tarea 2

charged particles on an x axis: !q "!3.20 # 10!19 C at x " !3.00 m andq " 3.20 # 10!19 C at x " $3.00 m.What are the (a) magnitude and (b) direction (relative to the positivedirection of the x axis) of the net electric field produced at pointP at y " 4.00 m?

653PROBLEMS

2 In Fig. 22-34 the electric fieldlines on the left have twice the sepa-ration of those on the right. (a) If themagnitude of the field at A is 40 N/C,what is the magnitude of the forceon a proton at A?(b) What is themagnitude of the field at B?

Module 22-2 The Electric Field Due to a Charged Particle3 The nucleus of a plutonium-239 atom contains 94 protons.Assume that the nucleus is a sphere with radius 6.64 fm and withthe charge of the protons uniformly spread through the sphere. Atthe surface of the nucleus, what are the (a) magnitude and (b) direc-tion (radially inward or outward) of the electric field produced bythe protons?

4 Two charged particles are attached to an x axis: Particle 1 ofcharge !2.00 # 10!7 C is at position x " 6.00 cm and particle 2 ofcharge$2.00# 10!7 C is at position x " 21.0 cm. Midway betweenthe particles, what is their net electric field in unit-vector notation?

5 A charged particle produces an electric field with a mag-nitude of 2.0 N/C at a point that is 50 cm away from the particle.What is the magnitude of the particles charge?

6 What is the magnitude of a point charge that would create anelectric field of 1.00 N/C at points 1.00m away?

7 In Fig. 22-35, thefour particles form a square of edgelength a " 5.00 cm and have chargesq1 " $10.0 nC, q2 " !20.0 nC, q3 "$20.0 nC, and q4 " !10.0 nC. In unit-vector notation, what net electric fielddo the particles produce at the squarescenter?

8 In Fig. 22-36, the four parti-cles are fixed in place and have chargesq1 " q2 "$5e, q3 "$3e, and q4 "!12e. Distance d" 5.0 mm. What is themagnitude of the net electric field atpoint P due to the particles?

9 Figure 22-37 shows two

WWWILWSSM

SSM

SSM

AB

Figure 22-34 Problem 2.


a

a

q4 q3

q1 q2

x

y

11 Two charged particles are fixed to an x axis: Particle 1SSM

10 Figure 22-38a shows two charged particles fixed in placeon an x axis with separation L. The ratio q1/q2 of their charge mag-nitudes is 4.00. Figure 22-38b shows the x component Enet,x of theirnet electric field along the x axis just to the right of particle 2.The xaxis scale is set by xs " 30.0 cm. (a) At what value of x % 0 is Enet,xmaximum? (b) If particle 2 has charge !q2 "!3e, what is thevalue of that maximum?


x

y

P

q q


of charge q1 " 2.1# 10!8 C is at position x" 20 cm and particle 2 ofcharge q2 "!4.00q1 is at position x" 70 cm.At what coordinate onthe axis (other than at infinity) is the net electric field produced bythe two particles equal to zero?

E net

,x (

108

N/C

)

2

4

0

2

(b)(a)

0

x (cm)

y

x+q1 q2

L

xs


q2

d

Pd

d

d

q3

q4

q1

p p

e

pe

y

x1

2 3

4


es esec

p

R R

z


y

Lx

q1 q2


moved to z " R/10.0. What then are the magnitudes of(c) at the protons location? (e) From (a) and (c)we see that as the proton gets nearer to the disk, the magnitude of

Ec:

and (d) E:s,net

increases, as expected. Why does themagnitude of from the two sideelectrons decrease, as we see from(b) and (d)?

14 In Fig. 22-41, particle 1 of chargeq1"!5.00q and particle 2 of charge q2"$2.00q are fixed to an x axis. (a) As amultiple of distance L, at what coordi-

E:s,net

E:c

nate on the axis is the net electric field of the particles zero? (b)Sketch the net electric field lines between and around the particles.

