Solutions Manual for Shen and Kong's Applied Electromagnetism

Solutions Manual for

Applied Electromagnetism SECOND EDITION

Shen

~

Huang

,

solutions

Manual

for Shen and Kong's APPLIED

ELECTROMAGNETISM

Second Edition

by

Liang C. Shen and Frank S. C. Huang

f]~

He

PWS-KENT Publishing Company Boston

PWS-KENT Publishing

Company

20Puk Pl~u Bolton. ~tusa,husms

Copyright : - wt}x

vectors:

1.23 Find tho phasor notation of the following vector: C(z.t) : (aUH) E(z. t)

where E is given in Problem 1.22(b). -1.24

From the following complex vectors, find CIt) in terms of wI: (0) C - )1 f - jkz)x I i exp (jkz)Y.

j(i - jY). and (c) C - axp

;9, (b) C -

Problems 1.25

19

LetA-~+W+(1 + i2)2,andletB=-~-(1 (c) A . B, ami (d) A x B.

j2)y

I jz.Find(o)A+B.[b)A-B.

z

u

y

Flgur. Pl.20

1.26 Find A . A • am} Re (A x B "j for the values of A and B given in Prohlem 1.25. 1.27 Sketch the trace of the tip of the vector A(t). where (0) A + j3y.

4x

1.28 Calculate A . B. given A 011111 times?

x + j2y

and B -

2x I

x-

jyand

where (b) A-

jY. AreA(t) and B(t) perpendicular

!.:!.

(4.) ~ ... b a

S''''J3

~-.k

OJ

hJ. .!= ,.,j'2.. I. IS' u,4,of"

!J. f1:. h!..

c II+j

b

J

B:..~. 2~.()8LI7S,'I', Soil!. c ~.tl L-I4-7.~J· He I ej4IJtJ • (DSlNtoJ I,." [ejwt'J • SI~W't ,I e,iutJ =1 (4) ye s (b> yes (c) yes

=:r. TPiI)

Z 'l:. 1-+j • .;r e i (27l'fr+ ~) ~ Z I: Z ~ e .; Z, s 2t+ eFt?, Zc. 1~ ei(lrSw'i.

£-3St~~Y

y

,,~

Ac

-

,., ~ )l~J2YJ

f:; ....t

AU);:

:.

#II.'" §'=2.X"JY coSW/;

,.

_ ... "A·g#.2-2=O

,,--

Atot) AlII/.. 8(-t) Ar~ ""(:

".,

..,£"

2 ~.swt x - S,"I.""'" Y p.rp."dicuJ-A.,.. IJ.t 4VlY -i,'me..

ac - '2 S,A.wt y

{3(*)

ta,nd.,

4

r:r

2.5

18

35

Poynting's Theorem E(t) - RclxEo e-th c""/-

Solution:

HI II

-

Rely ~

Eu

p,-Ikl

iEo cos Iwl - k7.)

e""/ - :Y ~

WJ,l

?

d te

SIt) - E x H =

z -WJJ.k Eg cos' (wI

(5) - -1 Re IE x H" 2

)-

r-

iE~

Ut• - TCOS

t-

Ufl

n d

k~E' - ~

2w'J,I

(U)_fE~ f. 4

3.

(till)

n

-

Eo cos (wI - hz)

WJ.l.

I-

2

~ - k, 2wJ.l.

(wI -

kz]

cos·lwl.

kz)

- kz]

Eo

0

Jc2

E~ 4w 101

-2-

)-

Problems 2.1

Let A - 5R + tiyzy +

X3~;

finn \I x A and 'ii . A.

-2.2 Let I/> ~ xyz: find \It/> and v . \11/>. 2.3 •C

-2.4

2.5

Let a - O,X I 0zY + and (2.121 Are true .

03Z

and b - b,x + b2y + bJz. Show that equations (2.9). (2.11a}.

Show that \I x (0 + b} - \I x 0+ 'il x band 'il . (0 I Show that \1(, "'2)-

, \I

,.2.8 Show that \I x (AI-

h) -

'ii . d

+ 'ii . b.

tf>z+ 2\I \ and that 'il . ( A) == A . 'il,.\ + Pv2? You must show that your proposed solution satisfies Maxwell's equations. What is the appropriate name for the theorem you have just proved?

+

2

36 _2.12

2.13

Maxwell's

Equalions

(a) It is known that the vector a is equal to zero at one point. Does that imply that ~ x a ~ 0 at that point? Give a counter-example if your answer is no. (b) Does E = 0 on a line always imply ~ x E - 0 on that line? Give a counter-example if the answer is no. (c) It is found that the E field is zero on a surface. Does it foJlow that aBI at - 0 on that surface? Show that equations and the conservation

(2.22c) and (2.22d) can be derived equation (2.23).

from equations

(2.22aJ, (2.22bJ,

~2.14 To represent time-harmonic fields, most physics books usc the factor e- ...• instead of e"", which most electrical engineering books use. For a time-harmonic real function A(x, y, z, IJ - alx, y, 7.) cos [wI + .pI, find the phasor notation that corresponds to the physicists' convention. What is the corresponding conversion rule hy which phasors can be transformed back to the real-time expression? 2.15

Refer to Problem 2.14 about the notation a-I"'I adopted in most physics Write the time-harmonic Maxwell's equations using that notation.

2.t8

Whal is the range of effective perrntttivtry of the Ionosphere at AM bcoadcastlng frequencies'? Use the following data: N - lO,a m-a and f - 500 kHz to 1 MHz.

2.17 Show that the dimension

+ jy)

2.20 Show thal S 2.21 Show that S 2.22

e

j2

books.

of each term of squatton (2,361 is watts pet' cubic meter.

'2.18 Indicate in watts, meters. and joules (0) E ' D. [b] H . B, and [c] S. 2.19 Let E _ (i

"

and H -

Iy -

ji)

the dimansions

R-".

of the following

quantities:

Find S in terms uf z and wt and find .

* Re {E x H ei"'}. * Re {E e"" x H d"'}.

Compare the energy stored in a cubic; region one meler on a side which has a uniform E field of 10' V 1m lo the energy stored in a similar region with a uniform B field of 10' G. (One C = 10 4 Wb/mz]. The medium is air.

2.23 Repeat Problem 2.22 for the case where F : 80 fO and JJ. : JJ.o for water.

the medium

is water

instead

of air, Use

VLF

5

2.a

V·

1..'1

g=

:=a» t;, ., 2

D = V' (2 X';) = 2.

-

V)(

C%,J

,{v

ej

0 oj

oJ

;

g :: -~ (0 ,J i -It ej fa, ) :: - j :§ '-l

~ = 0.3 ~

e ji.'

iJty,t): 1. ".t H -= J -

o,~" CM"((.Jt+4tJ);

:t,

~

./0

it: W'At..

=-;tB"

f:/x

Ez "'_

C:;"

E; •

~"'~1f;~H"J

=

i> = E. E

J := 0... V)(e,

;

£

0.]

+

V)(.lii •

V 'I. (ll,"'N,_):::

V, 8t =:

Q. (

tf· D~

o- c P, t Dz) ~ v·5, r'V,Dz.

Ir

040

v

~2. ) r:

VII. ii, +- If !7, K, + r:;'

X

E=

:;~

,'",fo t/,.'t

D

I

V·~.

~

.!k ..l:lltt

-t4t.J)

J,

~/',fl/'"

111'1

Ii it - It 5" • - Ie (6, ~~ ) • - k ~ = J, + It p, + ~ +-It Pi -(j, ~h) + ~ ( 5,,..li ) = r. + ~ ::; -

~

e;, =: e + 0 = 0

Of.

&1:' fVf-

fr/~

C

E,,..~ . Hoi: ~ Ji, .,. Hz. , ~. ~., 01 D~. ~ r Dz. SCk.,..' e s J~ '5", ~ jl. ~d fvt• fv, f Iv: .

.',

CMJ(lJt

v'D;=/!,

liD,., v,S;.

