Solutions Manual For Shen And Kong's Applied Electromagnetism 693842

AI-

h) -

'ii . d

+ 'ii . b.

tf>z+ <1>2\I \ and that 'il . ( A) == A . 'il

\I • A.

\I x A..,. \1 x A.

2.7 In a source-free region. H - zy +

yz.

Does D vary with time?

-2.8 What is the charge density in a region where D - ad? 2.9 Find the magnetic field B(y.L) associated with thE!electric field lows: E(Y,L)

= x 0.3

cos(wt

E(y,t)

gtven as fol-

+ kyJ

where wand k are constants. -~.10 Express k in of the magnetic permeability

medium when the electromagnetic region.

and dieieotric permittivity of the fields arc given in Problem 2.9 in a source-free

2.11 Let £" B,. H" lind D, satisfy equations (2.1j-(2.4j with given II and

at ~

Pvl' Let also E2, B~. Hz. and D2 satisfy equations (2.1)-(2.4} with given 12 lind Pv2' What are the electromagnetic fields due to a current /, and charge Pi;, where I, - 11 12 and Pw - (>,.\ + 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 uf z and wt and find <S>.

* 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

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V·

1..'1

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~ = 0.3 ~

e ji.'

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

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

:t,

~

./0

it: W'At..

=-;tB"

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Ir

040

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111'1

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HI

Z

orm Plane Waves

Problems

65

Figure 3.11 Various comet shapes drawn on silk found in China. These figures were painted between 24610 177 B.C. Below these figures 01'0 Chinese names for these comets.

)f mankind lur wing made beture 3.11). Mod-

Problems 3.1

, vary greatly in

dividual comet cornet's laiJ?

MHz].

3.2 Consider the sun as an isotropic radiation source. Calculate the total power radiated by the sun in the television channel-z frequency band (see Problem 3.1). The distance between the sun and the earth is approximately 1.5 x 108 km,

u

whether or not his observation 16

3.3 Assume that solar radiation

is isotropic. Estimate the total power radiated by the sun. The solar power density received on the earth is 1.4.kW/m~. See Problem 3.2 for other data.

sun that art!

the sun. Otherresent oxplana51 pa rlicles

and

3.4 Derive (3.Sb) from (3.Sa), assuming that E = Ex Jl:, and Ex is a function of z only.

pressure of the m the sun, and e ionized gases e plasma forms flow of protons t a speed

3.5 The star a Centauri is approximately 4.331ight.years distant from Earth. A light-year

is a unit of length that is the distance a light wave covers in one year. How distant is a Centauri in kilometers? 3.6 An electromagnetic pulse is sent from an earth station to the moon. and the reflected pulse is received 2.56 s later, How far is the moon from the Earth? [An electromag-

up La

netic pulse consists of a wide spectrum of electromagnetic frequencies. J

et's plasma and

1.

1964. pp.

waves at different

3.7 Find the 81 units of the following quantities associated with a uniform electromagnetic wave: (a) w. (h) k. (elf. (d) T. and (e) A.

Vol. 199, Octobor

4, April

Estimate the power density of electromagnetic radiation from the sun received on earth in the same frequency band as that of the VHF television channel 2 (54-60

r

laser emits light at a wavelength 6.328 x 10-7 m in air. Calculate its frequency. period, and wave number.

3.8 A helium-neon

3

66

Uniform Plane Waves

3.9 Figure P3.9 shows a dipole antenna,

II is VAry affective in receiving television signals when its length is approximately equal tu one-half the signal wavelength. What are approximate antenna lengths tor receiving signals for the following: (a) Channel 2 (f = 57 !vlHz) and (b) Chunnel l J (f = ;!1:3 MHz)?

r-

-

>./2 ----t~

I

I Figure P3-9

Two-wire transmission

line

. 3.10 The following set of slactromagnetic

fields satisfies the time-harmonic

Maxwell's

equations in free space: E - Eo e+1kz X and H-

f-lo

e

Ikzy

Express Ho and k in of Eo and

(0

and

Jl-o·

3.11 Do the fields in the previous problem represent

H uniform plane wave? III what direction does the wave travel? Find its velocity and determine the time-average Poynting vector (S).

3.12 The Federal Communications Commission of the United States requires a minimum of 25 mV/m field intensity for AM stations covering the commercial area of a city. What is the power density associated with this minimum field? What is the intensity of the minimum magnetic field H'i - 3.13 Study the following E field in a source-free region:

E

=

x Eo e-lkx

Does it satisfy Maxwell's equations? If so, find the k and the H field. U not. explain why. -3.14

Show that in 13.13J, if rf>. - 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 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

<-.

J:2--3I I

id circularly

I I

, I

I I I I

hand] of the h

I

I I

I I I ,

Figure P3.24

A I I I I

Ail'

W'W/'/////ff)71J#////;Mwj~ d. (Refer to sin a pair of wt - U, 10°,

(a) What is the difference the true altitude?

Snow Ground

between

thA Apparent

altitudo

measured

uy the

radar Ann

3

68

Uniform Plane Waves

(b) How much attenuation in dB does the radar signal suffer because Consider only the attenuation of the WHVt: ill the snow, anti neglect snow on the reflection at air-snow and at snow-ground interfaces. ure P3.24. Use f = 1.51.'0 and tan (j = ~ x 10 4 for hard-packed snow

of the snow? the effect of Refer to Figat 3 GHz.

- 3.25 The following data are given for a uniform plane wave in a dissipative medium:

(i) amplitude of E. at z = 0 is 1 Vim. [ii] phase of E. at z = 0 is zero, (iii) k = 0.5 - j 0.5 (11m), [iv] direction of propagation is in i, (v) intrinsic impedance of the medium is 1 + j ohms (a) Find the phasor expression for E, as a function of z. (b) Find the phasor expression for H as a function of 7.. [c] Sketch Ex at z = 0 and at z = 2 m. CIS Iunctiuns uf wI. (d) Sketch the time-domain H fields at z 0 and z 2 m as functions of wt. s

a

3.26 Consider that a small space vehicle with 100 kg of mass is located in outer space

where the gravitational field is negligible and the fuel has been exhausted. A searchlight of 1kW is turned on. with hopes that the vehir.1e will gain some speed. How much speed will it finaJly gain if the searchlight uan last 48 hours? Hint: The light wave carries radiation pressure, and there is a reaction force on the source of the light. 3.27 An icA particle of radius

Q is r distance away from the sun. Tho gravitational force acting on the particle is given by (3.46). The ice particle's mass can be obtained from its volume and its density which is assumed to be 1 gram/ern". The ice particls is also subject to radiation pressure which is given in (3.47). The force acting on the ice particle due to radiation pressure is approximately aqual to the crosssectional area of the particle times the radiation pressure. Show that, when the particle's radius is less than a critical value, the radiation force will be greater than the gravitational force. and this critical radius is independent of r, the distance from the sun. As a result. all particles with radii smaller than this critical radius lend to be blown out of the solar system. Find the value of this crttlcal radius.

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4

Reflection and Transmission of Waves

- 4.

Flgur .... 23 Tn rp..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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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 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 of III and the wave numbers. Assume all k's are real.

Fi,ureP4.0 z

k,

k,

1-/ __

---7-

(

7

-=-=--_,.....-=--1 Closs rod

'1~

Light beam

:-...

FI,ure P4.7

;7

4.7 Solid-state lasers (ruby or glass) are often fabricated of rods with the ends bevelled at the Brewster angle. Let ( = 2<0 for the rod. Sketch the propel' arrangement of the external mirrors and their angles. Indicate the bevelled angle of the glass rod. What is the polarization of the uutput of the laser beam? (See Figure P4.7.) 4.8 A parallel-polarized wave is incident from medium 1 on the plane boundary between medium 1 and medium 2. I:!othmedia are dielectrics with J.LI - !-t2 - !-til and real permittivilies EI and Ez. We know that. when the incident angle is larger than the

criLical angle 9e- no rime-average power is transferred to medium 2. Also. when the incident angle is equal to the Brewster angle 0b. the reflected power is zero. Now imagine a situation in which the Brewster angle is greater than the critical angle. A wave incident at the Brewster angle will not be reflected, because the incident angle is equal to Oh. nor will it he transmitted. because the incident angle is greater than 0,. Is this situation possible? Why? 4.9 Consider an electromagnetic

WAve of 1 MHz impinging at 60° on the ionosphere. This case is similar to that shown in Figure 4.13. Assume the the plasma frequency of the ionosphere iswp = 211' X 9 X 106 rad/s, and plot] E las a function of z [like in Figure 4.14). Mark the scale of z in meters. Solve only for the case of parallel polarization with Eo-1 Vim.

of Waves

Problems

95

ce shown irnes that I E I [volts per meter)

McdiuIII2 I possible

-ill not be .he cover lion. and

10 0.5

o

-3 -2 -J

Figure P4. 10

1

2

[meters]

-:--7 z

Flgur. P4. 11 ,.

-4.10

A perpendicularly

polarized electromagnetic wave impinges from medium 1 (characterized by !-t, = !-to and l, - 4foJ 10 medium 2 [characterized by J.l: - !-to and E2. = <=0]' This situation is shown in Figure P4.1U. v::::;; I

'I

...~.

fJ, - ~', ., ." (b) Let the incident angle be 1$0"; find h.linu k, in of ko - w~, (c) Find kl• in of ko. (d) In the second medium. find the distance ~a at which the field strength is l/e of that at z - 0 I. (e) Find] R,I and the phase shifllil',I! (Rrl, (a) What is the critica I angle?

n

-4.11

J

uniform plane wave in air impinges nuruially on Ii dielectric wall. The magnitude of the total E field measured in front of the wall is shown in figure P4.11. A

(a)

What is the permittivity

(b) What is the frequency

of tha dielectric of the wave?

wall? Assume

4.12 A uniform plane wave in air impinges on a lossless dielectric as shown in Figure P4.12. The transmitted wave propagates

respect to the normal. The frequency elled at I of the i. Whal

Figure

·i2ation

material at a 45" angle. in a 30° direction with

is 300 MH:l. x

Find fz in of f9• (b) Find the reflection coefficient RII• (c) Obtain the mathematica I expressions for the incident E field, the reflected E field. and the transmitted E field. (d) In hoth media, sketch the standing wave pattern of IEx 101"1 I as a function of z. (a)

undary Po and han the aen the o. Now ngle. A It angle than IIr• sphere. ency of

#2 ~ JI-o.

Figure P4.12

4.13

For two isotropic media wlrh s, :;t: Ji.z and EI :;t: (3. find the Brewster angle for both the perpendicular polarization and the parallel polarization.