12 Figure 22-39 shows an un-even arrangement of electrons (e)and protons (p) on a circular arc ofradius r " 2.00 cm, with angles u1 " 30.0&, u2 " 50.0&, u3 " 30.0&, andu4 " 20.0&. What are the (a) magni-tude and (b) direction (relative to thepositive direction of the x axis) of thenet electric field produced at the cen-ter of the arc?

13 Figure 22-40 shows a proton(p) on the central axis through a diskwith a uniform charge density due toexcess electrons.The disk is seen froman edge-on view. Three of those elec-trons are shown: electron ec at thedisk center and electrons es at oppo-site sides of the disk, at radius R fromthe center. The proton is initially atdistance z " R " 2.00 cm from the disk. At that location, what arethe magnitudes of (a) the electric field due to electron ecand (b)the net electric field due to electrons es? The proton is thenE

:s,net

E:c

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654 CHAPTER 22 ELECTRIC FIELDS


E x (

104

N/C

)

Exs

0

90 180

(b)

(a)

E y (

104

N/C

)

Eys

0 90 180

(c)

y

xR Bead 1

Ring

Exs

rP

d/2

+q

d/2

q

+ y

x


19 Figure 22-45 shows an electric dipole.What are the (a) magni-tude and (b) direction (relative to the positive direction of the x axis)of the dipoles electric field at point P, located at distance r d?!

15 In Fig. 22-42, the three particles arefixed in place and have charges q1 " q2 "#e and q3 "#2e. Distance a " 6.00 mm.What are the (a) magnitude and (b) direc-tion of the net electric field at point P due tothe particles?

16 Figure 22-43 shows a plastic ring ofradius R " 50.0 cm. Two small chargedbeads are on the ring: Bead 1 of charge#2.00 mC is fixed in place at the left side;bead 2 of charge #6.00 mC can bemoved along the ring. The twobeads produce a net electric field ofmagnitude E at the center of thering. At what (a) positive and (b)negative value of angle u shouldbead 2 be positioned such that E "2.00 $ 105 N/C?

17 Two charged beads are onthe plastic ring in Fig. 22-44a. Bead2, which is not shown, is fixed inplace on the ring, which has radiusR " 60.0 cm. Bead 1, which is not fixed in place, is initially on the xaxis at angle u " 0%. It is then moved to the opposite side, at angleu " 180%, through the first and second quadrants of the xycoordinate system. Figure 22-44b gives the x component of the netelectric field produced at the origin by the two beads as a functionof u, and Fig. 22-44c gives the y component of that net electric field.The vertical axis scales are set by Exs " 5.0 $ 104 N/C and Eys "&9.0 $ 104 N/C. (a) At what angle u is bead 2 located? What arethe charges of (b) bead 1 and (c) bead 2?

20 Equations 22-8 and 22-9 are approximations of the magnitudeof the electric field of an electric dipole, at points along the dipoleaxis. Consider a point P on that axis at distance z" 5.00d from the di-pole center (d is the separation distance between the particles of thedipole). Let Eappr be the magnitude of the field at point P as approxi-mated by Eqs. 22-8 and 22-9. Let Eact be the actual magnitude.What isthe ratio Eappr/Eact?

21 Electric quadrupole. Fig-ure 22-46 shows a generic electricquadrupole. It consists of two dipoleswith dipole moments that are equal inmagnitude but opposite in direction.Show that the value of E on the axisof the quadrupole for a point P a dis-tance z from its center (assume zd) is given by

in which Q (" 2qd 2) is known as the quadrupole moment of thecharge distribution.

Module 22-4 The Electric Field Due to a Line of Charge22 Density, density, density. (a) A charge &300e is uniformly dis-tributed along a circular arc of radius 4.00 cm, which subtends anangle of 40%. What is the linear charge density along the arc? (b) Acharge &300e is uniformly distributed over one face of a circulardisk of radius 2.00 cm. What is the surface charge density over thatface? (c) A charge &300e is uniformly distributed over the surfaceof a sphere of radius 2.00 cm. What is the surface charge densityover that surface? (d) A charge &300e is uniformly spread throughthe volume of a sphere of radius 2.00 cm. What is the volumecharge density in that sphere?