'1'1( ( ~ .,. Ez ) = V J( i,.,. V j( ~

s;

-i)l

/to,) '11,5,.0"

-Vtli1,cJ,+

~ 81. , v)(!lz"'.Jz

of

wm .

-1(=

I/'f

(wt

CAJ

H:: ;;

a

.sAt;r{y

G "'"

i11Ax4l1l1;

~!UA/"()'"

t:

.,.hUt,,,,,-.

TA/$ ., Me. S'jJtreos,'lt't;,."

2.:.!.:

(tl.)

2. If#.

s;....1

A..

NG).

"'.

'ii' ('V'ld )~C

=> q.j

A('](,'t. ),():

a(r.",})

+jWrJ·g

•

=0

y-~/!.,

.,11.

~

£ = [I'

al:

or:

( (-

S'OOKJh '

I MJlJ:

I

b~'t

£CDS~'" f

V"A: ..

:::L) ,-,':

E

W1' «t.

=

[,-/' 0

-I

INe.'

=() ~

UJ

""

yCoS}+-r"".s;t:'~o

VX·Ec;e.c5}~0

)w(-8-. fl·g )=0

I7·B=o -

V·Q =- tv

~

~

(J.(x.1,J)ej4

-

A(;a:.j.))

e-i ~

v,g.f.. __

J

=[-ltllt.",(I."./~-t'J'. :I':J J

"lTrl,,"JJf 3 f •

(:::;.co'i" /] = - 80.43 E.

E:: t. [/-

6

I:

c:nt.

(1?I.~.DIt.-..::!P

J) U5(-wt .. t/'J

::: a(~~.

q.""/~-J' ~.('1-1I./~'" )lJ = '·:2IT1f.5'JI,I()1 IYtE.

A

VICE'"

V'§60

Cl>5(I.JI.I-'J

17)(/i.-=J-t'wD _

21

b"'-t

}=o~lo."e/b.,;t

eiwt: Afx,'.J.t)-= 1f'e[a.(r,1,J)eiiJe;wt} e -i ..t : II (71'. V·l.'O. /?e [a..fx.j. D iife-ilJtJ E \lttE::iw ..

c». o, OJ

4ft

$0

3+ 2

D)

F= 2:.S,....)·Y$,,,,,X (C,) No, 'E= ~J'·"'_J::.o ~~ o~ ~t p'~e. V-CfJX¥.. )1:0 =;.jWV·S=O (0)

2./3

l£:.; r;""J .,.Y-S,'"",X+ ;

No.

A..t ce,«.

s,'4-x ID '1 lit

,

'1_

~.o

,7

ell:: II

[

r:

%,[iij= Aj"", H'~]::' v-.s~, A.:::

WI

L

(c)

.M :),

= 10...,,_ ...,J

ell

-

[~1':[ l

~

~

[ c» ~ e-)17 'J:

~

t"..tt

",..

w.:tt ,..,3

- .]'-.J._ -~

= 1)JJi.. ".,] -

..... ~ s(

"

it,. ""H

:~e. J~'

8] -..t!. v-s., •

• [~(.l.

,

w::.,

v~ ::. [..1. (.J. ...'H-)' 7:::

~ It1,

J

U! (4.) [ g ,j)] ::. ¥; ;,.. = (p)

r,

Wak- Su.

1'1""

(c l·E1 ::._m~ £ ,.!:L In [j.i] e _1L,L::. t>'I

[V'(EKN)j:::

r.

H J = ~ ~ = ':,~

!1 !=(;#Jr;e,jl .......E(O=;(()5(~~~)-;~(wt-J) B. = (r - j; )~-j J ~ #(~):o& 9Cb.s(c.lt-~)+; s;,.".(u-E - J) 5': Ese;r::

~ [CD$l(e.Jt'J)+

I=l~i"=(;.,j9)~(;oI-j:)· ~

t-e

]» EI. f j

=;

S'',..,'(4.Ji-IJ1 2Z

-+ c-fli'e{i,J=

t, _....F· ~ ~j Hr

e;rit,[ieilAJt]:::

Ji-Rc{!ej..,tJ ,,~(.,bS~-}lr$;A.wt +( E;xHr )s;,...'4Jt - (E,.'rliir'" ~)( ~) s,~wt GDSUlt

~CDSblt-Ezs;,...,t;

S ~ g Jl i1:.{E,. )ll4.)~SI4Jt BloAot g eiwt -= (NAUjwt -ilz ri"II,)~) +- j ( 17; j;~l4t

Ref. g 7C i eiraJtl 2.1 I L~t!.:1

11f.#j fz

=

uswt - ~ x iI

~J(~

f,-"'ItJ't -

+ilrGljc.Jt)

E;x#A

S,',..wt -

-(4~~)CAS"'wt-

•

(

~lC

~

-

tJ

I

--

((;;;-

TJ.l.H·H :: T;c.o·(3

(~,«J(

Ezy. #r)(

:. s I- I?t [ €.ljwtx B e jwt J U,.,. ~!.E.e::; X J:.,,~//)·~(lD./ =VI, :.

=

i

UA::

s.s .;«, 8t> 2 J[,7f

3,98)(10)

E.r)( #z t.oswt :f:

5

(.1~~,

X/D-9"'(IOf)t..

)

7

ii.r

+ lr~~) s,"-wt UJ5AJt lJt)s; ..}t.Jt - ( 4'11.Jl,r 0# Ez ~ N/l. ) S.~tlos",t (ill)/.

e..os'wt.- s: ........ wt) - ~(rp~ilz. .,.'lz~iI~) s,',w.)tU$w-q

4.42,)f.ID-,I-(JiJAi'/,w.J)

... VA/V! ~ 9xlo

Ue= -L£ 2.

AJtd.

ii· HI.. "j ill

.n.t

ff ejI. - 1/1., - -rr/2 and polarized .

0 -

b, the wave is right-band circularly

• 3.16 Find the polarization (linear, circular, or elliptical and left-hand 01'right-band) of the

following fields: (a) E = (ix +

y) e-11et

(b) E - ((1 I il Y I (1 - ili) e-lk• ( c) E - ((2 + ilx + 13 - il z) e-1ky (d) E ~ (j i + j2yl el/Iet 3.16 Show that. if

0 = h and t/J. - rI>lt - 11'/4. the wave is elliptically polarized. (Refer to (3.131.)Do not try to obtain an analytical expression for the locus. [ust obtain a pair of parametric equations similar to (3.141, calculate E. and Ey at ten points (wI = 0.10· . . . . , 90°). and sketch the locus.

m Plane Waves ~ television sig.al wavelength. e following: (a)

Problems 3.17

67

Show that an elliptically polarized

E - (ux

+ lJy}~)

polarized wave can be decomposed

left-handed and

waves. one

and solve fur - 3.18

H'

into two circularly Hint: Let

Jk',

where a and h are, in general,

e

E - [a'x I ia'y)

the other right-handed.

I'"

T

complex

(1/x - iL'y)

numbers.

Then, let

«=

and 1.1' in terms of a and h.

Show that a linearly polarized waves,

polarized

wove

can

be decomposed

into

two

circularly

3.19 A dipole an tenua is in the x-y plane And makes a 45° angle to the x axis. A receiver attached to the antenna is calibrated to read directly the component of the E field that is parallel to the dipnls. What are the readings when the fields are thoss given .. in (a)-(d) of Problem 3.151

ric Maxwell's

- 3.20 An electromagnetic wave in vacuum has frequency rn, WAvelength kg, and velocity Vn. When it entars a dielectric medium characterized what are the f, A, k. and v of the wave in this medium? ~3.21

>-0, wave number by fJ-o and E - 4f",

Aluminum has f - (0' jJ. = fJ-o. ann" - ~.54 x '107 mho/m. If an antenna for UHF reception is made of wood coated with a IUytH' of aluminum ann if its thickness ought tu be five limes greater than the skin depth of the aluminum at that frequency. determine the thickness of the aluminum layer. Is ordinary aluminum foil thick enough for that purpose? Use '1 Gllz Ordinary aluminum fuil is approximately 1/1000 in. thick.

r-

ave? In what time-overage sa minimum rea of a city. the intensity

3.22

Calculate the attenuation MHz. Take the following

3.23

Find tile power density Problem 3.22.

rate and skin depth of earth for 0 uniform plane wave of 10 data Icr the earth: fJ- - ILl)' f ~ 4Eu, and if - '10 "mho/rn. in earth

where the field intensity

is 1 Vim. Use the data in

3.24 Suppose that an airplane uses a radar 10 measure its altitude. l.At the frequency of the radar be 3 GHz. Suppose further that the ground is covered with a meter of hardpacked snow. Airplane

nol, explain

(rJ· $«., (t){)..];m 7= '1J. 2.//Xlp·'S'.su.. J = .2o/A ~ 9. 93}(10~

~ ) ::: 0 ~

e-jk~

hi

/0

[fJ·

(C)

rn ,

2S')cID·~:ZOlT

a#: b;

WI""-NS 10/. g MW

xlf)-U

4.o'l~ID"",

;I

-:-L -r.Vz.g.,.",.,;Itw;tt.t. €

JV~~

IJT":::.

