the wirfUI. The currents gonerate an J I equal 10
200 AJm. Thccore hasan liMa of3 x lU 'm 2.
( a ) What is the voltage induced in tht: sensing wire if the corA i.~ originll.l1y III the " zero" .o;la!f! (at point C)? Assume that swtlchlng from C to A is linear with time IIml !hat it is completed In a micro.!JflCOIUl. (b) What is the voltage imJucoo in the sensing wire If thA t:Ort' is originally 01 Ihe " one" stllle (at point AJ? Assume that .o;wlh.:hinl( {"om A 10 A' is linear wi th lime and thnt it is comptAtoo in U.5I/-s. This voltage is Ihe "noi~A " voltll.~tllrecause it would ideally be zero if the hysteresis loop were tI IJt'I'fecl square. H (WIlOOI'»
""r
lIqU
meter)
,.,
S#lTlslng wire
'z;.:;.-. 0.20 ....... ------
A'
A
" - 200
100
100
200
- 01
Flgur. P11.t hy~llIrll5llloop.
H (omperes per mf'ler)
C
t'erntll t;ore OIemory Ilnd III - OJ
18.10 Find the 10Ial expanllion fOfCtl acting on the surface of lin IIlr-core solt'noid that has 100 lurn.o; of coil and radius 0 - 1 em, length 1/ - 10 t.:ln, lind currenl l _ 10 A . t • • t1
HAptlll.t Prublem 16.10 for the CMIA in which 100 turns of coil are ....,ound OVt'f II. (ueromagnetic core with II- - 100011-0' The current hi 10 lilA, with 0 _ 1 em and f - 10cm.
t8.12 A r:npper IJipt: of radius 0 - 2 cm and Ihkkness d ... 0.1 em Is placed In a IIOIf!noid that has 200 turns per mAtflr and is t'xcitcd by n 1000 I ll. 10·A r:mrenL Thtl conductivity of the cOPllPr is 5.92 x 107 mhol m Ca1culatA the POWt'f per meter dissipaled in Ihe c opper pipe. t S. 13 A tran .• forlllt'r similar to Ihe one shown in Fil!lure 16.11 is made of a steel with relative
IJCrmeability equal to 1100. The effective length of the core density III B - 0.3 WhI m'. N l - 100, N , - 1,000, I , - 60 A,
I!
40 em. and Ihe flU)(
(a) Find I., assuming that the transformer 15 an Ideallrnnsfofmer. ( b) Find I,. using 116.37}. ( e) ComPQre the two an.w/ers.
ltie Fields -olating at
513
Problems 11. 1.
is rotating e angular -ct current tage. II is
Con!\ider II magnelic circuit lIilllilar to the one shown in Figura 10.11. The efreetivtlltHlMlh of the core is 0.4 m and its permeability is 2000 I'U' The cross-sectional area of the core is4 x 1O - ~m z.L.elll '" lOA, 12 - 24A,N\ ... 50.and Nz _ 20. {al Caicu18te the B field in the coro. Give both the direction and the mllgnllu.nA. (h) Jfllisa~ wilhf - 60Hz , whlltIl.I'fl I V ll llmI I V 21 ?A9sumelhalthcmo8nctic flux always staYI iUlhe core without any leakage.
t8.1s
in Figure ;ontcnt of , equal 10
The primary coil of II trAnsformer has 150 turus lind the seco ndary coil has 450 turns. The errectiV6 lCIlMlh of the core is 0.5 m ond the flux density in the core is 0.25 Wb/m 2. The transformer is similar to the nnA shown in Figure 16.11 . As~ullle that J1
...
60 A ann there is no flux lettkllMe.
(0) Find 12 , Ilssuming ideal transformer r:nmli tion.
Ill)' at the with time
(bJ Find V 2, assuming ideal lransformtlr condition and V I
Illy 1:11 tht! with time
ity equal to 1000 ~. (dJ The hysleT6!'1i!'l loop of thtl core material has Illl area equal to 90 Wb-A/ml. Willit i!l the power loss due to the hysteresis in the tran!'lformer1 A!'Isume that the core has a cross-sectionAl Are/! Aqua\ to 4 cm ~.