23 Figure 22-47 shows two paral-lel nonconducting rings with theircentral axes along a common line.Ring 1 has uniform charge q1 and ra-dius R; ring 2 has uniform charge q2and the same radius R. The rings areseparated by distance d" 3.00R.Thenet electric field at point P on thecommon line, at distance R from ring1, is zero.What is the ratio q1/q2?

24 A thin nonconducting rod with a uniform distribution ofpositive charge Q is bent into a complete circle of radius R

E "3Q

4p0z4,

!

SSM

a

a P

2

3

1

y

x

Figure 22-42Problem 15.


Bead 1

Bead 2

y

R

x

Ring

Module 22-3 The Electric Field Due to a Dipole18 The electric field of an electric dipole along the dipole axis isapproximated by Eqs. 22-8 and 22-9. If a binomial expansion ismade of Eq. 22-7, what is the next term in the expression for the di-poles electric field along the dipole axis? That is, what is Enext inthe expression

E "1

2p0

qdz3

# Enext?

z

qq +q+q

d

P++

d

+pp


Ring 1 Ring 2

P

q1 q2

R

R

d

R


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655PROBLEMS

rod and the component perpendicular to the rod.)

Module 22-5 The Electric Field Due to a Charged Disk34 A disk of radius 2.5 cm has a surface charge density of5.3 mC/m2 on its upper face. What is the magnitude of the electricfield produced by the disk at a point on its central axis at distance z!12 cm from the disk?

35 At what distance along the central perpendicu-lar axis of a uniformly charged plastic disk of radius 0.600 m is themagnitude of the electric field equal to one-half the magnitude ofthe field at the center of the surface of the disk?

36 A circular plastic disk with radius R! 2.00 cm has a uni-formly distributed charge Q!"(2.00 # 106)e on one face. A cir-cular ring of width 30 mm is centered on that face, with the centerof that width at radius r! 0.50 cm. In coulombs, what charge iscontained within the width of the ring?

WWWSSM

(Fig. 22-48). The central perpendicu-lar axis through the ring is a z axis,with the origin at the center of thering. What is the magnitude of theelectric field due to the rod at (a) z !0 and (b) z ! $? (c) In terms of R, atwhat positive value of z is that mag-nitude maximum? (d) If R! 2.00 cmand Q! 4.00 mC, what is the maxi-mum magnitude?

25 Figure 22-49 shows three circu-lar arcs centered on the origin of a co-ordinate system. On each arc, the uni-formly distributed charge is given interms of Q! 2.00mC. The radiiare given in terms of R! 10.0 cm.What are the (a) magnitude and (b) di-rection (relative to the positive x direc-tion) of the net electric field at the ori-

30 Figure 22-53 shows twoconcentric rings, of radii R andR% ! 3.00R, that lie on the sameplane. Point P lies on the central zaxis, at distance D! 2.00R from thecenter of the rings. The smaller ringhas uniformly distributed charge"Q. In terms of Q, what is the uni-formly distributed charge on thelarger ring if the net electric field atP is zero?

31 In Fig. 22-54,WWWILWSSM


+9Q4Q

+Q

3R

2R

R

y

x

Pr

+q

q

y

x



z

R

rods, one of charge "q and the other ofcharge &q, form a circle of radius R !8.50 cm in an xy plane. The x axis passesthrough both of the connecting points,and the charge is distributed uniformly onboth rods. If q ! 15.0 pC, what are the(a) magnitude and (b) direction (relativeto the positive direction of the x axis) ofthe electric field produced at P, theE

: Figure 22-51Problem 27.

P x

y

+q

q

center of the circle?