".,

.3. gl-)(/D 6m

I/::

:2. "3

= v" (E. e -jl):

- Aa.,-.e{.

4- JC

AL$t:>

£:(J';~Y)e·jl~ ,f('jAt

(b)

E.

({

Z

=

= J/.. 7+x /0 4-11

cI.,r~~I/t:>,.._ i

'IT

1

a

to'")

1

*E=

'j~'/l.ii

Jt

on«:»

'II 10'

6' "',,) 8

'14/0·,6 "'/"'~,

~/D")(D.')//O-I'x4rr)((,I.St!lO")1..

t.s

('1.·$'{,/l ) St(:

if (2. x S'7

;'1./0

tI (

S"4-

Ito K ~4)( J'S"I< ./..'3

Ii

c 3111D ~/ ".326110'"

:.

3./S

~b

[wJ =.ra4/su"

(A)

H. 3.9

mAu x

3)(loS

3,(,

sx

pOIlJ~" ~(.IIJ;-Iy .('0-'5'4)

=-,s.'

10'JiC O./' x /0-11

,.. (6D-5"4»)(

.f;.guvaey rrusg,t.

Vi.e.

I~

p'; (/'''- ~/01)1I -1-"

3.3

3.S

i.A, Sol -&'DMHJ

ri4At-ho,yJ.

Clrc"u..!ilrl,

p'{A/'U~,!-

e(~)::I-£~iljp(w-e-A.J)+;GC5(wt-~J)

pol.Lr'Jed.. ~ E(~):::.Yl£c()5't«-*x"'¥)+;UCtJ.s(u.)'t.-Ax-!) p"t-iju,

r; tDS(l,oJt-Al1'~.46)

(C)

g = [_(:J+j) X + (3- j) ~ J e -jk"

(d.)

L~f'f; - ha"..d .e.IUpll"A!l:; po.l~r'J CJL i.:: (jx+j2y)ejAI => E(-e)= -XS,~(I.)t."'AJ)-i~$,·t\,(.n.+~}) Lit1€4rLy p" LAn'3d ,

~

E(t)c ;

8

+

i(iD (,,()S(we -ki-O•

32.)

a

tXt -k J

rP a...

::0

wt

)

?a.-¢lIwl

~/a..:.b"

E,k(.t:)::

I

0

10"

E",'

a.

.geS"~

Ef

.7o'Ja..

0

4/ld.

Let

J..

.7./'a.

.6.f.3a..

.SA

.3424.,

.1'144

.906a.. .9"'4.

.9964.-

. 9'16el.

.'i6lJ~

.9,,6a..

. BIt/a.

9

,r.e",

••• a.' -r.E_' :.

L;,,~rl,

6

w)!~

pl>lAriJU)

e: f (g,. - j R.) ~ .!'= d (g..,. j k ) i( 4·;g) + Yi(j~.,. k.lj e ..j. ~ (Lt/6-AIJrv/u.) I.

a..re, : [; B·j!) - t (j~ - k. >J ( r'·9h.t -"~ed.)

9

wtW€,

gc(fA.+Yh)e"'jll}J

1~!J~t1i.rAf/~rm,

fNl.~

"u,..,buS. alW(

(:'2:'+yje/)~·J·~~+(£j/-Yjl2')e·j/q,J,

=* fd:,':

j(IJ:'-_!'):=b

... i)tc~""'I'P5d ~.s a ~jb) -

( x f(

.7070...

g-(;&"'YiJe;J·J.J.)

pDIA"'!J~)t(I&·~ltt

U·,-t;J.(./.4rt'j

i. :=(fa.+Yb)e·j~~=

E;~~.! '=a..

0

E. ' - 9 j .!') e -j1""I'l>Hd..

[ % f(

~

I

..~ t;i~t.JA.r1;J (R ~ of- 9E ) e -.iA} ::::(; e/ + ; j ~ '; e .j~$ + (; i.'..; j,i ') e. "J*}-

::(Le/t-hand..

}/«fJct/

5VoU*'A.L ~,..,..,

t:?l>D

1?u.""~erJ .

+ j ~') e oj RJ 4-

~I

eoD

?Oo

.Q4o... .fJ66a..

art. ~",~/,~

§:- ( ;

E,-a_CDj{tVt-~J+f/J4-;f)

a.~~ ~t; ANi. &1" a. (..II $ (l.o)~ - ~5 D) 2.030" 4-fJSo· 1>0·

pDh,.';Je4. wa.ve, ~

EIl,;pti.Ca.llJ ~

cx(t)=d.CCS(fNt-i.J+tP(4),

j

~~

~(a.-jh)

lt~(4-jb)"';/:(ja..+b)]

9 -}6a. -b)J ( ri'll..:f. - h~,,"u{ ). .

1JJt~!/c-f(a~j~)

e-jk;. (j'lt-luJ"eI.~"'-)

)

_ "/ =ttrr

···/g·A.

:!

.'. I g.$. I = II.~ r: 1.5'0 . I -E"~/I/r 2 {1 :: 2.IZ

•.

9

-Mtl.n. t:u'\d_

a., U1A.'

5.%0

i=f~

~

T/ ~(.UE::'

'""10'

/~-41"" ID7

1 >tJ'1f,(/o

<

7 = Yll-; J;( ~

-

4-

r

0

I

»1

1< s:>( -= 2"

L tI ~

r,

= 4. S'>i/~-

1

~O/t,

(~4,) c

~

I .':-

c

I-Jcr/41t E, 'l. • ~

m- e

J

-

~=

= 2.Ap

10"',.2\Y.

(

«I

.l: rFj (,1 ( 7.

I Fe' Tt;;;; =

:.

j~'()4f)-YL"" 2.

-,

".,

, tit- -,;;-1 ,.1(,..,) 0

-J-_I IF{; :: bt>"" T.

'5" ,"W/m"

~=)

(fUr- .,~C" t 2. [~';,;:. +ryllm) J = :2 (:. n. -I -#- (/.f = A~ ~ hW - It. • m -/ r:: o, '22S ,.., (It_",- 'Y'P4r.,.,/: Jtl./v.k) (b, E. ::::111/"0 (1.5[.) ( 1- j 9x/o-4)Ys.. - wJpp(I.Sf.) (1-j4.5'xID-f-) ~ {q.l = WI,M,t,JI.5 (4,5'IC/O· *)(~)

(a.)

Let lIoVt. c ve./ot.i..ty;"w

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4

Reflection and Transmission of Waves

- 4.

Flgur .... 23 Tn rp.cp.ivf) linearly polarized

electromagnetic waves. wire grating.~ rnayreplace metal plates Ior retlector antennas.

4.

Thus.

J. -

~(2~O)cos8e

(4.54)

Nole Ihal Ihe current flows in the y direction and that no current flows in the ~ direction. In fact. if the concluding plHtH is replaced by a grating of parallel conducting wires arranged in the ~ direction. these wires also serve as a reflector that is as effective as a solid conducting plate. Experiments have found that grates are effective when the spacing of the wire in the grate is much smaller than the wavelength of the wave. Grates are used instead of conducting plates to reflect linearly polarized electromagnetic waves because they reduce weight, save material. and decrease resistance to wind. Based on these considerations, some reflectors use wires to replace metal dishes for transmitting and receiving electromagnetic waves. An example of such a structure is shown in Figure 4.23.