4.14

II a wire antenna is attached parallel to the metallic surface of a vehicle and is insulated from the surface by a thin layer of dielectric material with a thickness approximately equal to 1 mm, would it receive an AM signal r f = 1 MHz)? Hint: Wire antenna interacts only with E field in the direction of the wire.

4

96

Reflection and Transmission

of Waves

- 4.15 It is known that the transmitting antenna of an FM station Is located in the direction perpendicular to a metallic plate, as shown in Figure 4.15. The frequency of the signa I is 94 MHz. (a) Where should a receiving antenna be placed to receive maximum signal? The antenna is II dipole that interacts with the E field. (b) If the amplitude of the incident E field is 1 V1m, what is the amplitude of the E field at this optimum position'? - 4.16

It is found that by placing a conducting plate 0.8 m behind a dipole antenna, the received signal coming (rom the normal direction is twice as strong as the incident field. What is the frequency of the signal'? What would the strength of the total E field be if tbe frequency of the wave is changed to 98 MHz while the antenna is still placed 0.8 m from the plate'?

- ".17 What would the r; field be if the receiving system in Prohlem 4.15 were Lo 'detect a wave coming in at an angle 10" off the normal? Assume that all other parameters remain thp. same. - 4,18

Derive (4.53).

- 4.19

For a parallel-polarized uniform plane wave impinging on a perfect conductor at an Angle O. find the electric and magnetic fields E and H for the incident and for the reflected waves.

4.20 Consider a 90° "corner reflector" shown in Figure P4.20. It is made of two conducting plates placed perpendicularly to each other. A uniform plane wave with E = "Eo expljkx cos 0 + jky sin 0) impinges on the structure at an angle O. Show that the total electric field is E - -24En sin(kx cos BJ sin (ky sin BJ. Hint: The field is the slim of four waves with four k-vectors shown in Figure P4.20. 4.21

Use the formula given in Problem 4.20 for the total electric field. Find the optimum position of a dipole antenna placed in front of the 90" corner reflector. The 0 angle of the incident wave is 30°. The frequency is 100 MHz. Express the position in x - y coordinates in meters. What is the "gain" of this receiving antenna? Gain is defined as follows: Gain - 20108,0

I~:I

(d8)

where E. is the E field at the antenna incident wave.

position

and Eo is the field strength

of the

Top view of a 90° corner reflector and the four k-vectors. P'lgur. P4,20

wes

Problems

97 x

tion

the

Medium 1

Medium 2

The (a)

the

Figure P4.22 7.

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!I an

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that

, the num :Ieof

z

z

-y med

)

f the

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-

-

z

z

>

>z

0.67

o

ler re-

'> z

(b) Figure P4.22

z

98 4.22

4

Reflection and Transmission

of Waves

Match the following descriptions to the standing wave patterns shown in Figure P4.22. The inr.idenL wave in medium 1 has an amplituda equal to 1 Vim. Note: There are three patterns that do not fit tiny of the following descriptions. Cross out these patterns. (i) Plot (If I E, .ntul I. with medium "1 being air, medium 2 having (2 = 4€n aud J.l2 = I~o' Normal incidence. [ii] Plot of I t::, .0.n.1 . with medium 1 being chaructertzed by €, - 4Eo and J.I, .. ~(o. and medium 2 being HiI'. Normal incidence, (iii) Plot of II';) LOIO' I . with medium 1 being characterized hy El .. 4(0 and 11..1 - !Joo. and medium 2 being air. The incidence angle is greater than the crttical angle. (iv) Plot of I E'loln' I . incidence angle is equal to the Brewster angle. (v) Plot of I Ez tOlal I . incidence angle is equal to the Brewster angle. {1 is greater than Ez. (vi) Plot of I Ey lulal I . Medium "1 is air and medium 2 is perfect conductor, (vii) Plot of IIIYlo(~1 I ("d. Medium 1 is air and medium 2 is perfect conductor.

4.23

Consider the CC:lSO of normal incidence of a uniform plane wave on a perfect conductor as shown in Figure 4.15. rt can be seen in (4.47) that an oscillating current is induced on the surface of the conductor. Therefore. the following expression may be written for the velocity of a charge on the conductor: v = ~ d q Eo cos(wl - kz) The above equation is exactly the same as Equatiun (3.39). Continue to work along this line and prove thai' the lime-average radiation pressure on the perfect conductor is twu limes tbat given in (3.4!i).

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:18

Using (5.45), we obtain

:al

VxA_;poAp_~

127

-ike

OZ

jk.

P

:b)

Example 5.8 ltl

Id

Calculate the total time-average electromagnetic power transmitted along a coaxial line when the fields are given by (5.4RJ.

.at

Solution:

The time-average Poynting vector (5) is given by

ra] (8)

= -1 Re[E

x H*)

=

2

Ib)

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=

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I

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2

P

p x

.V

cf> _! elk.

} = z_oV"

TIP

21)/

(11" V~/71)lln (b/a).

Ia rve :he

Ice

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 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

d 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 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

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...•.•

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l=O

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ithout

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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·

Repeat Problem 6.6 for a transmission line with an open circuit at the load (ZL = <10).

&.21

&.s. Repeal Problem 6.6 for a transmission line wilh a matched loaoll,. - 2.,J. &.8. A transmission line is terminated with a normaltzed load of O.R + j1.0. Calculate (aJ the VSWR. (bl the position of the voltage minimum, anti (e) the percentage of the incident power that is reflected by the load. Sketch I Viz) I as a function of z/): 6.10. Solve the problem discussed in Example 6.6 by using the Smith chart. Find the position of a shunt susceptance that can tune the line to have a perfect match. Determine the value (in mhos) of the shunt susceptance. &.11. For an open-circutted 5011 transmission lin a of length P,the input impedance at the other end is j33 n. Find the length Q (in X). 8.12.

Repeal Problem 6.11 when the line is short-circuited at one end.

6.13.

For the first waveguide in Tahle t of Section 5.2. design an iris that will give a j1.57 ittance ilt f - 8 GHz.

8.21

-n Lines

he one

Problems

181

8.14. From the Smith chart, find

rL for

the following

+

ZLII:(a) 1

jl.. (b) co, (c) 0, and (d) 0.55

- jO.38.

(6.58)

r

el

(bJ -0.3.

and

of 0.4 - jO.5. find the location of the first voltage minimum the first voltage maximum at the load end .

and

- 8.15. Use the Smith chart 10 find ZLn from the following

L

(a) 0.6

<;'.

(c) O.

period

8.18. For a load impedance

.-e.17. From the Smith chart. find the ittances 0.3 - jO.6 and (b) Zw = 5 + i3.

for the following

impedances:

[a] Z,

=

line is terminated with a normalized impedance ZI,n - 2 + i2. as shown in Figure 6.19a. The incident V. - 1.0. and ths characteristic impedance of the line is 1.0. Show that V ma> - 1mu - 1.62.1 VIOl 1- 1.55.1 V( - O.219A11- 0.78, V min Imln - 0.31:1,11(0)1 = 0.55, and If -0.219X) 1.45.

8.18. A transmission

1

,verage

8.1V. A shunt

ittance of Y Ln - - i1.57 is added to the transmission line that is terminated by a load ZLn~ 2 + i2, as shown in Figure 6.19b. The position of the shunt is 0.219Xg from the load, so that the line is perfectly matched. Let V • ~ 1.0 and Zo - 1.0 and show that V max - 2.08,1 VIOl 2.00, 1m •• - 1.B6,1l(0) 1- 0.71, and 1mln - 0.49.

ximum ~- 2

1-

x

8.20. In Example "-filled. 51 kW. ,nsity J. ::urrent 5.3b).

1-

6.B, find another

set of solutions

8.21. For the solution found in Example

6.B, how much total time-average power can be fed to the array without causing breakdown in the dielectric? Use the value lBl.000 V Icm as the breakdown strength of the dielectric, use a safety factor 10, and let a - 2 mm. Hint: consider the standing wave on the stub tuners as well as on the transmission lines.

-8.22.

'. Then for the

For the circuit shown in Figure 6.25a, let Zo - 50 n, RI. - 70 n, Hg - 50 n. 11- 2 m. 10' m/s, 6. - 10 9 s. and Vo - 1. Plot the voltage and current at z - 11./2 as a function time.

8.23. Calculate

the percentage of energy generated by the load in the circuit of Problem 6.22.

8.24.

L

=.:<:>1·

by the pulse generator

For a four-digit code system. design a D-A converter similar to that discussed in Section 6.5 using the transmission line shown in Figure 6.27a. Specify the value of R. the location of the sampler. and the time that a sample should be taken.

RL - 0.5Zo and Rg = 0.5Zo. 8.28.

Draw the voltage and the current reflection diagrams for the transmission line which is shcrt-circuited as shown in Figure P6.26. Plot V and I as functions of time at

z - 11/2. match.

a jl.57

of

that is absorbed

ind the

e at the

v_

-6.25. In plotting Figure 6.32. it is implicitly assumed that R,.> z.., And that H8> Zo. so that both rL and rs are positive numbers. Sketch a similar diagram for the case in which

ilate [a] , of the

of RI and Rz (in centimeters).

] ·1

'Igur. N.26

6

182

=

Transmission Lines

I-n

_r-i

V'l

z, Jz.

-,

Figur. P8.27

11--2--1

-6.27. Draw the voltage and the current reflection diagrams for the transmission line that is perfectly matched. as shown in Figure P6.27. Plot V and I as functions of time at ? - 11/2. 6.28. Refer to Figure 6.31. and let Rg = 220 and RL - O.57~" Draw the voltage reflection diagram for 0 <. t < 6T. and plot V at z = 3V4 Ior 0 <. t <.6T.

e.2e.

~

Refer to Figure 6.31. and let Rg - 2 Zo and RI• - 0.5Zo. Draw the current reflection diagram for 0 <. t <. 6T. and plot 1 al z - 3R/4 Iur 0 <. r <.6T.

6.30. Refer to Figure 6.38a, and obtain an expression for 1 . Sketch L(t) and I(z) versus? for the time period T < t < 2T. The sketch should be similar to Figure 6.38u ami Figure 6.38c. 6.31.

Derive (6.56).

7.1 Vector,

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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,

<1»

and (1".

(f,

4>') are

at P is 15 km away from a capacitor-plate anten na that is also placed vertica lly, as shown in Figure P7.8. The receiving an tenna measures an E field equal to 10 mV/ m. What is the va Ilia of E that the same receiving antenna will detect at a height 3 km above P? What must the orientation of the receiving dipole be to obtain a maximum reading? (A maximum reading is obtained if the dipole is parallel 10 the E field.)