JeC8USll
Ur
it
meier)
jltmpere! IJ'l'r rn..,tO!r]
-
cos(120wtJ.
(el Find 12 , taking into consideration that the core material has a finite permftahit.
18.18
Figure P16. 16 showlI/! magnetizl:ltiull curlle of a core used in a transformer. Notice thttllhtl hystere!lis is negligible in this case a nd that the curllA Is linear in the ranse o ~ I H I ~ 150 AIm but salural!Ki when H is incrtll:lyed beyund this nmge. Lei us now review Example 16.10. BllCllUse I VI I - wN t "+, we want to tll\e mllximum "+ in order to minimize the numbe r of coils in the Irllnsformtlr. UyiUK Figure P16.16, explain whlll will happen to the shtlpe of the '1'(1) and consequently to the s ha pe of VI(tl if 'I' is too high - for example, if "+ is 50 high liS 10 cOITespond to B _ 1.2 Wbfm2.
18.17
Estimate the approximllili power loss attri buted to hysteres is in the ferrite core shown ill Figure P16.9 if the core is switched back and forlh between W"lftmM and "oue states 1000 times in a second . Assume that Ihe core has I:In IIl1erege redius of 6)( 10 - 4 m and Ihllt itscross-sectiOIlHI area is 3 x 10 - 7 m'. M
B Iwebers per ~uart: meier) 1.0
1 !hlt! hCll>
\
05
lei over a ) cm and
- zoo
noid Ihal Ictillity uf ~ in the
~ relative [ the nux
- 100
100
- 0.5 -1.0
18.18
ZOO
/ilamperes per meter)
Flgur. P11.11
Show that the mecha nica l lorque required to drive an ac gener::Hor is nol constant with time or, 10 he eXllct, that il consists of II (;onstant limn lind a term Ihat varies sinu:wit.il:llly with timt;! with a n angular fr!XIucncy lw. What is the time average of the torque? Express the torq ue in te rms of the Mila of the winding A, tht! currt!nt I , the magnlilic nllx density B, and tltt! phastl angle" between Ihe voltage and the current. Plot T as a function of t for tt - O.
514
16
Mtlgnetoquasistatlc Fields
11.18 Fi)lure 16.15 depicts an fling rulilleu in a constant mtlgnellc fiflld . It illustrates Ihc operating principle or Ii single-phase ae generator. Let us nuw consider
tl.20 WhAI is the lulBlmcchanical tOrqu8 naeded lu dri\IQ tho three-phRSfi generatur that you hove designed for Problem 16.19? Expross this torqllA liS 8 fun ction of time in terms of the appropriele parameters. Pial'/' AI II ruuc tiuu of timo, and compatfl it wilh that obtained in Problem 16.18. 15 tho \nslantRnooU8 1Ilt!Clllmical torque "smoother" [riDe!! its time-averago value Aucluale ll:!ssj compared witn that for R "lngle-phllsl:! Htmerator?
App.
11.21 Design a coil configuration similar to the one shown In Figure 16.17. Desig" it [n such
a WllY that it will produce a rotating magnAtic field in the armature-slator air gap and tha t the field will haVA an anguillf speed equal to w/ 2 when fflf.l with the 3-phase current "hown in Fixure 16.18. Prove Inal your cl ~i)ln is correct by draWing inHtlllltlilleous-currenl diagrams .'Iimilar 10 those shown in Figure \6.17. 18.22 Show qllalilativtlly thai the torque genera led by an induction motor may be varied by
chlm)ling the resistance ('If IhA w indlllKS of the rotor Figure 111.2 \ .~hows lorque cllrves versnll v,lv, wilh three different values of rotnr conductivities. What are the fflillt ivtl lllllgnitudes of "1'
tOfllUt: angle is negative-that i.e;, when the position of Ihe rOlor magnAtic-moment is ahead of Ihe magnetic field? 11.24 Consider Ihc colis of Ihe magnepllllltl's track shown In VlgurA 16.25. For the
magneplane 10 travel lit a speed of 250 kmlh. what IIhould btl thl:! distance 2, in melAl'II? AssulIle that the power Is providAd by a thrl:!l:!-phose SO·Hz power line,
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Appendix E
Answers to Odd-Numbered Problems
ChapterS Chapter 1
l ,l(a) 5t;3 (b)ll+ j (C) -26+;2 (d)
1 .5 !: 2 1140. ''') 1 .7Proof (e) 3 cos ( WI ...