Problems -4.1

ThA E field measured ill oil"just above a glass plate is equal to 2 Vim in magnitude

and is direct at 45° away from the boundary, as shown in Figure P4.1. The magnitude of the E field measured just below the boundary is equal to 3 V1m. Find the angle 8 for the E field in the glass just below the boundary.

3Ves

93

Problems

x

/ \Vz

Flgur. P4.1

- 4.2 The H field in air just above a perfect conductor is given by H, - 3i

I 42: amperes

per meter

as shown in Figure P4.2. Find the surface current conductor. The conductor occupies the space y < O.

J. on

the surface of the perfect

the following descriptions with the figures shown in Figure 1'4.3. Fields are near the interface but on opposite sides of the boundary.

4.3 Match (a)

54) the

(b) ( c) (d) (e) ( t)

medium 1 and medium 2 are dielectrics with E, '» Ez medium 1 and medium 2 are dielectrics with E, < E2 impossible impossible there is a positive surface charge on the boundary between two dielectrics medium 2 is a perfect conductor .t

lIel

..... r

sa lVe

Figur. P4.3

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/es ,d. .tal , of

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~

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~:1. de :Ie

t-, e.., I

t;,

94

4

Refleotion and Transmission

of Waves

4.4 Calculate the critical angle 9. uf an air-glass interface similar to the interface shown in Figure 4.8. The dielectric constant of glass at optical frequencies is ~.25 times that of air. - 4.5 A pearl is emherlded at the middle of II cubic heavy-load glass [s,

= 3.6). Is it possible to cover a portion of the surface of the cube so that from outside the pearl will not be seen at any viewing angle? If so. find the shape and the m.inimum area of the cover [in terms of the cubic surface area}. Hint: consider conditions of total reflection, and neglect multiple internal reflections.

4.6 In the three-media configuration shown in Figure P4.fi, the wave numbers are k,. K,-, and 1>3' Find the transmission angle in medium 3 in terms of III and the wave numbers. Assume all k's are real.

Fi,ureP4.0 z

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Light beam

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Problems -5.1 Show that the complete fields of the TE wave in a parallel-plate waveguide are given as follows: he

::ut Ice ice

-k Eo H. = --'sin lI.x e-I~.,

ith

H,

ors

.ild .ild

the

WIJ. =

ik.Eo --cos k,x e /'k

,I

WJ.I

E. - 0; E, - 0; H) - 0

5.2 Find the complete fields of the TM wave in a parallel-plate

waveguide.

-5.3 What is the lowest frequency of an electromagnetic wave that can be propagated in the TE mode in the earth-ionosphere waveguide? Model the latter as two perfectly conducting parallel plates separated by RO krn. - 5.4 Find the surface-charge density P. on the upper and the lower plates of plate waveguide for (a) the TEIll mode. and (u) the TMm rnoda,

me

8

parallel-

- 5.5 Find the mathematical expressions for a TEM wave in a parallel-plate waveguide that propagates in the z direr.tion (see Figure 5.11. Sketch the parallel-plate waveguide, and indicate the directions of E, H, and T•. -5.6 A rnicrostrip line has the dimensions

a - 0.15 cm and w ~ 0.71 ern, and the permittivity uf the substrate is e = 2.(; (u. I-' - f.Lo. (1 - O. Estimate the time-average power that is transmitted by the line when I E 1= 10i V1m.

128

5

Waveguidefl and Resonators

'5.7 The breakdown voltage of the dielectric substrate used in the stripline described in Problem 5.6 is 2 x 10' V1m. Use tI safety factor of 10 so that E is less than 2 x 10°

I I

V1m everywhere in the line. Find the maximum time-average stripline can trnnsmit. Neglect the ohmic loss.

power that the

5.8 With the fields in a rectangular waveguide. find the surface current 1., on the top f y - b) of the waveguide. we want to cut a slot along z, where should the slot be cut in order to minimize the disturbance it will r.ause? Assume that only the TE,,, mode

rr

exists in the waveguide. 5.9 Show that. if the wavelength of an alectromagnetlc

wave in an unbounded medium characterized by 11 and f is greater than 20. then this WtlVAcannot propagate in the rectangular waveguide (shown in Figure 5.8) with the dielectric inside the waveguide also characterized hy Il and c. Exhaust air duct

t

4 HI meter

Figure P5.10

I

5.10 An AM radio in an automobile cannot receive any signal when the car is inside a

tunnel. Why'( I.Atus assume that the tunnel is the Lincoln Tunnel. which was buill in Hl3911nder the Hudson River in NAW York. Figure P5.10 shows a cross secticn of the Lincoln Tunnel. * 5.11

find the frequency ranges for TE,n mUUI:!operation for those rectangular waveguides listed in Table l.

5.12 Design an air-filled rectangular

waveguide to be used for transmission of electromagnetic power a12.45 CHz. This frequency should be at the middle of the operating frequency band. The design should also allow maximum power transfer without sacrificing the operating Irequency bandwidth. Find the maximum power the waveguide can transmit. Use a safety factor of 10. Neglect ohmic loss, The breakdown E in air is assumed to be 2 x 106 Vim.

5.13 Repeat

Problem 5.12. but assume thai a dielectric material is used to fill the waveguide. The material is characterized by f - 2.50lo. Il = J.Lo. and u = O. The breakdown E fisld in the dielectric is J07 Vim.

5.14 Consider the size of a,rectangular waveguide to explain why it is not used to transmit electromagnetic waves in the VHF range. (Take 100 MHz.)

r-

*G. E. Sandstrom. Tunnels, New York: Holt. Rinehart & Winston. 1963, p. 242.

Dnators

129

Problems

riueuin 2

l(

106

hat the the top Jt be cut ,. moue

Figur. P5.16

Air

o nedium e in the ~ wave-

z

fields associated with the TElo mode propagating in the z direction are given by (5.23). Find the electromagnetic fields associated with the TRIO rnude propagating in the - z direction, with maximum electric field equal to E1•

5.15 The electromagnetic

a rectangular waveguide shuwn in Figure P5.1S. For the region z < 0, the medium is air and for z > 0 the medium is characterized by ~2 and 1'-2' A TEw mode with maximum E·field equal to En impinges on tha boundary from lhfl left. The result is that some power is reflected and some is transmitted. Assume that the retlected wave is also TE,o. with maximum E-field equal to E I' and the transmitted wave is TElo mode with maximum E-field equal to E2. Find the ratio EllEn in terms of Q. w, 1;0' /lU' E2' and 1'02' 5.17 The corner refter.tor studied in Problem 4.20 requires the solution 5.16 Consider

E - -z4Eo

inside a s built in )n of the /cguides

electroperating without -wer the JSS. The

• fill the - O. The

sin [kx cos OJ sill (ky sin 0)

Show that although the coordinates art! different thts solution is in fact the resonator mode that we studied in Section 5.2. Placing conducting plates at x - a and y - b to form a cavity resonator as shown in Problem 4.20. what (Ire the restrictions on the incident angle 8? 5.18 (a) Find the real-time expression of the fields of the TElOl mode in the rectangular cavity shown in Figure 5.9. (b) Find the total stored electric snergy in the cavity as a function of time. Find the (c)

corresponding total stored magnetic energy. Show that energy is stored alternatingly in electric and in magnetic fields. that the maximum stored electric energy is equal to the maximum stored magnetic energy. and that the total stored electromagnetic energy in the cavity is a constant independent of time. Note that these properties are similar to those of the low-frequency LC resonant circuits.