7.8 A vertical receiving dipole antenna

----__---

I

_---

_----

~""",""-T

I I I

Figur. P 7.8

I

-~ --------------p~ 7.0 The power lost on a cylindrical conductor that is ~z long and that carries I amperes

of current is given by

Pnhm

=

%12 R. ~z

where Polun is the loss due to finite cond.uctivity of the wire. R. is the surface resistance given by 1/(ud.2?ro}, and d. is the skin depth. The efficiency of the antenna is given by '10 -

Power radiated Power radiated. + Polun

Assume that a short antenna of length ~7. has an efficiency of ten percent. Is the efficiency improved. by increasing the length to 2 ~z while maintaining the same current and, if so, by how much? Assume that the antenna is still a short antenna after its length is increased to 2 ~z.

Antennas

223

Problems y

e following

x

Ig vector A ..Igur. P 7.10 '.8) can

be 7.10.

Consider the antenna system consisting of two short dipoles arranged perpendicularly to each other in space, as shown in Figure P7.10. These dipoles are driven by the same amount of power from a common source. However, the current on the x-oritmted dipole has a -90 phase with respect to that on the }I-oriented dipole because of a phase shifter inserted in the transmission line that leads to the former. Find the total radiated electric field on the z axis. that this antenna system radiates a circularly polarized wave in the direction. Is the wave left-hand or right-hand circularly polarized? 0

, 0', 1/>') are acitor-plate ngantenna e receiving lion of the is obtained

z

7.11.

Find the expression of the total radiated electric field on the x axis that is due to the antenna system discussed in the preceding problem. What is its polarization on the x axis?

7.12.

requires that a field strength of 1V 1m be maintained at a point loco ted in free space. What power must be fed to the antenna if it is (0) an isotropic antenna, [h] a short dipole, and (c) a half-wave dipole? Neglect ohmic loss. An isotrupic antenna radiates an equal amount of power in all

A certain application

1 km from an antenna

directions. 7.13.

Tho current at the center of an antenna is 100A; what is the E field 1 km away from it on the horizontal 10 - 90 plane at 10 MHz if the antenna is (0) a dipole with hI hl - 0.5 m, [b] a capacitor-plate antenna with D.z - 1 m, and [c] a half-wave dipole? Q

)

[ amperes

ne surface ia antenna

cent, Is the Ithe same rt antenna

7.14.

Show that if the radiation field pattern shown in Figure 7.4 for the infinitesimal dipole or the capacitor-plate antenna is plotted in x-z plans in linear scale the pattern is exactly formed by two circles.

7.15.

Find the directivity half-wave dipole.

7.16.

Find the radiated electric field of a linear antenna that is 3 m long operates at 100 MHz in air. plot its radiation pattern.

7.17.

Consider a uniform linear array of two half-wave dipoles that are 1.5 wavelengths apart. The currents on these two dipoles are in phase. Sketch the radiation pattern in the horizontal (8 = 90") plane. Show clearly the number of lohas in this pattern. Also, estimate the beam width of each of the major lobes. The beam width is the angle between two directions in which the radiation intensity is one-half (-3 dB) the maximum value of tho boom.

uf (a) an isotropic antenna.

(b) a capacitor-plata

antenna,

r~

=

and [c] a

3 m] and that

Chapter 7

224 7.18.

Antennas

IQgure 7.23(bJ shows the array factor of a two-element array separated by 20>-.Find the beam width (in of the angle between two adjacent nulls) of this array factor near rP" 90° and 0 .. 30°. (a) Use the approximate formula given by (7.49). (b) Find the exact value starting from (7.45).

7.19.

Find the directivity of the two-wire transmission line shown in Figure 7.27 with radiation fields given by (7.42J.

7.20. Find the field pattern of a two-element array with d - >./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

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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

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qx

= -

(qpx - -

2p

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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

ing 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 es light polarized in the ic direction and absorbs light pulnrizcd in the 5' direction. The film at B es 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

the orrnngemant shown ill Figure P8.'lo. This figure differs from Figure P8.15 only in the placement of a third Polaroid film at D between A and B. The absorption axis of the third film is 45n Irum either the x or the y axis. Nuw, can the observer at C sap. thp. light? (If you do not believA in your answer, do an experiment with three pairs uf polarized sunglasses and see for yourself.] Figure PS.16 l1fjO

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286

Electrostatic

Fields

Problems g.1 Consider the dipole arrangement shown in Figure 9.20. Let q = '1.11 X 1O-1Y C. Find at:

(a) x-O.l,y=O.l,Z-O.1 (b) x-l,y=l,z-l

Use the exact formula (9.14) first. Then use the approximate formula (\1.17),and find the accuracy of the latter, The medium Is air.

e.2 Three point charges are located on the x axis with q, -

Q at x - 0, th - 2q at x = 1. and q, - -3q at x - 2. Find the position(s) on the x axis where ~ is equal to zero.

e.3

Four point charges are located on the corners of a rectangle, as shown in Figure P9.3. Find the planes on which the potential is equal to zero, Sketch these planes. y

..

q

..I

----+TI 2m

I

y

sm

I I -Q

+11

-2q

x

>

x

Figure P9.4

9.4 Two point charges arc separated by

0

meters in air. as shown in Figure P9.4.

(a) Find the potential function lx, y, z). (b) Calculate at x - 1000, Y = 100a. z - O. (c)

Show that. for distances much greater than potential is approximately given by

U

away from these charges, the

-Q 1 ,.,------~ 4'11'Eo (XZ + y2 + r)1I2

WhArA(x~ + y2 + Z2JI/2 >:> o. USA this approximate formula to calculate W(1000,1000, 0), and compare it with the result obtained in [h). g.5 Find the E field in air due to a point charge of 106q.(Q. - -1.6 x 10 diagram similar to Figure 9.4.

10

C). Sketr.h a

9.6 For the charge distribution given in Problem 9.2, calculate E. at (aJ x - -1, [b] x [c] x - 2.5, and (d) x - 3.

0.5,

9.7 Apply E

= - 'V to [9.14Jto find the E field located at the origin and prod need by two charges + Q and -Q located at ro, 0, l1Jand (0, O. h], respectively. as shown in Figure 9.2a. where h = 1 em. Show that E - -!(q/2'11'Eh%j.

9.8 Solve the same problem as in 9.7, but use (9.19c).

g.9 Skelch the direction of the E field located at the center of a square shown in Figure P9.9. The E field is produced by four charges at four corners of the square OABG. tbese four charges carry Q. q. -Q, and Q coulombs, respectively.

Ic Fields

Problems

287

y

:. rind Flgur.pe.e

and find

x

-1. and He P9.3.

8.10 A line charge 2h meters long is located along Inc z axis as shown in Figure 9.6a. The charge density is Pr coulombs per meter.

(a) Calculate the electric field at f>

-

O.lh. - 0, and z - U using ths exact formula

r~.21J. (b) Calculate the electric field at the same point using the assumption that the line is infinitely long. (c) Find the percentage error of the value obtained in [b] as compared with the exact value. D.11 For the same line-charge described in Prohlem 9.10.

(a) Calculate the electric field at P - 20h. - 0, z - U using the exact formula. (b) 1)0 the same using the assumption that the line is a point charge at the origin. (c) Find the percentage error of value obtained in (b). 8.12 A plane charge of p, coulombs per square meter is located on the x = 0 plane, and another plane of -p, coulombs per square meter is located on Ihe x - 1plane. Find the total electric-field in the region (al x '> 1.(b) 1> x » 0, and (e) x < o. 8.13 Consider the problem discussed in Example 9.11. Assuming that everything is the same except for the Iact that ths total charge on the conducting shell is now equal to

zero, calculate E everywhere. and sketch E, versus r similar Figure 9.15.

:es.lhe

10

the sketch shown in

9.14 A charge distribution of the following Iorm is set up in air [spherical coordinates): 0

o;

), 1000.

O<.:r<.:o R

10-

=

(

o

U

< r

b< r

ketch a (a) Find the 1J field for 0 (b) Find D Ior a < r < 0. (c) Find 1J for b < r

x= 0.5. by two

9.15

Figure

charge distribution

r

<.

c.

uf the following form is spt up in air:

P" - 10 " . e " coulombs per cubic meter

Use Gauss' law to find the E field everywhere. Hint: To find IhA total charge in a Gaussian surface. yuu must rio the integration because the charge is not uniformly distrihuted f lowever. symmetry still exists with respect to t/> and O.

Figure DARC,

A

<

9.16

Electric charges are distributed uniformly in the region 0.1 < x < +0.1 with density 3 6 C/m • Elsewhere. the density is equal 10 zero. Ftnd the E field everywhere.

P, - 10

9

288 Plot E. versus x. Find the potential the origin. 9.17

difference

Electrostatic

V. - Vo for a point x with respect to

Find the potential difference V A - Vo for two points A and B located r - 1 in the E field obtained in Problem 9.15.

9.18 TIIIl solution for the electric field of an oscillating frequency w is given in (7.14) as-follows;

E - {;

jkI

::-'hr {r [;~r (i:r 2 I

)2]

cos 0 +

Hertzian

8 [1 +

dipole

at r = 0

and

with angular

;~ + (j:rIZ] sin

oJ

Derive the solution (9.20) for a static dipole by selling w - O. Notice that k and I ~z = apia! - jwp. 9.19

Fields

=

w(lJtfZ

In the electric field E - 3:i + 4y - 5i. find VA - Va if A is located at (1,1,2) and 1J is at the origin. Does the difference depend on the path of the integration?

9.20 Consider the spherical-shell problem shown in Figure 9.14. Find the potential r » 0, and (d) r - o. Assume (rJ versus r. 9.21

Repeat the preceding problem for the case in which the total charge on the conducting shell is equal to zero while all other conditions remain unchanged. You may want to use the result obtained in Problem 9.13.

9.22 Consider the coaxial line shown in Figure 1-'9.22. The inner conductor is a solid conducting cylinder with a radius equal to 0.1 m. The outer conductor has an innet' radius equal to 0.4 m and an outer radius equal to 0.5 m. The medium between tha inner and the outer conductor is air. The inner conductor carries a net charge of - 3Eo Clm and the outer conductor carries a net charge of -lSEo elm. The symbol '0 used here represents 8 constant equal to 8.854 x 10-12• (a) (b) (e) (d) (e)

Find E" in the region 0.1 m < p < 0.4 m. Find Ep in the region 0.4 m < p < 0.5 m. Find E" in the region p > 0.5 m. Find atp; O.2m,knowingthat - Onlp -1m. Sketch Ep as function of p for 0 < p < 1 m. Ma rk the scale for Ep and p.

Figure P.9.22

-ostatlc Fields vith respect to at r - 0 and

j

with angular

1.2) and B is at

rtential <1>( r] at inity Plot (r) harge on the changed. YOIJ :tor is a solid tor has an inlium between ; a net chargo 1m. The sym-

lp.