(b) - lOR
-t
t.9 (a) .Jicos (WI
... 4 cos{wL ... 0.6) 1.11
139 - 4.t (e) -55 {dl Zlt
1
~ (5t
1.15
~)
89'" zt)
2.2
-t
lal 229
j t.41.3 r.olwl;s in wt;1
!~)
(b)4 cos(wl + 0.6)
- GI ... 59 ... 21 I
141 1.13 Proof
1.17 Proof ollliltild 1.19 Proofomttl&d
v9'
1.21 Proof omitted 1.23 iwllJ -14) t ... 8(1 ... Jlt] 1.25 lal( - 1 ... j:l)9 ... (1 ... j3) t (b)2:t+/1-119+(1+J1t (e) 5 (d)4t - Il +j3)1.( - 1+j3)t 1.27
Sketch umittod
Chapter 2
2.1 - 6yt 3x2,.6z 2.3 Proof 2.5 Proof 2.1 No 2.9 B{y,tj .. O.3(Ie/w) cus (loll t icy) I: 2.11 £ 1 ... E z• H I + Hz 8 1 t 8 2 and OJ ... Dl I IUptlrposilion theorem 2.13 Proofomiued 2.159 )( E . 1...,8 , V)( H _ r - ;",0. V' B .. OandV' 0 .. P. 2.11 Proof omitted 2.191, f: 2.11 prooromittcd 2.23 UIII V~ .. 1.13 x 10'
Chapter 3
3.1 3.6)( 10 · II" Wlm 2 3.34 x l ~W 3.54 .1 )(1013km 3.7(a)rtu.llloc (b)m - ' (el ll8c - ' (d) soc (e) III 3 .9 (al 2.63 m (bl 0.704 m 3.11 Yos, f direction.
~. - .1 ~ E~f 3.13 No, MRxwell's equations not satisfied 2 .uo &.s~.) Rixhl-hand c ircular polarl7.11f1on (b) Rixhl-ha nd circular polarl7.11lion v /Ag!, _
(e) Loft-ba nd elliptical polarization (d) Linear poiafi7.Atlon
Chapter.
Chapter
fi ,!.pruof
3.19 (61) 1 (hI 1 (c) 1.58 (d) 2. 12 3.211 .34 )( 10 - ' m , aluminum foil is aoout9.5thick 3.23 2.65mW/m 2 3.25(aIE._c- o.h e -I'u. (bJII .. f(o .5 - ;o.5) 0 - 0-,. e - /I.I ·51 Ie) Skfttch omitteu (d) Sketch omitted 3.270.6)( lU e In omitted
Chapter 4
.4.3 (II c (iI) r (iii) b (Iv) a (v)d (vile " .5 Yes. circulaf shape, 0.30' on each surface [0 il{ the !e nsth of t=ach side of Ihe cube,) " .7 hevelled tin,lj!e - 35 -; , 1 . mirror making 70- with i: axis; i polarized 41·. 1Eh 1- '21 f'.. lIl - t= .,. , c ... ·I. 4.1 61.9
0
11:: 1- ~1,.:.lIl + e-pu·cil;,' I.IEh l_OS II-:' llt - e-.a'· le -·",..,.".I Ebl_ ;'! 1,.;.11 1 + e -rlu' lc -'-"'~' ...11(a) 9t, (hI 75M Hz 11
520
Chapter
521
Answlln to Odd-Numbered Problems 4.13 cos-'
...,t"", . -
p , f,l
cos
' Ilp! - p! ) •
I
t , l~ztJ
-Pllt:
ttl
4.15 lal 80 cm in
front of the plate (b) 2V/m 4.171 ·9'.)61r..1 4 .19 H...:JI/~""-·~ E' _ Ii c058 - .f: lIinS) H,.qe ,.. H' - ;-1 I~ I~- . " -
E' _
Ii cos/J + f sinS)
H.,,~
".-.,•..-.
where" -
~~
4.Z1 x .. 0.87 m. y .. 1.5 m, 12.04t!1I 4.Z3 Proof omitted
Chapter 5
5.1 proof 5.3 1.875 kHz 5.5 E - i: on lower platt!,
J...