5.19 Find the lowest resonant frequency of the TE,u, mode in an air-filled rectangular cavity measuring 2 x 3 x 5 ern". Note thai there are three different choices for

designating the z axis and that these result in three different TElol modes. 5.20 Electromagnetic waves in air with wavelengths ranging from 1 to 10 mm are called

transmil

millimeter waves. Millimeter waves may be guided by dielectric slabs. Consider a dielectric slab with f, - 10to and tz - flf• as shown in Figure 5.12. What should its thickness be in order that only the TEo mode may be excited for frequencies up to 300 GII7.'~ -5.21 Use direct substitution into Maxwell's

equations to show that the fields given by (5.48)are solutions of Maxwell's equations in cylindrical coordinates,

5

130 5.22

Use the formulas of divergence 'il . 'il x A ~ 0 for any vector A.

5.23 Find the rectangular coordinates p - 1. 4> ~ 30°. and z = 2. 5.24 Find the cylindrical x, y. and z,

coordinates

5.25 Show that the differential

volume

Waveguides

and curl in cylindrical

coordinates

of a point P where the cylindrical of a point

Q

where the rectangular

in the cylindrical

and Resonators

coordinates

to prove

that

coordinates

are

coordinates

are

is pup

dcp dz.

5.26 To convert a vector expressed in cylindrical components into the same in rectangular components, or vice versa, it is convenient to prepare R table fur dot products between unit vectors in these coordinate systems. For example . p - cos f/J. as shown in the fullowing table. Complete Lhe table.

.x .

Dot Products Between Cylindrical and Rectangular Unit Vectors

I

p it

c/J

z

-

cos r/>

E 5.27

Use the above table to find the rectangular located at p - 2. cf> = 30°. and Z ~ 3: A =

5.28

8p +

components

of the following

vector

44> - 3 i

What is the maximum time-average power 1:1 coaxial line can transmit without causing breakdown? Assnme Lhal the coaxial line is air-filled and that the breakdown E of the air is 2 x 106 Vim. lise a safety factor of 10 so that the maximum E field anywhere in the line does not exceed 2 x 105 V 1m. The dimension of the line is 20 - 0.411 ern and 2b - 1.14:1 ern. Neglect ohmic: loss.

-5.29 Consider the coaxial line shown in Figure P5.29. Half uf the line (z < 0) is filled with air, ann half of it (z > 0) is filled with tI material characterized by EI and 1-11' The electromagnetic wave incident Irum the left bas thfl following fields:

E'

• Vo =p-

e-'''-''

p

I

•

VII

i~

H --e floP

The fields of the reflected V'

E' _ p~ei~ p H' _ t/J• __- V:, eik.,. u

floP

wave may be expressed

as follows:

131

ators

Problems

~ that

(a) Write down the fields of the transmitted wave in z > O. What wave number k should he used? (b) Find V ~ and the amplitude of the transmitted fields in terms of Yo. 711' and 710 by matching the boundary conditions at 7. - O. Compare your result with the reflection and transmission coefficients obtained in Chapter 4 for waves reflected from dielectric boundaries.

!S are

!S are

gular ducts

t/!. as

---+-; ------~ air

l

E,.

$"

.

...•.•

.....:'

l=O

zector

ithout

ireak.wn E line is filled nd J.ll'

/11

.,.....,.·~·:I

>z

Figure P 5.29

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6

Transmission

Lines

We can find the reflected wave by carrying out an analysis similar to the one for a transmission line with a capacitor. The result is as follows: (6.58)

Figure 6.39b shows the voltage V(z) on the line during the time period T < t < 2T.

6.1

.6.1 &.1

Problems &.1. What is the voltage in the stripline discussed in Example 6.1 when the time-average power being transmitted is 10 kW? &.2.

Consider ths coaxial line discussed in Example 6.3. Calculate the maximum time-average power that may be transmitted in the line, I Jss the hrea kdown E - 2 x 10' Vim and a safety factor ofl0.

&.3. Two coaxial lines have equal characteristic impedances: 50 n. Both art: air-filled, ThA first line has a power capacity of 1 MW. and the second line's capacity is 1 kW. Find the ratios %z and bl/bz. Consider only the breakdown voltage. -&.4.

&.1

Use (6. tb] and Lheboundary condition (4.3) to obtain the surface-current density J. on the lower plate of the parallel-plate waveguide. Then calculate the total current on the lower plate. Compare the current wilh the definition of I [z] given by (6.3bl

~.5. Find the surface-current density J. on the inner conductor of a coaxial line. Then calculate the total current on it. Compare the total currant with I [z] defined for the coaxial line.

8.2

...8.2:

&.2:

~8.8. A transmission line is short-circuited (Zl. - 0)

-6.7.

(a) Find tha expressions for] V(z) I and I liz) I as a function of kz, Zo, and V•. (b) SketchIV(z)landll(z)1 (c) Find VSW R on the line.

&.2·

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Problems

annas

221

to Puerto Rico. This radio telescope system is called the Very Long Baseline Array. The angular resolution will be of the order of 10 9 radian, which is smaller than the angle spanned by a dime located in New York City when it is viewed from Los Angeles. In ordinary arrays, individual antenna elements are connected by transmission lines. For the very long baseline array, such physical interconnection of antennas is not practical. Instead, the signal received by each antenna in the array is recorded on magnetic tapes which arc later transported to a central facility where the tapes are replayed simultaneously. The key to such processing is a very accurate time standard for all recorded data. At present. time synchronization of the recordings is provided by hydrogen masers which are accurate within 20 nanoseconds*.

Problems - 7.1 Find the rectangular coordinates of a point P where the spherical coordinates are (r - 1.0 = 600• ¢ = 30°). - 7.2 The rectangular coordinates.

coordinates

of a point Q are (1. 2. -4). Find its spherical

7.3 Show that \l . \l x A - 0 in spherical coordinates for any vector A. differential spherical surface element is equal to ds Hint: ReIer to Figure P7.4.

r2 sin H do dr/l.

-

7.4 Show thatlhe

-

7.5 To convert a vector in spherical coordinates to the same in rectangular coordinates. it is convenient to prepare a table for dot products between unit vectors in these

LJ-1_

»

that find

.49)

:l=

:.It

vaii

OK. r. Kellermann and A. R. Thompson. "The very long baseline array;' Science, Vol. 22f1. No. 4709. July 1985. pp. 123-130.

7

222

Antennas

coordinate systems. For example, X • f' - sin 0 cos t/J. as indicated in the fullowing tahle. Complete this table.

sin 8 cos

X

-7.6

the table prepared in the preceding problem to express the following vector located at (I' - 1.0 = _00°. = 45°) in rectangular coordinates: USA

+ 88 -

A = 12i' _

A

sJ

function Ir - r'] that appears in (7.7) and (7.8) can be expressed in spherical coordinates as

7.7 Show that the distance

Ir - r' I

Z

=

1'2

+ r"

cOS'Y= cos 8 cos

(J'

2rr' cos 'Y

+ sin (J sin (f cos(tP -

.p')

where 'Yis the angle between the vectors rand spherical coordinates of rand t', respectively.

r' and (1', 8,

./4and", - O. Sketch the field pattern on the x-y plane. 7.21.

Find the field pattern of a four-element array with d - >.!4 and ",-0. Sketch the field pattern on the x-y plane. (a) Use (7.37) to obtain the field-pattern formula. and [b] use the result obtained in the preceding problem and in Figure 7.16 and the patternmultiplication technique.

7.22. Write a computer program to plot field patterns of a ten-element phased array with d - >-/4 and varying phases. 7.23. A uniform linear array consists of 6 short dipoles. The spacing between adjacent Alements is ),,/4, as shown in Figure P7.23.

(a) What should the phase shift", be, in order to point the maximum radiation in the 4> = 90· (that is,.9')direction? (b) Supposo that the E-field due to the first element (the dipole at far loft) is given as follows: E

Bo

1000 -jJ.:r • 0 e sm r

= --

Calculate I £81 of the entire array at point A(0,1000,0), point 8(1000,0,0), point qo, -1000.0), and point O( - 1000,0,0), separately. All positions are given in rectangular coordinates in meters. Use the phase shift found in (a).

lc:) Sketch the field pattern of the array in the x-y plane. (d) Sketch the field pattern of the array ill the x-z plane.