Problems

289

9.23 Model the dome of a Van de Graaff generator as a conducting sphere. The dome is charged to hold the maximum amount of electric charge Qm before the air surrounding the dome breaks down. Use the following data: radius of tho dome - 0.11 m. breakdown E of air = 3 x 106 Vim. (a) Calculate the maximum Qm accumulated on the dome just before the breakdown. (h) Calculate the voltage of the dome ill reference to the potential at infinity just before breakdown occurs. [c] When the dome is charged with the maximum charge Qm. a person uses a conducting rod to discharge the electricity. Assume that the discharge takes 0.01 seconds to complete. how Iltrong is the discharging current (on the average)?

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332

Electric

Force and Energy

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 typical spacings between them are approximately 2 mm.

*

Problems 10.1 A point charge of Q coulombs is located at the origin (0. O. 0). and a second point charge of Q' coulombs is at (1. O. 0). A small test-charge is placed at [3. 0, 0). and it is found that the total force on the test charge is equal to zero. Find q' in of Q. 10.2 Two identical small balls are attached to weightless strings 15 COl long. Each hall carries 10 9 C of charge. and eoch has a mass of 1 ~. They achieve an equilibrium state under the influence of electrostalic force and gravitational force. as shown in Figure Pl0.2. Find the angle a. Hint: a is sma 11.

Flgur. P10.2

+Q

10.3 Consider a long line-charge with fl' - 10-' C/m. Find the force particle carrying -10 9 C, 1 rn away from the line charge.

acting on a dust

10.4 A line charge with p, ~ 10 fi C/m is located in air at x - 1. Y - O. A plane charge with 9 P. = 10-6 C/m1 is located at x - O. A positive point charge of 10- C is at (112, 0, 0) in rectangular coordinates. What is the total force acting on this point charge? 10.5 Charge and PI'

is uniformly distributed - 0 for r > u.

in the spherical

volume

r oS a with

6

PI' - 2 X 10-

(a) Use Causs' law to find E for r :s o. (b) find the force acting on a I~sl charge uf 10 I~ C at r - (1/2. (c} Is the force obtained in [b] to be changed if the charge distribution

extends

l

C/m

to

r=

20 instead of being limited to r -~ o? 10.6 In a seed sui-ting machme, ulluesiroble seecis are deposited with an electrostatic charge while they an automatlr. colur-sensitive or size-sensitive monitor. The good seeds are ed uncharged. All seeds are dropped between a high-voltage parallel-plate region to sort out the undesirable seeds. Let the charge on the undesirable seed be q. its mass be Ill, the voltage batween the parallel plates be V. and the plate separation be J. Assume thai the seeds enter the parallel-plate region at velocity vo, and find the displacement y of the bad seed as a function of x. Figure P10.6 illustrates this situation. Consider only IhA trajectory inside the parallel-plate. *P. Lorrain and 0 R. Corson. EIClt.'ll'omogOl!lism (San Francisco: W. II. Freeman and Co, 1978). p. 189.

d Energy

Problems

p calcuihedded approxi-

V (volts)

III

~

+

nd point and it is uf 1.1, .ach ball

ilibrinm hown

-

333

100

y 0

I I

+ Figur. P10.e

+

[ruilllseccnds]

-lOU

T

Figur. P10.0

J

in 10.7 At room temperatura (ZO°C) and standard atmosphere. what should be the size of the corona wire if b ~ :l cm. V, - 10 kV. and the roughness factor of the wire is equal to 0,8? (Refer to Figure 10.4,) 10.8 What should the lowest voltage on a Van de Craaff generator be in order to have it produce corona on its surface? Assume that F:c - 4 x lOij V1m and that the radius of IhA metal sphere is equal to 0,6 m. 10.0 Refer to Figure 10.1. If the voltage applied to the parallel shown in Figure P10.9. find the locus of tht: electron located at x - 20 cm. 10.10

a dust gewith 0,0) in

; 10 rostatic r The -oltage 10 the be V, region Figure

plate.

plate is the sawtooth signal on the fluorescent screen

For the cathode-ray tnhe shown in Figure 10,8. what should the voltages V. and V, bs in order to make the electron beam trace a circular path on the screen at 60 revolutions per second? Assume that the vertical and the horizonta I deflection plates are identical.

10.11 An electron is accelerated by a difference in potential of 1 kV between the anode and the cathode. It enters the parallel-plate region with this kinetic energy, Its velocity makes a 5°
x

--3I:m_1 ,1

r

. 41110V

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Q, - -1.60 x 10

I·e

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Pigur. P10.11

of I.

10

334 10.12

Electric Force and Energy

Consider the ink-jet printer shown in Figure 10.10. Define qd - charge on the ink drop md - mass of the drop Vo - deflection-plate voltage d - ddlt:ction-plate spacing Vd - velocity of the ink drop at entry to the deflection plate ~dP - deflection-plate length zp _ distance from the deflection-pistil entry to the print plane

Show that the vertical displaoernent flrl VO~dp

Xd- --.,-2

mrluvd

10.13

(

Zp -

of the ink drop is given by

1)2

-~dp

Find the capacitance of the spherical capacitor shown in Figure 10.13 by using (10.42) and (10.50). Start from E - ~ 411'Ef

r for b > t »

0

and show that your result agrees with (10.47). 10.14

Find the capacitance of the cylindrical capacitor shown in Figure 10.14 by using (10.42) and (10.50). Start from

E_~

p for h e- o »

u

211'fP

and show that your result agrees with (10.49). 10.15

Consider the parallel-plate capacitor shown in Figure 10.12. What is the maximum capacitance one can obtain hy using mica as the insulator? Let the area of the plate be 10 ern" and the voltage rating of the capacitor be 2 kV, with a safety factor of 10. Use Table 10.1 for the value of ( for mica.

Consider the cylindrical capacitor shown in Figure 10.14. What is the maximum capacitance one can obtain by using oil as the insulator? Ta ke a - 1 em, h - 2 em and the voltage rating - 2 kV. with a safety factor of 5. Use Table 10.1 for the value of e for oil. 10.17 A parallel-plate capacitor is filled with two dielectric materials in a configuration shown in Figure Pl0.17. The total area of the plate is A. (a] Find the capacitance C in of A, d, f,. and f2' (b] Suppose that the positive plate carries Q coulombs of charge. and find Q, and Q2 in of Q. where Ql and Q2 are charges on the 1eftand on the right-hand sides of the plate, respectively. Neglect fringing fields. 10.18

10.18

Consider the capacitor shown in Figure Pl0.17. Let fl - 3to. Cz - 5f.n, J - 0.6 mrn, and A _ 20 ern", The potential between the plates is 300 V. Plnd the total stored electric energy In this capacitor.

I"lgur.P10.17

wi;!

w/2

Problems

Force and Energy 10.19

335

Find the capacitance per unit length of II coaxial capacitor with two layers of insulating materials, as shown in Figura to 15c. Express CI I, in of 0, b, C, {I' and Ez·

10.20

Find the capacitance C of 8 parallel-plata capacitor with two layers of insulating materials, as shown in Figure PIO.20. Express C in of A (the area of the plate), dl, dz. El' and E!.

'igur. P10.20

13 hy using 110.42)

10.21

Refer to the capacitor shown in Figure PI0.20. Lel E] - 3Eo, ~z = 5Eu, d, - 0.3 mm, d2 ~ = 20 cm '. The voltage across the capacitor iii 300 V. Find the total stored electric energy in this capacitor. 0.3 rnrn, and 1\

10.22

Derive (10.t!2j.

10.23 A parallel-plats

re 10.14 hy using

capacitor cernes + Q on one plats and - Q on the other plate. The area of each plate is A and the separation between the plates is S. The medium is air.

(a) Find the total stored energy V t,; in this capacitor in of Q. A, S and '0' (bJ What is the clcctrostatlc force Acting on the plates? Is it attractive or repulsive'? Hint: find the change in UE with respect to S, is the maximum 'ea of the plate be factor of 10. Use is the maximum h - 2 cm and . the value of E for em,

1a

configuration capacitance C in 5 Q coulombs of arges on the lefting fields.

J = 0.6 rnm, and al stored electric

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368 Conductor

Solution Techniques

Conductor

Conductor

Figur. P1t.1

4>, - II

CaS!' ,

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Case II

<1>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 <1>" and the solution of Case II is <1>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 <1>3 in of 'I>, and ....a-

11.2 Consider the three boundary-value problems shown in Figure Pl1.2. The solution of Case T is <1>,. lind the solution of Case II is <1>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 <1>3 he expressed in of <1>, 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 is independent of rand q; when spherical coordinates are used. Find the potential "'(0) in the region OJ s. IJ s. 90°. Hint: J (l/sin 0) dO = In [tan 0/2). Find the surface charge density on the cone. The upper plate of the parallel-plate capacitor discussed in Example 11.1 is maintained at 100 V. and the lower plate hi at 80 V. All other conditions remain unchanged. Find <1>. Model a de vacuum-tuba rectifier as two parallel plates with a space charge in between. as shown in Figure 11.3. Let the separation be 1 cm. Find the voltage needed tu produce I A/m2 current. Find the surface charge distribution on the vertical and the horizontal conducting walls for the case discussed in Example 11.8. Plot p. for 7. :> 0 and x = y _ O. Let q = 10-6 C. and a - b = 1. Find the images of a point charge near a corner of a conductor similar to the one shown in Figure n.ll except that rPo = 45°. Find the electrostatic force that acts on the point charge q at (0, 0, d) and is due to induced surface charges at z - O.as shown in Figure 11.5. Calculate the capacitance per meter of a 12-inch (0.3048 ml-diametsr located 6 ft [1.R!! m I above and parallel to the ground.

steel pipe

Example 11.10 states that the maximum electric field on the surface of the conducting cylinder is located at the point nearest the ground. Show the validity of this statement by plotting out E, on the surface as a function of (p. Use the following data: V, - 100 V. h - 2 m, and a - 1 m. 11.13 For the point charge q located d meters from a grounded conducting sphere shown in Figure 11.14. find the surface charge distribution as a function of 8. 11.14 Repeat the preceding problem for an isolated conducting sphere carrying no net

charge.

11

370

Conductor Flgur.

P".'6

~igur.

Solution Techniques

1<1> - 0)

P".i.

11.15 Equation (11.42) ).!ivl::sthe potential dus to a point charge in the presence

of II grounded conducting sphere. Equation (11.44)gives the potential due tu a point charge in the presence of an isolated sphere carrying no net charge. From these results. finel the potential due to a point charge Q. cl meters from an isolated conducting sphere carrying a net charge of 1111'

11.16 A llna charge PI is inside a conducting tunnel of radius a, as shown in Figure P J Ll6.

Notice that the linA charge is b motors off CAnter. Find the potential function in tht! tunnel. lIint: This is a complementary problem of the one shown in Figure 11.12. 11.17 Calculate the force per meter acting on the line charge in the tunnel shown in Figure PI1.16.