En~. H
.. -
9 ~ c'". J.. -
E c IU on upper plAte 5.7 89.33•kW i ~
E.,
-f.-alb:
5.9 proof
•
•
5.11 5.26- 10.52 GHz for 2.85 x 1.2ti2 (em) wavogulde. 21.1-42.2 GHz for 0.711 )( 0.355 (crn] waVAguide 5.131 .318 MW 5.170 .. tAn - 'lnnlmbl 5.195.83 CHz 5.15 Ey _ E\ sin (·uJ(JJflk.. H ~ ~ (P.,k.tw~) sin (rxlo) elk,. H... UE ,rlw~a) cos (rxlaJe'k.. 5.21 Proof omitted 5.Z3 (0.866, 0.5. 2) whcrok. _lw 2 III _1 ...10)21112 V 5.25 proof omitted 5.Z7 A _ 4.93:i of 7.469 - :It 5.Z9 (II) I I - P....!e-ik ,.,
,
+ J3l ",7
•em
Chapter 6
: 107
'. Chapter 7
,n foil is
60
••
jI."a
C
2rVo _ ,I - - - e Jh
21~.IISillb l
•
7.1 (0.75, 0.433. 0.5) 7,3 Proof omitted 7.5.t · .. sinS sin¢>. y . 6 • coshin¢>,9 . ;, - COSIP, t . r
6.. cusO coS
.
. ~ .. - 8i n~, t . ~ .. 0
B- - sinS, i
.r
."
7.7 Proof omitted 7.9 Yell, imlJrovcd to 18% 7.11 I! _ (_ P) jkJaz'le-Ib. lineAr 7.13 (8)0.314 Vim (b) 0.028 Vim (cIa VIlli 7.15(8) 1 (b) 1.5 (c) 1.ti4 1.17 Six lobes: beAm width . 19 .2° olong41 .. 0; beam width .. 2ti.4 ° 010ng¢>_4·1.8° 7.190=4 7.Z1Sketchomilled 7.23(a)-!:IO- (b) 6: 1.414:0: 1.4 14 Vim (cl Sketch omitted (d) Skelr:h nmitl~d
- jO.5)
r,
Va
(u,lbJ ...... 1000 6.5 I.- t -
(bl Sketch omitted (c) QI:I 6.9 t_}2 .96 (b) z .. - 0.35).. (c) 24.5% 8.110.342)" 6.13d _ 0.2Scm 8.15(_)1 .26 t ;\.61 {bI0.54 (cll 8.17 (a) 0.67 + ;1.33 (h) 0.15 jO.09 6.19 I'roar omitted 8.2114 .2 kW 6.Z348.6'lb 8.25 Sketch omitted 6.27 Sktltch omitted 6.29 Sketch omitted 8,31 Proof omiH9d
I)m - I
Ingle
~
..
8.7(8) IV{zll - 2 I V.llcoskz l,l l{zl l"
).