-I

1-),,/4 y

Figure P7.23 x

8.1 Raleig

CHAPTER. '7

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252

8

Topics in Waves

Figure 8.17 shows a typical arrangement of a liquid-crystal display.* 11 is operated in the so-called "distortion-of-aligned-phases" or DAP mode. Figure 8.17a shows the normal state of the crystal before activation. The light entering the crystal is polarized and then transmitted through the crystal with no alteralinn in polarization. The second polaroid absorbs all the light, and no light is transmitted. 10 its activated state, the crystal changes the polarization of the transmitted light. which propagates through the second polaroid and becomes visible.

Problems 8.1 The derivation

of 18.5) only considers the electric field. Why is the magnetic field neglected? Hint: Cumpare the magnitude of E. with 1)H" near the sphere. or the stored electric-energy density (1/21~ IE I~with the stored magnetic-energy density (1/2jlll HI··

8.2 Wby is the rising

01'

setting sun red?

8,3 The smoke emitted

from engines of boats contains fine particles. Against a dark the smoke looks blue but agains! a bright background it looks yellow.

background Why?

8.4 Explain the appearance

of shafts uf sunlight through breaks in a cloud-covered

sky.

8.5 Shuw that

II -

f:

e'~ dx - ,fi

8.8 A

If;

Hint: I~-

f.'

e'

Then. transform integration.

dx . X-)'

d,

t:

re

e-I'" dy

coordinates

into cylindrical

coordinates

to perform the exact

8.6 Show that I~ -

t •

,.,

exp( - tr«:

+

.J.fir4X) ux

p exp (t/) 4 z p

e,g 0 hi fe th

cl

te

8.10 A

Hint:

22 -p x +

qx

= -

(qpx - -

2p

Then, use the result obtained varia hie from x to px - 4/211.

s~ Sf

q~

)2 +-. 4J)"

in problem

8.5 after transforming

the integration

that on earth a microwave beam of 10 GHz is radiated by a zo-mctcrdiamp.tAr disk antenna aimed Citthe moon. Estimate the size of the microwave heam on the moon.

8.7 Assume

"See R. W. Curtler and C. Maze. "I.iquld Crystal Displays." Il::t:(; Spectrum, November 1972. p. 25.

Problems

::s in Waves

253

Iisplay. * It ·AP mode. I. The light :he crystal J the light. ranges the he second

gnetic field

ere, or the rgy density

ins! a dark oks yellow . •vered sky.

8.8 A person leaving his home by train mails a letter home every day. Suppose that the train travels 200 miles per day and that the mail moves at a speed of 200 miles per

day. How frequently do his letters arrive home? Try to solve this problem by simple reasoning, not by substituting numhers in some formula. 1

Ihe exact

8.8 On a foggy day, the driver of an automobile stepped at a railway crossing hecause he heard a whistle from a moving train. The sound of the whistle came from his left. A

few seconds later he heard the echo, and the pitch of the first sound was lower than that of the echo. Assume that the echo was due to reflection from a nearby mountain close to the track. If you were the driver, would you cross the track-that is. could you tell whether the train was approaching or leaving you? (See Figure PS.9.) 8.10

A Doppler radar sends a signal at 8.8UOGHz, and the receiver displays a frequency spectrum of returned signals as shown in Figure P8.l0. What CClIl you say about the speed of the targetls]? I' Amplitude

of the returned sign ..1

ntcgratlon

20-mcler..ave haam

~ovember

-

10 kHz

15 kHz

a.aoo

r.Hz

Flgur. P8.10

rrequenC"y

Topics in Waves

8

254 Absorption axis I I

I

Randornlv

Passing axis

pnlllri7.pcl

IIgltt

~

~~/A ,\Jl

Z

~~/.;)-Ctobserver]

Figur. P8.15

to.

8.11 Fur the FM-CW Doppler

radar discussed in Section 8.4. aSSUJll~ that the upper frequency of the rad a I'. is 8.8 GJ 17..Suppose the radar is to measure target speeds ran~in)! from 0 to 3 Mach and distances from 1 km to 10 km. Find the system's approximate frequAnci bandwidth and the time interval the system must be ahls to resolve. in Figure 8.JGa. and if reflections at interfaces z - 0 and z - cl are negligible. a linearly polarized wave incident from the left will become a circularly polarized wave, as discussed in the text. What is the polarization of the exiling wave if the roflections at these interfaces arc not negligihle?

8.12 If d - >'0/4, as shown

8.13 II d - >'n/2

AS shown In Figure 8.16a. what is the polarization of the exiting wave if the incident wave from the IAft is circularly polarized?

8.14 For a quartz crystal.

I, - 2.41 to, and ( = 2.aSEo. Find the minimum quartz quarter-wave plare for a li)(ht having>. - 6500 A.

8.15

thickness

of

II

In FigurA PB.15 the Polaroid film at A is oriented such that it passes light polarized in the ic direction and absorbs light pulnrizcd in the 5' direction. The film at B passes y-polarizerl light and absorbs x-polnrized light. A randomly polarized light source. such as a flashlight. sheds light from the left along z, Can an observer at C sec the light? Explain.

8.16 Consider

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368 Conductor

Solution Techniques

Conductor

Conductor

Figur. P1t.1

4>, - II

CaS!' ,

rj>l -

0

Case II

3 = 0

Case III

Problems 11.1 Consider the three boundary-value

prohlems shown in Figure Pf t.t. The solution of CAse r is " and the solution of Case II is 2' In Case III, the charges 4, and qa arc the same charges that appear in cases I and II. and they appear in exactly corresponding positions. Express 3 in terms of 'I>, and ....a-

11.2 Consider the three boundary-value problems shown in Figure Pl1.2. The solution of Case T is ,. lind the solution of Case II is 2' In Case III. the charges (/, lind qz are the

slime charges that appear in cases [ and Il, and they appear in exactly corresponding positions. Note the differences in the boundary conditions for the three cases. Can 3 he expressed in terms of , and 2? If so, obtain the expression. If not. explain why. 11.3 The radius of the inner conductor of a coaxial line is a and that of the outer conductor is b. The potential of the inner conductor is V and that of the outer conductor is zero.

There is no volume charge density between band u. Start from the Laplace equation to obtain the potential in the coaxial line. 11.4 Two concentric conducting spheres have radii u and b, respectively [h > 01.The uuter sphere is at zero potential, ann the inner sphere is maintained at V volts. There is no

space charge he tween the conductors. Start from the Laplace equation to obtain the potential {rl for b > r » o. Couductnr

Conduclnr

Conductor

Figur. P11.2

~~ 4>~ - 0 C':ilse I

Casp. II

Case 111

rechniques

Problems

369

1.1

In Figure Pl1.S a conducting cone is at a potential Yo. and a small gap separates its vertex from a conducting plane. The axis of the cone is perpendicular to the conducting plane. which is maintained at zero potential. The angle of the cone is (J,. BecaUSAof the symmetry of this problem and the fact that the boundary conditions on the potential approaches zero 1: 'i?l/~~'l=0 4. f

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0.Q1mho/m. Find the potential at point B shown in the figure. 12.8 A point electrode puts out I amperes of current above a conducting plane. as shown in Figure P12.8. (8) Find 4>{x. y, z] for z > o.

(b) Find the current density] ,Ix. yJ at the surface of the conductor. (c) Sketch the paths of the current flow.

-rents

389

Problems

T !!

3m

r I

00.