11.18 A point charge q is inside a spherical cavity of a conductor. as shown in Figure Pl1.1S. The radius of the cavity is 0 and the cavity is filled with air.

(a) (b) (c) (d)

Find the potential 41in the cavity when b s O. Find the surface charge density of the cavity wall when b = O. Find the potential 41in the cavity when b = en, Find the surface charge density of the cavity wail when b = 0/2.

11.19 Calculate the aleetrostatic force acting on the point charge in thA cavity shown in Figure Pl1.1S. 11.20 Sketch the E lines due to a point charge near the interface of two dielectric media. The situation is similar to the one shown in Figure 11.17, except that (2 - 0.5f1·

11.21 A rectangular conducting trough of width a and height h is maintained at zero potential, as shown in Figure Pll.:.!1. The potential on the top plate. which COVArs the trough. is known to be 4l(x, b) - 200 sin(211"x/u)volts. Find the potential 41in the trough. There is no volume charge in the trough. 11.22 Three sides of a rectangular conducting pipe are grounded, while the Iourth side is maintained at 100 V. as shown in Figure Pll.22. find thApotential in the pipe. TherA is 110 volume charge in the pipe. FigureP11.22

Flgur. P11.21

200 sin [2 ..

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'1

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x "gure P11.24

of a rectangular conducting pipe are shown in Figure in the pipe. There is no vulume charge in the pipe.

11.24 Consider the boundary value problem shown in Figure Pll.24. The upper and the lower conducting plates are maintained at :Gt!I'U potential. The plats tit the left is maintainecl at 100 V. Two gaps insulate the side plate from the ground. There is no volume charge in the region and cI> approaches zero <18 x approaches infinity. (a) IJse the method of separation of variables tu obtain two ordinary differential equations. (b) Solve the differential equations. (The function involving y must be a sine function.) (c) Match the boundary conditions, and find the final solution. 11.25

/\. spherical capacitor is filled with a dielectric and with another material of f~ in the remaining

material of 1;1 in half of the space space, as shown in Figure Pll.25.

(a) Find the potential function (r) in the region a < b. The potential at r = a is Vo and it is zero at r = h. Hint: The potential satisfies the Laplace equation (J 1.2) and it may be assumed that it is a functiou of r only. (b) Find the electric field in the region a < r < h. (c) Find the D field in the region c c r-c b. Hint: The D Fiald in medium 1 is differAnt from that in medium 2. Note: the boundary conditions on the tangential E and on the normal D fields are sarisftsd using the suggested approach. (d) Find the total charge on the inner conductor and the capacitance of this capacitor.

era the the

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42

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43

rrrsnts

Problems

Solution:

387

The resistivity read hy the sonds will he will nol read Pr - 1U n-1I1. although it is expected rending. we must first calculate potentia I problem in Example 12.5. In the z h - 16 in. x (2.54/100) m/in. = 0.406

t: _ 1 (0.1 0.1

influenced hy medium 2. Thus, the sonde located entirely in medium 1. To find the the potential detected at IJ. We solved the present esse. we have (11 - U.1. x = u. y = 0, rn, Z f h - 32 x 2.54/100 m - 0.813 m, and

0.01) _ 0.8181

+ 0.01

Therefore, according Lo (12.22t1), <1'0 -

[1

1 -471' x 0.1 0.406

Substituting

+ 0.818] -0.813

=-

1 (34.7)

471'

the above value into 112.2I1J. WA obtain

Pr - 471' x 16

2.54

x -

Ino

x -

1

41T

x 34.7 - 14. I nom

Problems 12.1

12.11. 't and

A parA1IAIplate is filled with two materials in tI configuration shown in Figure P12.1. The total area of the plate is A. The dielectric constant and the conductivity of one material are f, and 1T" respectively. Those of the other material tire tz and C1Z' Find the equivalent circuit for this parallel plate, and express the circuit parameters in of A. d. t" 0" t2' and tJz.

12.2 A parallel

plate is 1'i11I~d with two materials in 0 configuration shown in figure P12.2. Find its equivalent circuit, and express IhA circuit parameters in of A, the area of the plate. and U1• U1• f" tz. (1" anti. (1z, which are defined in the figure.

ldalV ly. a~ exact arAnt

12.3

A coaxial line has two layers of insulation.

(a) IIIl! potential

t for h < f1 < c (c) the resistance of a section of such a line Q meters long

Flgur. P 12.3

Figure P 12. 1

y

tool

dif-

boun1 re-

alive

Figure P12.3 shows the geometry. Find

[::·:::r:~:: ]1 1"-W/2-~W/2_1

r p

~igu,. P12.2

[: ,: ::::~n,

lid'

I

I I

L/1

I

\

I.....

I I I I I

........ __ ---.-I _-./

I I

, I 1-

I ''\ I

12

388

'lgur. P12.4

Direct Currents

-- 1'1 Xn~

Flgur. P12.6

/

Perfect conductor

y

A

"uur. P12.7

if -,',20 /. ,~~ ,10 cm //'

',./

,/

/

em

"

;;;

o

//'8 I

l10cm

x

~I-I0A

~

I 120cm I

I

12.4 A spherical conductor of radius a is inside a spherical conducting shell of radius c.

Two materials are used to fill the space between these conductors. The dielectric constants and the conductivities of these materials are EI• 0'1' Ez, 0'2. respectively. Figure P12.4 shows the configuration. Find the equivalent circuit of this system, and express the circuit parameters in of a. b, c. £,. Ez. 0'" and 0'2' 12.5 Two oil wells are 1 km apart. The resistance between two steel pipes in these wells is

measured at 3.4111. What is the conductivity of the )(round near these wells? Use the following data: the length of both pipes - 1 km. and the diameter of hath pipes - 10 cm. 12.6 A current electrode is near a perfectly conducting plate that is bent to form a 90·

corner, AS shown in Figure P12.6. The output from the electrode is I amperes, and the material filling the space has a conductivity equal to 0'. Find the potential function 4>( X'. y,

7.1.

12.7 A current electrode is near a perfectly conducting plate that is bent to form a 60· corner. as shown in Figure P12.7. The electrode produces 10 A of current, and the material filling the region defined by 0 < t/> < 60° is water with conductivity equal to

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 es 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

y

B A

IT =

0.01mhn/m

x

-70m-

FIgure P12.15

13.1 Magn

12

CHIi PT~R

U

hAs t.

we

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VT

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45

For

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looks

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46

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---

1.2·12

(i)

I~r Jo>l/2

j

A

;:c -Lr:.Z ~6= 4"'1iL-r+

s:

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B

tMtJil.

]

Z '.. :;::

II

'

,

r

'~ol ~f ..·4711~/I. ~[I+r~~)/2(Wl>J 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"" "..,1<:1.:...... ',

~[1· ISj].->

f4.'~1f.tFIJ/z·

~~r~

~:~.r

~li+('-;~)/24ffi

'i ~;

I'

I

"

I

1

i--.J "

I

~'

I~

48

... AI

", 12.15 -

FfTJ,..

Pro' ~

I

t./~,)=4n/V-'M ~')/r

= L!;l:.I I(/~~'i.,-

___ .______

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I,"J,(~"'J."AI) +~

_

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LBI

,

• 47

I

~,

-•

13

422

Magnetostatic

Fields

Because magnetic field is present in the coaxial line. we know that magnetic energy is stored there. The magnetic field is given by (13.7): I -

Hoi, ~ { Substituting

o

2 Jl

-

elsewhere

the above expression 1

UH

b>p>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 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 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 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 ed 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

Three infinitely long parallel wires each carry 10 A of current in the 1 direction, as shown in Figure P13.22, Find the force per unit length acting on the 113wire due to the magnetic fields produced by the other twu wires. Give the numerical value of the force. its direction, and its unit.

static Fields

Problems

427

aers d « o. ier carries I nt: Because hat is at one field due to rt is needed

T 1m

ect current. sly equal to Ice a maxithe loop in

j

#2

~--lm--...j

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x

Figure P13.22

Flgur. P13.21

13.23

The magnetic field in a coaxial line is given by H~

= (Ol/P ~

forO.lm < P < 0.201 elsewhere

The medium is air. What is the total stored magnetic energy per unit length in the line? Give the numerical value and indicate its unit. 13.24

(a) Calculate the stored magnetic energy por unit length of the parallel-plate conductors shown in Figure 13.5. (1:» Ifthe parallel plate is used as a capacitor to store electric energy, find the voltage Va for which the stored electrtc energy is oqual to the stored magnetic energy found in (a). Let l = lA. w '" 10 em, and a - 1 em. Express Va in volts. The medium is air.

13.25

Calculate the inductance

pel' unit length of tho coaxial line shown in Figure 13.3a.

13.26

Calculate the inductance Figure 13.5.

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-hown in

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14

Magnetic

From

B(O" we find the corresponding

f1101(Figure

11

101 ~

Materials

and Magnetic

Circuits

14.17b):

2400 Aim

The iterative method calls for substituting the above value into (14.23) to obtain the "first-order" approximation of B, which is denoted as BP':

nPI = (1000 The corresponding HilI _

- 2400 x O.121)J.lo_ 0.178 Wb/rn! 0.005 11111 may be read from Figure

2100 Aim

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-

'14.17b:

approximation

of B by substituling

the above H(1l

(1000 - 2100 x 0.121},.,.0 Wt I . - 0.1117 () m0.005

This procedure can be repeated to find the n-th iterative result of BInI. When a digital computer is available, the magnetization curve can be approximated by a standard polynomial-curve filling and stored in the computer memory, A simple program may be written to carry out the iterative procedure, which requires very little computer time [see Problem 14.7). The problem at hand can also be solved by a graphical method. Note that (14.22) or, equivalently, (14.23) is an equation of a straight line on the B-H plane. As shown in Figure 14.17b this line intersects the B axis at 0.251 Wb/m2 and the H axis at 0264 AIm. It also intersects the nonlinoar magnetization curve at B ~ 0.'19 Wb/m2• This result agrees fairly well with the result ubtained by the iterative method.