28 Charge is uniformly distributed around a ring of radiusR ! 2.40 cm, and the resulting electric field magnitude E ismeasured along the rings central axis (perpendicular to theplane of the ring). At what distance from the rings center isE maximum?


P P+Q

+Q

R

(a) (b)

R


z

D

R'

P

R


x Pq

L a

a nonconducting rod of length L!8.15 cm has a charge &q!&4.23 fCuniformly distributed along its length.(a) What is the linear charge densityof the rod? What are the (b) magni-tude and (c) direction (relative tothe positive direction of the x axis)of the electric field produced atpoint P, at distance a ! 12.0 cm from the rod? What is the electricfield magnitude produced at distance a !50 m by (d) the rod and(e) a particle of charge &q!&4.23 fC that we use to replace the

+ + + + + ++++

L

R

Py

x


rod? (At that distance, the rodlooks like a particle.)

+ + + +

R

P


32 In Fig. 22-55, positivecharge q ! 7.81 pC is spread uni-formly along a thin nonconductingrod of length L ! 14.5 cm. What arethe (a) magnitude and (b) direction(relative to the positive direction ofthe x axis) of the electric field pro-duced at point P, at distance R !6.00 cm from the rod along its per-pendicular bisector?

33 In Fig. 22-56, a semi-infinite nonconducting rod (that is,infinite in one direction only) hasuniform linear charge density l.Show that the electric field at pointE

:p

P makes an angle of 45' with the rodand that this result is independent ofthe distance R. (Hint: Separately findthe component of parallel to theE

:p

gin due to the arcs?

26 In Fig. 22-50, a thin glass rodILWforms a semicircle of radius r! 5.00 cm.Charge is uniformly distributed along the rod,with "q! 4.50 pC in the upper half and &q!&4.50 pC in the lower half.What are the (a)magnitude and (b) direction (relative to thepositive direction of the x axis) of the electricfield at P, the center of the semicircle?

27 In Fig. 22-51, two curved plastic

E:

29 Figure 22-52a shows a nonconducting rod with a uniformlydistributed charge Q.The rod forms a half-circle with radius R andproduces an electric field of magnitude Earc at its center of curvatureP. If the arc is collapsed to a point at distance R from P (Fig. 22-52b),by what factor is the magnitude of the electric field at P multiplied?

"

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tric field of magnitude 1.00 ! 103 N/C, traveling along a field linein the direction that retards its motion. (a) How far will the elec-tron travel in the field before stopping momentarily, and (b) howmuch time will have elapsed? (c) If the region containing the elec-tric field is 8.00 mm long (too short for the electron to stop withinit), what fraction of the electrons initial kinetic energy will be lostin that region?

41 A charged cloud system produces an electric field in theSSM

656 CHAPTER 22 ELECTRIC FIELDS

an electric field. Determine the field (a) magnitude and (b) di-rection.

47 Beams of high-speed protons can be produced inguns using electric fields to accelerate the protons. (a) Whatacceleration would a proton experience if the guns electric fieldwere 2.00 ! 104 N/C? (b) What speedwould the proton attain if the field ac-celerated the proton through a distanceof 1.00 cm?

48 In Fig. 22-59, an electron (e) is tobe released from rest on the central axisof a uniformly charged disk of radius R.The surface charge density on the disk is"4.00 mC/m2. What is the magnitude ofthe electrons initial acceleration if it isreleased at a distance (a) R, (b) R/100,and (c) R/1000 from the center of the disk? (d) Why does the ac-celeration magnitude increase only slightly as the release point ismoved closer to the disk?

49 A 10.0 g block with a charge of "8.00 ! 10#5 C is placed inan electric field What are the (a) magni-tude and (b) direction (relative to the positive direction of the xaxis) of the electrostatic force on the block? If the block is releasedfrom rest at the origin at time t$ 0, what are its (c) x and (d) y co-ordinates at t$ 3.00 s?