15/111

12.9 For the case shown in Figure 12.9, find the pArr.AntagAof tha currant emitted from the electrode crosses the boundary and ental'S in medium :.1. 12.10 A source 4 meters below an interface of two conducting current. as shown in Figure P12.10.

media emits

2 A

of direct

(a) Calculate the potential at point B. (b) Calculate the potential at point C.

ius c. ectric ively. I. and ells is ;6 the •=

10

12.11 A well-logging resistivity tool similar to the one shown in Figure 12.12 is near a boundary between two beds, as shown in Figure P12.11. The boundary is making a 60° angle with the well. Find the apparent resistivity measured by tins tool at the

position shown. 12.12 Refer to Example 12.6. Obtain Po (the apparent resistivity measured by the tool) as a function of tool position for Zo - ~ 160 in. to Zo 160 in .• where Zo is the position of the center of the tool (the midpoint between electrodes A and B) relative to the boundary. Calculate Po for at least 21 points, lind pial Po versus ~o. 12.13 Repeat Problem 12.12 for the situation shown in Figure P12.1l.

a 90° d the

.ction H

60°

d the ual to lawn

Figure P12.11

12

390 12.14

Direct CUrrents

A point electrode is located at (0. Yt' 0), and a perfectly conducting sphere of radius a is located at (-i, 0,0) as shown in Figure P12.14. The electrode gives I am-

peres of current. The conductivity of the medium is (1. Find the potential ~ on the y axis. Hint: usc (11.44). 12.15

Consider a well-logging resistivity tool similar to the one shown in Figure 12.10. Let the spacing between the current electrode A and the potential electrode B be 6 rn. The tool measures the conductivity of the earth formation as it travels in a well. Assume that the well passes near a mineral deposit modeled by a perfectly conducting sphere, as shown in Figure P12.15. Find the apparent resistivity measured by the tool as a function of y. Use the. following data: (1 a 0.01 mho/m for the ground; the radius of the mineral deposit '" 50 m; and the distance between the center of the sphere and the weU = 70 m. Plot O'epparenr versus Y for -70 < Y < 70. Hint: use the result obtained in the preceding problem.

FIgure P12.14

x

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x

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(l(.r(,~Jf ..

E. _

~(f).

I, =

=r+: ;~

E.,,~

(A.)

CUt Ji

+W# i"'fuf~"t

'5

h,u

t:M..rr'C tJ t

fAt

{3~CA.""'t

J.

n .a., C,

4,

C,

€r"

l!:J: G

".,..t. ~Sf:'(J./A.)

to

3

J

;.(.

==

1# '"~

7roL

I;. (,

c r-"II

1T',o(/0'

use. IM~se Wlt.~d..: ~(" If 1);: J_ (.J.. - ...L 'i'..J.. _ _J_) 4rr~

I~'

A.,

R~

R, • [t'Z-%.{+(

W~vc,

/I~ •

1

"'"..

At,

a.. 5".".1._,

#If J

UIJ.·'(S'-LS'" ..... 'J

'~~-'(~/4.).(f-.

o:»

h

I

"J.JT

(i.

_J_ ~41

K:

'1.2+~/#·# "'~./""

1

'1- '1.)2+ 'J 'I J ...

[(,,+~.)1...('1_¥.)·.l'

R, • [(h%.l .. {~.V.)I

..

J~

vs ~

~ • [CX-x.)'1"'('~'1.)·",'J Ji

.E.:.1

use

meH.D4, :

i"",~t.

1~ «t • J)- -L(....!...-1 4-fTtr Il, ,.., t.J~"t. IJ eli; (..r /J,/ < ih; 11;",hluk.._ 2 .,,1/. B"~ ""~i.4 ... ,./ 1"= cr,~&. If;tr. I ~

(, .. J

r

4:;'). =

h J. ('-.t;,;

i6 c

~:z(r,!,.) * t. -41f.l£/r A ~

c:

:r '.

8 i"" "..,1p>a

27rp

12. d¢ I' n

0

in (1~i.~i4). we obtain ]2

pdp

47r2pl

Jl12 -

(b)

47r In ~

This result is the stored magnetic energy per unit length or the coaxial line. Consequently. we can calculate the inductance per unit length or the line from 113.41):

L

= __t:_

27r

In (~)

(13.49)

11

This inductance per unit length also appears in the transmission-line representation of the coaxial line in (6.19) of Chapter 6.

Problems 13.1 Find the magnetic field " at tha CAnter uf a square of the square loop is b meters long.

a current I. The side

loop carrying

13.2 A circular loop that has radius a and that curries a current I produces the same magnetic-field strength at its center as thai at the center uf a square loop thai has side b and that carries the some current I. Find the ratio of b to o. 13.3

Consider a larga conducting plate of thickness d located at -d/2 :; y :; d/2. as shown in Figure P13.3. Uniform current of density Tis flowing in the direction. Find H in

z

all regions. 13.4 The earth's magnetic field at the north magnetic pole is approximately 0.62 G (1 C .. 10 • Wb/m2). Assume that this magnetic field is produced by a loop of currant flowing along the equator. Estimate the magnitude of this current. The radius of the earth is approximately 6,50U km. y

FIgure P13.3

Ids

Problems

423

lat

Figure P 13.6

Figure P13.5

ie. ne Pigur. P 13.7

,91 le

13.5 An infinitely long tubular conductor of inner radius 0 and outer radius b carries a direct current of I amperes. as shown in Figure P13.5. Find Ihe H fiald at o; where (a) p s o. (b) 0 S p s b. and (e) IJ s: p.

Ie 1e

-Ie -n III

11 11 Ie

13.6 All infinitely long tubular conductor has outer radius b and inner radius 0 offset by a distance c from the axis of the outer cylinder, as sbown in' Figure P13.6. This Accentric tubular conductor carries 0 direct curren I of 1 amperes, Find the H field at point A shown in the figure. Hint: Consider the tube 10 be 0 superposition of two solid cylinders that have radii b and a and thai carry uniform current density 1 in opposite directions, 13.7 An infinitely long wire is bent to form a 90° corner, as shown in Figure P13.7. A direct current I flows in the wire. At point A find the H field due 10 this current. Follow the steps given below. (a) Use the Biot-Savart law to express the H field at A due to a typical segment of wire dyon the wire axis. Express the field in rectangular coordinates. (b) Jntegrate the result obtained in [a] to find the H field due to the semi-infinite wire Note: to facilitate integration, let y - a tan II, so that dy - a sec" 0 dO. (c) Find the H field at A due 10 the semi-infinite wire BO. (d) Add the results obtained in [h] and (c) to yield the total field at A due to the current in the wire BOC.

oe,

13.8 Follow a similar procedure 10 the one r!escribed point 1\', as shown in Figure P13.7,

in Problem

13,7 to find the H field at

13.9 Consider a circular loop currying a current I counterclockwise. as shown in Figure 13.11. Plot the magnetic field Ll, on the z axis for -0/2 < z < o/z. Find the value z, in terms of a. such that, if z Zo, then H, is uniform within 10% of the value of H. at thA center of the loop.

I 1<

13

424

Magnetostatic Fields

Figur. P 13.1 0 Helmholtz coils

x

13.10

13.11

To improve the uniformity of the magnetic field along the axis of a circular loop {see Problem 13.91,one may use two identical loops separated by a distance equal to their radii. as shown in Figure P13.10. Such a pair of current-carrying loops is called Helmholtz coils. Find Hz as a function of z on the axis of the Helmholtz coils. Plot H, for a < z < o. Find, in terms of 0, the value Zo such that. within the range I z I < Zo. H, is uniform within 10% of the magnetic field at the middle of ths two coils. Compare your result with that obtained in Prohlem 13.9 for a single loop. A square conductor

loop 2u meters long on each side carries a direct current 1 as shown in Figure P13.11. [a] Calculate the magnetic field B at (b,O,O).Express the magnetic field in terms of 4 integrals, where each represents the contribution from the current on each side of the square. Use the Biot-Sevart law. Du not try to integrate those integrals. (b) Assume that b is much greater than o, Now, evaluate the integrals approximately to obtain an approximate value of Bat (b.O,O). Ylt-.

e

T

,

I

2u

[b. u.

UJ

x

Figure P13.11 _"-

A

8

13.12 A surface charge of p. C/m2 is uniformly

distributed on a record disk. The inner radius of the disk is 0 and the outer radius is b. The record disk is turning at a constant angular velocity w rad/s in the clockwise direction. Find the magnetic field at the center of the disk due to the surface charge on the turning disk. Ignore the presence of the metal post on the turntable.

Fields

"2

Problems 13.13

The earth's magnetic field at the equator is Approximately B late the cyclotron frequency of the electron in the ionosphere.