Problems 14.1

Refer to the magnetization curve shown in Figure 14.3. The material is a nonlinear medium because p. depends on the magnitude of H. For magnetostatic fields, 1.1 is equal to the slope of the line ing the origin to the (H, B) point on the magnetization curve. In this way. Figure 14.3b is obtained from Figure 14.3a. Now. if the material is placed in a time-harmonic field, the effective p. will be different from the J.lfor the magnetostatic fields. Consider a field H - Hu + HI cos (wt + 4», where Ho is the bias magnetostatic field and HI is the amplitude of the lime-harmonic component of the total field. Let HI « 110: then the effective permeability of a material is the slope of the tangent of the magnetization curve at Ho. Sketch the effective J.l versus Ho for the curve shown in Figure 14.3a. Compare it with the rnagnetostatic /Jo shown in Figure 14.3b. and show that the !L'S in these two cases are equal to each other at 1'3'

14.2 Point out the differences between the following pairs of : (a) diamagnetic paramagnetic, (b) remanence vs. retentivity, and (c] coercive force vs. coercivity. 14.3 What are approximate shown in Figure '14.9?

values

of the retentivity

and the coercivity

vs.

of the ferrite

:ic Circuits

Problems

453

14.4 Consider the carbon steel. alnico V. and eunico materials listed in Table 14.2. Which has the highest permanent magnetic-field strength? Which has the most difficulty in losing its permanent magnetism once it is magnetized'? ohtain the

14.5 A permanent magnet of radius 1.5 cm and thickness 0.3 em is put in a magnetic field that is parallel tn the disk, as in the situation depicted in Figure 14.7. The torque on the disk is equal to 1.2 x 10-3 N-m. and the magnetic field is equal to 10-' Wb/mz. What is the remanence of the permanent magnet? 14.6 To write "one" in the memory core X2YJ shown in Figure 14.11, how should current pulses he sent along the wires? Specify the polarity of these pulses.

the

14.7 Consider the magnetic-core memory sketch Ad in Figure '14.11 and the corresponding hysteresis curve for the cores shown in Figure 14.9. Now suppose that, because of malfunction in the circuitry, a positive pulse of amplitude I, which alone is capable of producing the switching magnetic field slrength HI, is sent down the line yz and that simultaneously an identical pulse is sent down the line Xl' Assume that all cores arc initially in the "zero" state, which corresponds to having the magnetic flux circulation pointing either toward the upper left or the lower left (using the righthand rule). What are the states of nil of the cores after these pulses have ed through?

above Hili

n a digital standard gram may computer

14.8 Compare the hysteresis loops of two ferrites shown in Figure P14.8. The curve labeled ttl is "thinner" than thai labeled tt2. Which ferrite core requires less switching current? Which ferrite has a better ability to withstand magnetic interfcr-

hat (14.22)

ancss?

As shown H axis at .9 Wb/m2•

14.9 Consider the magnetic circuit shown in Figure P14.9. The material is steel, and Figure 14.17 shows its magnetization curvs. The flux density in the air gap is 0.5 Wb/m". Find the current I needed to produce this flux.

ad.

14.10 The magnetic circuit shown in Figure P14.10 is made of a material Find the flux densities Al and B2• and indicate their directions.

with ~ - 600~).

B ill

( on linear slds, p. is on the . Now. if ent Crom j, where armonic lity of a etch the vlth the ases are

112

_l

t H j

J

Figur• P14.8

Hysteresis loops for two

ferrite s.

Flgur. P14.8

tstic vs. vity. !

0.5 em

lengths.

eross-seetiona I areas:

PIP, - gem p,p. - P,P, - 10 em P2P,p. - 26 em P,Pu - gem

PzP, - tz cm' all other branches - 9 em'

ferrite P,

Figure P14.10

14

454

r-8CIlI

Magnetic Materials and Magnetic Circuits

6cm-1

1

Pigure P14.11

tz cm

J

14.11 To produce a magnetic flux of 0.5 Wh/rn" ill the air gap of the magnetic circuit shown

in Figure P14.11. what should be the magnitude of the current in the coil? Take 200~. The cross-sectional area of all branches is equal to 4 ern",

IJ.-

14.12 Write a computer program 10 carry out the iteration procedure outlined in Example

14.5. First, approximate the nonlinear curve in Figure 14.17b by a polynomial of fifth order. Then carry out the iteration five times to obtain the fourth-order epproxtmation for B. 14.13 Find the approximate value of B in the magnetic circuit shown in Figure 14.178 for excitation current J = 15 A instead of 10 A. All other conditions given remain unchanged. Carry out the iteration a sufficient number of times to obtain an accuracy to tha third digit.

15.

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15

468

Fields

Flgur. 15.0 Poynting's theorem for quasistatic fields.

p - -):[ ~

=

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= -

- -.#do

.

<1>

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A

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t

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fl.

do •

J) -

t.

V"1"

where In represents the currant flowing into the volume through the surface Am whose surface normal is pointing outward. Thus, we have established that the circuit theory's concept of power input is valid only when the displacement current is negligible.

Problems parallel plate shown in Figure P1S.l, with width w, length II.and separation n. Finn Ihp, zaroth-, Iirst-, second-. and third-order electric and magnetic fields. Show that the sum of the quasistatic solution is Aqual to the full wave solution as presented in Chapter 6 for a short-circuit transmission line. Assume that the current at z - 0 is 1(1)- 10 COS(wl) and that all fields are functions of t and z only.

15.1 Consider the short-circuited

15.2 Calculate the total zeroth-order stored electric energy in the parallel-plate region shown in Figure 15.1. What is the zeroth-order stored magnetic energy in the same

region'? 15.3 Calculate

the total first-order stored magnetic energy in the parallel-plate region shown in Figure 15.1. What is the first-order stored electric energy in the same volume? Denote UW" the maximum total first-order stored magnetic energy in the

ic Fields

Problems

quasi-

469

r

.~---.--/1

_l_¢=========~~

w

l----/-~I/

Figure Pt5. t A parallel plate with short10 (;US (wI) at z = O.

circuit current T -

region and U~~ the maximnm total zeroth-order stored electric energy in the slime volume. Show that Umn

urur

u~

?

~

(k2J"

F.m

:rface ished :1 the

15.4

Calculate the total second-order stored electric energy in the parallel-plate region shown in Figure 15.1. Compare it with the zeroth-order stored electric energy. If 2 is 0.1'>' long. what can you say about the relative magnitudes of the zeroth-order stored electric energy, thc first-order stored magnetic energy. and the second-order stored electric energy?

15.5

Find the total zeroth-order stored electric and magnetic energies in a parallel plate with Ii short-circuit current I - 10 cos(w!) at z = 0 (refer to Problem 15.1).

15.8 Find the higher-order

stored electric and magnetic energies in the parallel-plate region shown in Figure Plri.l up to the third order. If Q - O.lX. compare the relative magnitudes of these stored energies. Use the total zeroth-order stored magnetic energy found in Problem 15.5 Ior comparison.

15.7 A coaxial line 2 meters long is filled with a material characterized

by f and CT. The radii uf the inner and the outer conductors arc U lind b. respectively. The voltage between the coaxial conductors is Vo cos[w!). find the zeroth-order electroquasistatic field E10). the current 1101, the charge QIOI. anti the first-order current 1111. Express these in of the parameters Yo. c, h. Q. f. CT. and wI.

th w,

ectric efull sume and z egion same 19ion iarne 1

the

15.8 Two concentric spherical electrodes of radii a and b. respectively. are filled with a material characterized by e and 11. The voltage between the electrodes is VO t;OS (wI).

Find the zeroth-order electroquasistatic field F:;IOI. the current J'", the charge the first-order current Ill). Express these in of Vo. a, b. f. (J', and wI.

QIDI,

and

15.9 Show that the time needed to charge a Van de Craaff joIener'iltOI' shown in Figure 9.24a with radius It to a maximum voltage of V mo. by applying a charging current I is equal to 41!'EoRVmA.lI. Calculate the charging time I if R. - t rn, VIIIax = 10RV and 1- lO-s A.

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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 -

$

-

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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

V,

I

L.

I I

/ -

<,

3kO

'-''-) I. 1/ 0'

O~z

.> I

,-----------, I I

I I

I '-'J

Flgur. Pie.S

_

j

,..,,../'

,/

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/

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4.5 kO

/

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v~ \

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,/

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I

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Figur. Pie."

V$

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I

J

Problems

511

~----~kO

---

<,

I p~

" PI

I

I I I

I I

1

kG

\

ne.

on

\

\

:he lar

1 kO Figure P16.5

t a

he Four resistors form a circuit 8S shown in Figure P16.5. The total magnetic flux linking the circuit is increasing at a rate of 0.5 Wb/s, in the direction painting out of the paper,

16.5

is

(a) Find the direction and magnitude of the induced current in the circuit. [b] Find the readings of the voltmeters Vt and V2• Two resistors are connected by wires to form B circuit as shown in Figure P16.Ga. ThA magnetic nux linking the circuit varies with time. Figure P16.6b shows the time variation of the magnetic The positive value of the flux corresponds to the flux directed into the paper. The magnitude of the nux is for a single turn of a circuit loop that encircles the magnetic flux.

16.6

nux.

(a) Plul the current 'It) versus lime. Be sure to mark the scale of the current. (h) Plot the voltage VeL) versus time. Mark the scale.

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

1(1)

I

\}I

(Webe,·s)

I

I

'kilO

7 kO

I I

1

0.5

VIIi

Seconds

-0.5 1

I

I L

...J

(b)

(a) Figure PUI.S

512

Magnetoquasistatic

16

Fields

16.7

What is the EMF induced on 0 propeller hlade that is 1.5 rn long and is rotating at 10,000 r/min in the earth's magnetic field (0.5 x 10 • Wb/m2)?

16.8

Find the voltage induced in the rectangular loop shown in Figure 16.1 if it is rotating about the axis parallel to tha z axis located at x - d + ~. Assllme that the angular frequency of the rotation is wand that the infinitely long wire carries a direct current of I amperes. Show that thA induced EMF is not a pure sinusoidal voltage. It is approximately sinusoidal when d » a.

16.9

A magnetic coro is made of a material whose hysteresis loop is shown in Figure P16.9. Note that this hysteresls curve is not a "square loop." To read the content of the core, two pulses al'e applied to the wires. The currents generate an Ii equal to 200 AIm. The core has an area of 3 x 10 -7 m2• (a) What is the voltage induced in the sensing wire if the core is originally at the "zaro" state fat point C)? Assume that switching from C to A is linear with time and that it is completed in a microsecond. (b) What is the voltage induced in the sensing wire if the core is originally at the "one" state (at point A)? Assume that switching from A to A' is linear with time and that it is completed in U.5 J.l.S. This voltage is the "noise" voltage because it would ideally be zero if the hysteresis loop were 1;1 perfect square. B (webers per square meter]

Sensing wire

t

0.220.3

0.1

-200

-tOO

100

-0.1

Figure P16.9 Ferrite core memory and Its hysteresis loop.

200

H (amperes

per meter]

-0.3

16.10

Find the total expansion force acting on the surface of an air-core solenoid that has 100 turns of coil and radius a - 1 cm, length ~ = lU CII1, and current I - 10 A.