I
8.12000V 6.3 (allol )
Chapter 8
8.1 Uw/U E - {krf « 1 8.3 When bockground is dl:lrk, one sees light sr:attered by the smoke particles. Blue light Is scattered \IIure strongly than red light. Against a bright background. one Sflelllight IJl:Issing through tha arnow. The blue Iighlgelll !lCaucrod. and r&d and yellow lights suffer llt!!1 scattering. 8.5 proof omittoo 8.7360 km in radius 8.9 Train lA movinK toword the inler'lection. 8.11 Band· width _ 59 kHz, 6.7 jd for t kUl resolution 8.13 Circuitlr but opposite hand 8.15 No
• An.wen to Odd·Numbered Problems
522 Chapter 9
9.1 Exact: la1 5.5302 x 10 · l 0y (bI 5.54244 x 10 U y Approxlm~IB : laI5.5426" 10· 111y (bl 5.54256 ,,10 ' u V 9.3 x _ 1m plnne, y .. 1.5m pillllc 9.5 ( - PI 1.44 II 10 '1 L 'I Vim 9.1 Prooromitted 9.9 SKtllCh omitted 9. 11 Cal (J 40'll'th
.-!!.- (0.05) VIm 40nh < c. E .. ' ~ ruru '"
(0.0499) Y/m (bl (J oandh " r
4rf~
(2 _ e '(? .. 2r
t
21J
9.11 1~ -1
9.19 3 V. ind!!pondonl of path
Chapter 10
r<
9.11 (a)
h~ndr ;;J
c 9.15 10"
,r'
1) lIint: I drt! '(1 -t- r)J? .. - e "lr
(30- 1
1) 9.23 lal 4.03 x
Idl .!L( l -t- ! 4 '11'l a e
(eI 0.2% 9.13 E .. 0 for r <
-.!L (bl
~
IC1 .!L(1
Chal
1b)
t!
4'11'te 4T,r: 4n r e 10 - 8 G (b) 330 kV Ie) 0.4 rnA
b
10.3 1.8" 10 -~ N (allraclivt:ll 10.5 (a) 112.4'11' x 10') Vi m (hl n)( 1.2'11',,10 · aN(repulsivtl) (e) No 10.70.36mm 10.9z .. '" 3. 14 cm 10.11 (a) Vo .. I.R74 X 10' m/ s. vo..- 1.867)( lO ' m/s. v u• - _ 0 .163)( lO ' m/s (hI X(I) .. !S.78 )( 10 14 )11 - (0.163 x 10')1 Ill, :.:(1) .. ( I .Sfl7 x 10' )1 III (e) x .. - 3.52
.! fj
10 .1 _
•
x 10 4 m. z .. 3" lO 'l m A ( tl -t- l2) 10.13 I' ruoromilted 10.15 5.3 x 10 10 farad 10.17 (a) d - , (b) QI .. - <,Q - , Q, • - .,Q 10.19 2'11' t \ '(2 f)-t-f l 10.214 .97 x 10 A I 10.28 (a) ciSl2Afo
Chapter 11
11.1 . 3 .. . )
,
f l
t2
-In !clu) -t-
(hI
..
1l.1l 17.5(,IJ. F/ m
Yot:4r'n (taU~) Sin9] 2
11.13 - Q{d' - a }1(4Talo
(
11.541(6) .. Y, In ( 'lin
)'"
"
+ rl ' - 2ud coser"] _ II'" + d ' _ 2rd c0s9 )'IJ.
11 .17 IPrJ'/ 12rtfd -
11.19 (d/ olq l/14Tfjd - bfl. where d -
i )1
11.7 12.3V 11 .9 Sketch omitted l
11 .15 ~ t _ -d' ) t q(o/d) I CJo. where fl, 4Tt HI Rl 4l1'tr
" ,' case H, _ r l I d ' - 2 d
]
In (hIe)
- (j/2Aco (allnclive)
<1>2 11.3'" - Yin (blp)lln (binI
t
In (lan~). p.
1['
ChI
bll whe re d - cr/h, "I\ractiv~
cri b, aUrar.li ve 11.21 200 slnI2'11'x/ u)
sin hI2Ty/ o)/ 8inh(2Tb/ ol
/ oj sinh(n'll'y/oj sin[nry/b1 !linhf nn/ b)l + . / 2::- -1n [Sin\ nTxsinh(nrb/ o) Innh(nTo/ bl .-
..
]1.23 (400 '11') 11 .2.5
V,
(-)
ChI
(; ;)
('
;
(e) 0 " • ~
_.!V!!:"'''-'_
(~ ~) ~
(d) Q
V""
ChI
Chapter 12
12 .1 C .. A\,) -t- tJ/2d. C _ A{o) + cJl )l2d; C in parallel wilh G 120.3 (_) lin {olp)l I (bIn) , In (clb)] (2To\t) Ibl lill (aJb)l(2'1fcJ1 t) + lIn (blp)l(21rotl) (c) -
2""
I'n
0\
(11
•
523
Anlwers tn Odd·NumbeNNl Problems tz.$O.92 x 10 ' mho/m 12.7100V
12.9
(~\ jJl
12 15 p
.