13.14

Aecause natural uranium contains a slight amount of Uranium 234, the electromagnetic isotope separator can also yield 2l4U.If the radius of the circular path for 2JUU particles (see Figure 13.14) is equal to 10 rn, where should one place collectors for 235Uand 234U particles? Express spacings in meters.

13.15

Refer to Figure 13.17. The magnetic field is changed from 5 x 10-4 to 10-3 Wb/m2• All other parameters remain unchanged. Find the following:

coils.

::>(see their allad ot Hz z; H, ipare .t J

425 =

10-4 Wb/m2• Calcu-

(a) the position of the electron at the exit siele of the magnetic-field region (b) the exit angle (the angle between tho trajectory and the x axis after the electron

has passed through the magnetic field) 13.16

Consider an electron having initial kinetic energy IIIe v~/2 and entering a region of uniform magnetic field, as depicted in Figure P13.16. This situation Is similar to that shown in Figure 13.17, except that the electron in the present case is inclined at an Q angle with respect to the x axis. (a) Show that v, and v, of lite electron "'fter it enters the magnetic field are given by

AS

V. ~ VII

of each inte-

rns

vz -

cos(w,t ~ (Xl Vo

sin[wcL+ o]

where We = 1./J3.lm. And t - 0 corresponds Lothe moment the electron enters tbe magnetic field. (b) Find the coordinates x and z of the electron Attime t. Note that x - 0 and z - 0 at

-roxi-

t - O.

(c) Find the point where the electron leaves the magnetic field. Assume Vo = 2 X 107 m/s, a-50• Wi' - 8.77 X 107 rad/s. and d - 4 em. (d) Find the angle between the x Axis and the trajectory of the electron after it has left the magnetic field. Sketch the entire trajectory, and compare it with the one shown in Figure 13.17. z

-d-j

------,

Electron

x

)(

x

)(

I

x

x

:

:

r

x

x

FIgure P13.16

~_:~i---__ ~x

x .:_x_x_l ner at a

etic are

13.17

1'wo parallel wires are carrying 100 A of current in opposite directions. On each wire find the force per unit length due to the magnetic field produced by the other wire. Is the force repulsive or attractive? Assume that the lines are 1.5 m apart.

13

426

Magneloslatic Fields

13.18 Two identical circular loops of radii 0 ara separated by a distance d, where d « Q. One of the coils carries I amperes of current clockwise, and the other carries I amperes counterclockwise. Find the force between these coils. Hint: Because

these coils are close together, you can approximate the magnetic field that is at one coil and is produced hy the current on the other as HI = 1~/l2?1"d), the field due La an infinitely long wire. Let 0 - 1 m and d = 0.05 m. How much current is needed to produce a force of 9.8 N? 13.19 A circular

loop of radius 0.5 m and 100 turns is excited by fI 2 A direct current. This loop is placed in the Earth's magnetic: field, which is approximately equal to 5 x 10-5 Wbfmz pointing north. How do you orient this loop to produce II maximum torque? What is the value of this torque? Find that orientation uf the 1001J In which it experiences no torque.

13.20

The square conducting loop ABCD shown in Figure P13.20 carries 2 A of direct current. Each side of the loop is 0.1 m long. The loop is placed in a uniform magnetic field B. Find the force on eacb side of the loop and Lhe torque on the entire loop if: lal B = k 0.2 Wb/m2 (b) B = - Z 0.2 Wb/m2 y D

c

~T U.1 m

1=2 A x

Figure P13.20 -'--

A

13.21

An infinitely long conductor Figure P13.21.

B

of radius

0

carries a direction current I as shownin

(a) Find the H field in the region O-laJI, 14, ..

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Solution:

Magnetoquasistattc Fields

From (16.721. w~ find 2 x 47r X 10-7 X 2 x 1O~ x 2 x 10' x 1.0 />'eoll -

$

-

7r x 0.5

64 x 10 N

- 6.5 metric tons

Problems 1«1.1

A small circular loop of 5 mm rad ins is placed 1 rn away Irorn a 60-Hz power line. The voltage induced on this loop is measured at 06 microvolt. What is the current on the power line?

16.2 Assume

that the current on the infinitely long line shown in Figure 16.1 is the triangular pulse shown in Figure P 16.2. Find thp. induced voltage on the rectangular loop. Use the following data: a - 2 ern, U - 4 ern, and d - 1 cm.

the network shown in Figure P16.3. The magnetic flux is increasing at a rate of 0.5 Wb/s in the direction pointing into the paper. Find the readings of the voltmeters shown.

16.3 Consider

16.4 Find the readings increasing

at

a rate

of the voltmeters shown in Figure PJ6.4. The magnetic of 0.5 Wb/s in the direction pointing into the paper.

flux is

1 fampflrAS)

3

I [microseconds]

Flgur. P1e.2

,..-----------, I

S kfl

v~O\ - \

V'/

4.5 kfl

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V$

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J

Problems

511

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AppendixE

Answers to Odd-Numbered Problems

ChapterS Chapter 1

1.1 (a) 5 + j3 (b) 11 + j [c] - 26 + j2 (d) - 2.2 - j1.4 1.5 ± 21/4

e

i(w/8)

+~)

(e) 3 cos (wt

(b) -10i

1.7 Proof

1.9 (a).J2 cos (wt

+ 4 cos(wt

+ 0.8)

r~)

1.3 Cooswt; sin wt; 1

(b) 4 cos(wt + 0.8)

1.11 (a) -6x +

+ 13y - 4Z (e) - 55 (d) 23x + 22y + 142

59 + 2! 1.13 Proof

8,

~(5X + 2i) 1.17 Proofomilled 1.19 Proof omitted '193 1.21 Proof omitted 1.23 jW[(3 - j4) x + 8(1 + j) 2] 1.25 (a) (-1 + j3) 9 + (1 + ;3) Z (b) 2 i + (1 - j)ji + (1 + j) 2 (e) - 5 (d) 4i - (1 + j3H + (-1 + j3) Z 1.27 Sketch omitted 1.15

Chapter 2

2.1 - 6yi - 3x2y,6z 2.3 Proof 2.5 Proof 2.7 No 2.9 B(y,t) = O.3(k/w) cos (wt + ky) z 2.11 £1 + Ez, H, + H2 B1 -t B2 and D] + D2, superposition theorem 2.13 Proofomilled 2.15V x E = iwB, V x H = J - iwD,V' B - o and v : D = Pv 2.17Proofomitted 2.19t,t 2.21Proofomitted 2.23UII/Ue"" 1.13 x 107

Chapter 3

3.13.6 X 10-1U W/m2 3.34 x 1026 W 3.54.1 x 10'Jkm 3.7 (a) rad/sec (b) m" (e) sec-' (d) sec (e) m 3.9 (a) 2.63 m (b) 0.704 m 3.11 Yes, -z direction, f 1 ,v !loJ,~

~ 2

j

E~z

Chapter

Chapter'

3.13 No, Maxwell's equations not satisfied

J.lo

~1a) Right-hand circular polarization (b) Right-hand circuJa.r_polarization (e) Left-hand elliptical polarization (d) Linear polarization @)roof omitted 3.19 (a) 1 (b) 1 (ell.58 (d) 2.12 3.211.34 x 10-5 m, aluminum foil is about se thtck S.232.65mW/m2 3.25 (a) E. = e-o.Gze-Ju.~z (h)H. 9(0.5 - jO.5) e-O.51 e-}U'~' (e) Sketch omitted (d) Sketch omitted 3.270.6 x 10 6 ill

Chapter 4

4.lfi1.9° 4.3 (i) c (il) f (iii) b (Iv) a (v) d (vi) e 4.6 Yes, circular shape, 0.3 ff on each surface (0 is the length of each side of the cube.) 4.7 bevelled angle _ 3So; mirror making 70° with z axis; R polarized 4.91ElK 1~ ~IEol11 - e-,0ll>3' ei""I. 1El.l-

v'31 EoII 1 + e-/

I Eo111 + e- '1 1Z53

520

Z53

'

2'

e-

JII