16.11

Repeat Problem 16.10 for the case in which 100 turns of coil are wound over a ferromagnetic core with J.I. = 10001-'0' The current is 10 mA, with a = 1 em and P = 10cm.

16.12

A copper pipe of radius a - 2 cm and thickness d ~ 0.1 em is placed in a solenoid that has 200 turns per meter and is excited by 0 1000 Hz 10-A current. The conductivity of the COpPHf is 5.92 X 107 mho/m. Calculate the puwer per meter dissipated in the copper pipe.

16.13

A transformer similar to the one shown in Figure 16.11 is made of a steel with relative permeability equal to 1100. The effective length of the core is 40 em, and tho flux density is B - 0.3 Wb/m2• N1 - lOa, Nz - 1,000, I. - 60 A. (8) Find 12, assuming that the transformer is an ideal transformer. (b) Find 12, using (16.37).

(c) Compare the two answers.

itic Fields ~otating at

Problems 16.14

is rotating .e angular -ct current tags. It is

m

18.15

The primary coil of a transformer has 150 turns !U1dthe secondary coil has 450 turns. The effective length of the core is 0.5 m and the flux density in the core is 0.25 Wb/m2• The transformer is similar to the one shown in Figure 16.11. Assume that 11 " 60 A ami there is no nux leakage. (a) Find 12, assuming ideal transformer condition. (b) Find Vz, assuming ideal transformer condition and V, = cos(120wt). (c) Find I2, taking into consideration that the core material has a finite permeability equal to 10001-'0' (d) The hysteresis loop of the core material has an area equal to 90 Wb-AlmJ. What is the power loss due to the hysteresis in the transformer? Assume that the core has a cross-sectional area equal to 4 em",

illy at the with time illy lit the with time :Jecause it

(amperes

Consider a magnetic circuiL similar to the one shown in Figure 16.11. The effective length of the core is 0.4 m and its permeability is 2000 1-'0'The cross-sectlonal area of the core is 4 x 10-4 mZ• Let I] = 10 A, I2 = 24 A, N, = 50, and Nz 20. (a) Calculate the B field in the core. Give both the direction and the magnttnda, (b) If I] is a-c with f= 60 Hz, what are IV 1 1 and 1V 21 ? Assume that the magnetic flux always slays in the core without any leakage.

in Figure contont of 1 equal to

are meier)

513

16.16

Figure P16.16 shows a magnetization curve of a core used in a transformer, Notice that the hysteresis is negligible in this case and that the curve is linear in the range o s 1 H 1 :S 150 AIm but saturated when H is increased beyond this range. Let us now review Example 16.10. Because 1 VI 1 = wN1'lt, we want to use maximum -v in order to minimize the number of coils in the transformer. Using Figure P1B.16, explain what will bappen to the shape of the -V(L) and consequently to the shape of Vt(L) if v is too high-for example. if -v is so high as to correspond to B = 1,2 Wb/m~.

16.17

Estimate the approximate power loss attributed to hysteresis in the ferrite core shown in Figure P16.9 if the core is switched back and fortb between "zero" and "one" states 1000 times in a second, Assume that the core has an average radius of G x 10-4 m and that its cross-sectional area is 3 x 10·' m2,

per meter]

B [webers per square meier) 1.0

1 that has \.

0.5

,d over a 1

em and 200

mold that ictivity of ed in the h relative 1 the flux

-0,5

-1.0

16.18

H (amperes per meier)

Figure P16.16

Show that the mechanical torque required to drive an ac generator is nnt constant with time or. 10 he exact, that it consists of a constant term and a term that varies sinusoidally with time with an angular frequency 2w, What is tbe time average of the torque? Express the torque in of the area of the winding A, the current I. the magnetic flux density B. and the phase angle a between the voltage and the current. Plot T as a function of t for a - 0

514

16

Magnetequaaistatic Fields

16.19 Figure 16.15 depicts an ac generator with a single coil being rotated in a constant magnetic field. It illustrates the operating principle uf 8 single-phase ac generator.

Lei us nuw consider a three-phase ac generator. How would yon physically arrange three sets of coils in order to generate three-phase electricity? To illustrate your design, sketch a diagram similar 10 Figure 16.15. 18.20

What is the total mechanical torque needed La drive the three-phase generator that you have designed for Problem 16.19? Express this torque as a function of time in of the appropriate parameters. Plot T AS a function of time, and compare it with that obtained in Problem 16.18. Is the instantaneous mechanical torque "smoother" (does its time-average value fluctuate less) compared with that for a single-phase generator?

Appe

U5.21 Design a coil configuration similar to the one shown in Figure 16.17. Design it in such

a way that it will produce a rotating magnetic field in the armature-stator air gap and that the field will have an angular speed equal to w/2 when fed with the a-phase current shown in Figure 16.18. Prove that your design is correct by drawing instantaneous-current diagrams similar tu those shown in Figure 10.17. 16.22 Show Qualitatively that the torque generated by an induction motor may be varied by changing the resistance of the windings of the rotor. Figure 1fl.21 shows torque

curves versus v Jv ; with three different values of rotor conductivities. What are the relative magnitudes of crt. CT2' and "3? 16.23 Refer 10 the synchrunous

motor shown in Figure 16.22. What happens when the torque angle is negative-that ill, when the position of the 1'0101' magnetic-moment is ahead of the magnetic fieW?

Sym A

A R

16.24 Consider

the coils of the magneplane's track shown in Figure 16.25. For the magneplane tn travel at a speed of 250 km/h, what should be the distance 2, in meters? Assume that the power is provided by IJ three-phase 60-Hz power line.

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dEc: l/Sd.r

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At t,;..,c

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v=-;!

~

rh

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= o./-

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..

oiJe

..

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IoII.IJt ~ ..) ~ •

"-4 I< JW./"'"

,KltJ-r·-O.12

IJB II: -~&1

vrUAfC-,

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WS,,.,z,

-41!A

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+ • C.S~ , - U,;;U

+

d"

S; ......

it ~ Af',,.,,~,;-..J~':Is,,, ..,~~..t iJ

r

-t....~ of:

r ,.~..Jt)'s·,"w t

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•

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~

L =~tl1.""'a.'J.; F- !t~_.f~p.,tm:ll-z'P.nZ1f"l..

Lft;.l.!

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~

FI"(),..

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/I

(/f#.II-),

x (':::

)".

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a:. (".AIl,)

,'-

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1

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(4.)

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(0)

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to.)

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= J-I J

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(a.)

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=: ::

• "i'ir On.

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«

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£ c. twsr'::;

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.... '1

-')

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....

W ~

(

= j-o"lTTx,.Jto.n,'x¥'l.lo·r; = o· ff V ao ~At, IV.'I :. h

O=200tH'I'f1l,;"1

Il

o· yr

=

o.

JI V

59

c l.cK ~"JI!

)

..

.ro=

~./~'

W./tttl

",5

}l,I,=JJ.lt_

(<:J.)

(~)

V~/V,::a.

(C)

"',I,

~

J.I../tI,

=

Va :.(,#,o!6.JeoS(,UfffJ

7·r

,.s

(Ih'lt)

=H(=(O!A).l

-Jl,l'.

i-.to :r~

-+

oA

=

11=/~ox'o/flh

= Iro~

60 - «.sr

« e.r:

I(

IOU

7 JI'Jl1X 10- ).

r, ~ If. 71 ~ (J)

.,

=

0.1'1

1t.['l-tl1t>-Y"

: /.01

"/0

.,J

(1'.Il4~..

('lro/u-:~ ) B'~·I...")

8

....

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a.

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=;::=== ~

~~ "'h

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60

0.1"