Chapter 13
+
.. 600
y
13 .1 ! 2
oJ
)C
100% 12.11 12.3 O·m
C! _ lts cll(y'"
12.13 Sketch omiltAd
50
t
6 - roY1J'f I pO - 7OrJ d)lJIIl I12 where d .. 1t70)2 ... V11I2 , fO .. (501~1f{70)1 t YI
.J2 fI{ .... b) 13 .3 H T
0 .270
13.11 H All .. t
2!. 4..-
1
II .. ( - .tlJ(d/2) for Y ;::. d/2.
13. 5 (a) 0 y}lI'l] (b)( -I)U(4TO) (clQ (d)(
t
tjll{4Tnl
m
-G
dx
J
[0 + Ih -
xft
12
Ib - a 4.
H ili.:" t
~,
)
2 2 (bl I(l - (l2)/I2rP(b - (II)] (cjIlZ,.-p
< (- dl21
H ... J{d/2J fur y
13.7(1I){ - t)fo{dyJ/14';uJ 13.9 Zo"
.. (- Illy fur I y I <
50
d[(y ... fil~ + {7otl' !2j'
arl llI
IICD .. H"D HIM .. t
Jib ... 1,1) 4..-
I' II '
(b +
__
0)1)112
13,132.8MHz 13.15(a) x .. O.04rn,z. - O.OO725rn (b) - 20.S' 13. 17 1.33:0: 10" N/m Impulsive) 18.19 I.} Loop should be placed horizontally or vertically in the Mst·\WSt direction (b1 7.ISS)( 10·) N-m (cl Vertically in nurth-south direction
13.21 {II} H • ~
.Je.... 2TlIt
(hI U H
•
I' 8 E(eI 5)( 10 . Him 13 .28 2.14)( 10 • 16w
4
c In (db)
'" [ 1
Joulelmeter 13.za - + In (blu) ~ ~ zt 2... 4 (c-h)
Chapter 14
Chapter 15
-
c!
c!
1
J
+ b ~ + t 1 cb 4fc- h)
14 .1 Sketch 14.30 21 weber/ fll 3• 100 AI m 14.5 0.11 wel~r/mJ 14.1 x,y, - 1. X'Yl _ O. x,y, _ I. x.y. - O. x,)') · 1. xlY) . 1. x.)'t - 1 14.!14.233A 140 .11 t B.32A '4.13 0.29 webe r/ m 15 .1 [I" _ 0, 1-I1t)
_
- Yl,cos(wl}/w. [1'1.
il..,/oI7.lIin(wIVw. ~ 1I. 0: ~ - O.
tl~V( Hl Hl r sin(wIVw; K II 15.3 uW _ ~ ... l\1 (~) sln1lwl) (i) owi·. U~l - 0 15.5 U':' - o.
Hill _ Y IfI-J'f/oi
Hl t CQII(wIVw:
£III -
0
lf
<,)
• I' uw --.....; cos1lwtXwm!1 2w
15.7 £~'I • v . cO!l(wtVIP In (b/ ull. lit) _ Z..-V",,, cos(wlj/ln [b/o~ ott)_ 2... V s in[wlj/ln (bIoi 15.9 11.1 sec
2... Vaf i cos(""tj/ln (bIoi. 11'1
Chapter 16
tI"""
16.1 101 A 16.3 VI 0.25 V. Vt _ -0.15 V. V, - -0.25 V 16.5 (.}O.12~ rnA (clockwise) (hi V, . 0.25 V; VJ 0.625 V It1.7 58.9 mV l 16.9(.) -O.llV (b) -0.012 V 16.11 0.395)( 10 · N le.13t_1 6A (b) 5.91 A 16 .15(1I)20A (b)33U cos {120..-tl (c)19.78A (d)1 .08W 16.110.18mW 18.19 Sketch umitted 16.21 Sketch omitted 18.23 It b8C1l1l11:19 a generator.