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I+CDs(2uJ't)

~~~ .. _wt

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• A;r [3~S-'+

~

(1- fIi./V;)

~

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z50

a.

It,../",

{.~

sJ.p",U ht. s,...Ulu"~

wlt,,, v;./~

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it = 2" If/w = V/.; = '9.H/'()

oIttII.

+'U· )

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~

c.1t1st

*' I,

== ~9.

-,./s7

#

~

Ii >6';

5~.~"'· II:

C()S'licit

A:r (3t.DStI..) i,.J~4At

Teet.a.,)

F/'rI,.. fyj ..rt. I'.~/J /6.13

+c.tJS'

('OS(2~+"''''/Jo·J(.tGDj/Z/l+'»)..

h .. ",..,(lb.'7)~ O-(I-1If./lIS)=-kd/(Ul..VS)

HI" lar.Jt.r tr;

Z4D·).J

"'A

M

61

~~.:-".,..

*/". >6j.

v."'f14C,· ,10• .../«'

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to

r.

we.

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

elk..

l. 1E

2,

1- 0.51 1';u1l1- e-'2~.a·le-~""'.1

4.11 (a) 9Et, (h)

75 MHz

Ezzi ~

.J3

160

Chapter

Answers to Odd-Numbered 4.13 cos

"

Jl.l(Il'lE~

Z

-

}l,

2

521

Problems

f',l • c OS

1

tl(Jl.Slz

Jl.lflJ

l~J 4.15 (a) 80 cm in front of the plate (b)2V/m 4.171.996IEol 4.19H'- "llop'-I."inl-/k .. ooll. Ei _ (x cosO - Z sinB) HOlle-I"'.'n'-lk ee'O,;: H" _ Ylloe-1k"''''-lkzco,B. f](}l~ -

Et

(x cosO + z sinB) Holle

=

4.21 x Chapter 5

}l,)

=

J.ll(li

,h

where

11 -

~~

0.87 m, Y - 1.5m. 12.04dJi 4.23 Proof omitted

E • Eo 5.3 1.875 kHz 5.5 E .. x Enelkz, H = - y ~ elkz• J. = -z-P.' '1] Eo '~ '1] on lower plate, Is = i - e' Z on upper plate 5.789.33 kW 5.9 Proof 5.1 Proof

'kz

1/

5.11 5.26-10.52 GHz for 2.85 x 1.26:l (cm) waveguide, 21.1-42.2 GHz for 0.711 x 0.355 (cm) waveguide 5.131.318 MW 5.15 Ey = E1 sin ('IrX/O) elk," 5.170= tan-l(na/mb) 5.195.83 GHz Hx = (l':lk./WIL) sin (1I'x/o) ejkt~ . Hz = (j E 11I'/WJ.lO)cos (1I'x/o) ejk.~ 5.21 Proof omitted 5.23 (0.866. 0.5. 2) where k~ = [w~ ILE - (11'/0)21'12 V 5.25 Proof omitted 5.27 A = 4.93 i + 7.469 - 3t 5.29 (a) E' = P 4-,1:]·, P

+ j3)

H

27

I

A

= c{> -

Vt

e

jk.

_

1

I

(b) Vo

=

Va (111 - 1/oJ/('I11 + Ilo).

V,

•

=

2VO'1]I/(7]1

+ 1/0)

'1]lP

Chapter 6 em

6.12000V

rrr:':

6.3 (°1/°2) '" (b,lU2) - "1000

Vu Jkz 6.5 j.=! - e- '.1

27rVa --e-'

kz

"I

I sin kz I Zo (b) Sketch omitted (e) 00 6.9 (a) 2.96 (b) z = - 0.35).. (e) 24.5% 6.11 0.342).. 6.13 d = 0.25 cm 6.15 (a) 1.26 + jl.61 (b) 0.54 (e) 1 6.17 (a) 0.61 + jl.33 (b) 0.15 jO.09 6.19 Proof omitted 6.2114.2 kW 6.23 48.6% 6.25 Sketch omitted 6.27 Sketch omitted 6.29 Sketch omitted 6.31 Proof omitted 6.7 (a)

)=

: 107 I)m-t 1,

Chapter 7 )D

I V (z) I = 21 V + II cos kz 1.1 I (z) I =

1/a

=

2~

7.1 (0.75,0.433,0.5) 7.3 Proof omitted 7.5 ~ . 8 = cos8 cosc/l.~ . ' t = sinOsine, y . 8 = coso sin4>,$>. ~ = 1;084>. Z . i' • cosO. i . 0 = - sinB, Z . ~ = 0 ·kID. -IJoe 7.7 Proof omitted 7.9 Yes, improved to 18% 7.11 E = (- y) J ZIl8 • 811'x

l

linear 7.13 (a) 0.314 VIm (b) 0,628 VIm (e) 6 VIm 7 .15(a) 1 (b)1.5 (el1.64 1.17 Six lobes; beam width = 19.2° along c/l= 0; beam width = 26.4° along e = 41.8° 7.19 D=4 7.21Sketchomitted 7.23 (a) -9Uu (b) 6; 1.414; 0; 1.414 Vim (c) Sketch omitted (d) Sketch omitted

foil is

- jO.5)

mgle

f3 60

Chapter 8

8.1 UH'UB - (krJ~ «1 8.3 When background is dark, one sees light scattered by the smoke particles. Blue light is scattered rnure strongly than red lighL Against a bright background. one sees lighl ing through the smoke. The blue light gets scattered. and red and yellow lights suffer less scattering. 8.5 Proof omitted 8.7360 km in radius 8.9 Train is moving toward the intersection. 8.11 Bandwidth = 59 kHz. 6.7 lAoS for 1 krn resolution 8.13 Circular but opposite hand 8.15 No

Answers

522 Chapter 9

to Odd-Numbered

Problems

9.1 Exact: (a) 5.5302 x 10-10V (b) 5.54244 x 10 12V Approximate: (a) 5.5426 x 10-10y (b) 5.54256 x 1O-12V 9.3 x = 1m plane, y = 1.5m plane 9.5 (- i) 1.44

10-3

X

• z

Ylm

9.7 Proof omitted

Pi (b) P __

r

(0.0499) VIm

(0.05) VIm

9.9 Sketch omitted

9.11 (a)

9.13 E - 0 for r

(c) 0.2%

P __

pi

40nl1

<

401!'ffl

<

o and b ~ r l2 - e-'(r2

e, E = t _3_ for a ~ r 411'€l'2

+ 21' + 2)1

9.19 3 V, Independent (d)

..sL(:! 1. _ 1.)b

Chapter 10

1~-1i (3e-1

9.17

of path

h and

I' ~

e

9.15 10 -6 fT2

1) Ilint: S dre '(1 +

9.21 (a) _q_ 4?rcc

(b) __iL

4n

+

I'

1. e

1.)b

(c) 0.4 rnA

e

i

4

Q 10.31.8 x 10-5 N (attractive) 10.5 (a) 1'(2.411" x 10 ] VIm 9 (b) a x 1.211"xlO-8N (repulsive) (e) No 10.70.36 mrn 10.9 z = ::t: 3.14 cm 7 10.11 (a) Vo = 1.1374 X 107 mIs, vo. = 1.867 X lu7lnls, Vox = - 0.163 X lU mls (b) x(t) = (8.78 x 1014)e - (0.163 x 107)1 m, z(t) a (L8n7 x 107)L III (e) x = - 3.52 3 x 10-2 m . x 10 4 m, 10 . A + 10.13 Proof omitted 10.155.3 x 10farad 10.17 (a) J -2-

10.1 -

Z- .

= ~.

(b) Qj

1

10.214.97

+

~l

(tan~).

fl

J

F/m

(1

q

11.3~

q,2

a)

,Jo

1

uu~

u'

(Lan t) SiDO]

11.13 _Q(dZ

-R -

411't

VIn(blp)lln(b/o)

=

P. = - VOfo/[r In

11.1117.5p.p.

11.15 -

[lin

10.19211" / (clal + 1 In (hIe)] + €z El E2 10.23 (a) Q2S12Afo (b) - Q2/2Ato (attractive)

f2

x 10-11

11.1~3 = In

(€I ~2)

Q2 = ~

~I

Chapter 11

-e-'/r

1')11'2 =

..sL(~

(c)

4?rft;

9.23 (a) 4.03 x 10-0 C (b) 330 kV

+

411"~u

<

+ q(o/d)

-

f

4wtr

rei

Ra ~ ( 1'2 I IF - 2 CfcosB 11.19 (d/o)q2/[4wE(d

-

)1/2

11.5~(6) - VoIn(tan~)/

11.712.3 V

11.9

Sketch omitted

o2)/[41ro(a2 + cF - 2ud COsB)3/Z)

qo . where

Rl - (r2 + dt

11.17 (pd/1211'E(d -

bfl, where d -

o2/b,

-

2rd eos6)1/2.

blJ where d -

attractive

at/h. attractive

11.21200 sin (2wx/aj

Cha

sinh(2'lTyl o)/sinh(21!'bl c] 11.23 (400Iw)

Cha

-1{sin1nWX/a] sinhtney/c] L _n sinhln-rh/u]

sin(nwylbj ~inh(nWXlb)} sinhf ns-u/b]

+ __;:,.__:.._,....:.--.:,....-_....:..

fl-udd

11.25 (a) Vo

(~ _~)

(1. _ 1.) (b) r

b

Vo (~ _ ~) ri

Ch, Chapter 12

12.1 C = A(Fl + Ez)/2d,C (hulf)

=

A(u1 + (2)f2d; C in parallel with G 12.3 (a) lln [o/p)1 1 [b/o) In (e/b)] (b) I ln (a/b)/(2'lfoli) + I In (blp)/(2'lfu2iJ (e) I 211'f °1 U2

lin

523

Answers to Odd-Numbered Problems 12.50.92

x 10 ~ mho/m

12.9 ( --

112

x 100% 12.11 12.3 n-m

~ 11

+

111

= 600 f.!

12.15 p

_

l6

u

where d Chapter 13

12.7 10{jV

.J2 IJ( 7I"b)

12.13 Sketch omitted

50

d[(y + 6 - foyld)2 + (70 - 70ro/d)2]112

= ll7012 + i]1/2. To = (50f/f(70f

I'

+

50 ) d[(y + 6f + (70)211125'

lll/2

en.

13.3 H - (- ilJy for I y I < Q, H

= (-x)J(d/2) for y > 2 2 H _ ](d/2) for Y < (-d/2J 13.5 (a) 0 (b) 1(/ - o2)/[2'Jlp(h2 - 0 )] (e) Ii21rp 2 13.7 (a) (-z)Io(dy)/l411'(a + y)3rl) (b) (-i)I/(4?!'o) (e) 0 (d) ( z)I/(4?!'0)

13.1 Z 2

x

13.9 zo

! 0.27u

=

13.11 HAB

H

= z.£!_ r

dx 411' J -a [02 + (h _

_.

I(b - 0)

BU -

IT

47r

Z

z

_ DA

a

/(b + u) 47r

-

X)2}3/2

fa J n II +

r

J

u -0

_

Ii +

dx [b

0)2]3/2

dx (b + 0)2t/2

m

13.13 2.8 MI-l7. 13.15 (a) x - 0.04 m, Z = - 0.00725 (b) - 20.5° 13.171.33 x 10- 3 N/m (repulsive) 13.19 (a) Loop should bs placed horizontally or vertically in the east-west direction (b) 7.85 x 10-3 N-m (e) VerticaJly in north-south direction

_..1L

13.21 (a) H

(b) UH

27r02

p

[1

I6'1r

Joule/meter

fJ.o

13.232.74 x 102

4

c in (db) 13.25 - + In (blu)·~ 2 22 2?!' 4 (c - b ) •

8

(e) 5 x 10-8 HIm

~ ~

c -

-2--2

c - b

2

+

+ b

(;2

2

4(c -

2

J

b)

Chapter 14

14.1 Sketch 14.30.21 weber/rn', 100 Aim 14.50.71 weber/rn! 14.7 X3Y, - 1. X~Y2 - O. x,y, - I, X~Y4 = O. XIY2 - 1. X2Y2 = 1. X.Y2 - 1 14.94.233A 14.11 8.32A 14.13 0.29 weber/m'

Chapter 15

15.1 EIO) _ 0, Hlol HIZI _

Y Iow2tp.

15.3 uW

- ~

(~)

__

y 10 cos[wtJ/w:

7! cosfwt)/w: Ei31 -

W2E2V~

(~)

Ell) ~

-x IowlJ.Z sin(wtJlw. HIli

X /rI"lllE

0; EIZI

-

H) (~)t sin(wt)/w: H1

sin2(wtl(~) uwR~. U~I - 0

3

1 -

=

O.

0

15.5 UWI - O.

!l [2

UI~I - - -; cos2(wt)(waRl 2w 15.7 E~ol = Vocos(wt)/[pIn (b/a»), 1(01 = 2;rVu'fPcos(wt)/ln (b/o), Qlol_ 2?1"Vo& cos(wl)/ln [b/o), 1111 - -2;rVuEPwsin(wl)/ln (b/o] 15.911.1 sec Chapter 16

16.1101 A 16.3 VI ~ 0.25 V. v, = -0.15 V. Vs = -0.25 V 16.5 (8)0.125 rnA (clockwise) (b) VI = - 0.25 V; V2 = 0.625 V 16.758.9 mV 16.9(8) -0.l2V (bl -0.012V 16.110.395 x 10-JN 16.13(a)6A (b)5.91A 16.15 (a) 20 A (b) 330 cos (1201l't) (el 19.78 A (d) 1.08 W 16.170.16 mW 16.19 Sketch omitted 16.21 Sketch omitted 16.23 It becomes a generator.

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