50 At some instant the velocity components of an electronmoving between two charged parallel plates are vx $ 1.5! 105 m/sand vy $ 3.0 ! 103 m/s. Suppose the electric field between theplates is uniform and given by . In unit-vector nota-tion, what are (a) the electrons acceleration in that field and (b) theelectrons velocity when its x coordinate has changed by 2.0 cm?

51 Assume that a honeybee is a sphere of diameter 1.000cm with a charge of 45.0 pC uniformly spread over its surface.Assume also that a spherical pollen grain of diameter 40.0 mm iselectrically held on the surface of the bee because the bees chargeinduces a charge of #1.00 pC on the near side of the grain and acharge of "1.00 pC on the far side. (a) What is the magnitude ofthe net electrostatic force on the grain due to the bee? Next, as-sume that the bee brings the grain to a distance of 1.000 mm fromthe tip of a flowers stigma and that the tip is a particle of charge#45.0 pC. (b) What is the magnitude of the net electrostatic forceon the grain due to the stigma? (c) Does the grain remain on thebee or does it move to the stigma?

52 An electron enters a region of uniform electric field with aninitial velocity of 40 km/s in the same direction as the electric field,which has magnitude E $ 50 N/C. (a) What is the speed of theelectron 1.5 ns after entering this region? (b) How far does theelectron travel during the 1.5 ns interval?

"

E:$ (120 N/C)j

E:$ (3000i # 600j ) N/C.

SSM

Module 22-6 A Point Charge in an Electric Field39 In Millikans experiment, an oil drop of radius 1.64 mm anddensity 0.851 g/cm3 is suspended in chamber C (Fig. 22-16) when adownward electric field of 1.92 ! 105 N/C is applied. Find thecharge on the drop, in terms of e.

40 An electron with a speed of 5.00 ! 108 cm/s enters an elec-

37 Suppose you design an appa-ratus in which a uniformly chargeddisk of radius R is to produce anelectric field. The field magnitude ismost important along the centralperpendicular axis of the disk, at apoint P at distance 2.00R from thedisk (Fig. 22-57a). Cost analysis sug-gests that you switch to a ring of thesame outer radius R but with innerradius R /2.00 (Fig. 22-57b). Assumethat the ring will have the same sur-face charge density as the original disk. If you switch to the ring, bywhat percentage will you decrease the electric field magnitude at P?

38 Figure 22-58a shows a circular disk that is uniformlycharged. The central z axis is perpendicular to the disk face, withthe origin at the disk. Figure 22-58b gives the magnitude of theelectric field along that axis in terms of the maximum magnitudeEm at the disk surface. The z axis scale is set by zs$ 8.0 cm.What isthe radius of the disk?

44 An alpha particle (the nucleus of a helium atom) has a massof 6.64 ! 10#27 kg and a charge of "2e.What are the (a) magnitudeand (b) direction of the electric field that will balance the gravita-tional force on the particle?

45 An electron on the axis of an electric dipole is 25 nm fromthe center of the dipole. What is the magnitude of the electrostaticforce on the electron if the dipole moment is 3.6 ! 10#29 C %m?Assume that 25 nm is much larger than the separation of the chargedparticles that form the dipole.

46 An electron is accelerated eastward at 1.80 ! 109 m/s2 by

ILW

P

z

(a)

P

z

(b)


z

(a) (b)

Em

0.5Em

0z (cm)

zs



e

air near Earths surface. A particle of charge #2.0 ! 10#9 C isacted on by a downward electrostatic force of 3.0 ! 10#6 N whenplaced in this field. (a) What is the magnitude of the electric field?What are the (b) magnitude and (c) direction of the electrostaticforce on the proton placed in this field? (d) What is the magni-tude of the gravitational force on the proton? (e) What is the ra-tio Fel /Fg in this case?

42 Humid air breaks down (its molecules become ionized) in anelectric field of 3.0 ! 106 N/C. In that field, what is the magnitudeof the electrostatic force on (a) an electron and (b) an ion with asingle electron missing?

43 An electron is released from rest in a uniform electricSSM

F:g

F:el

field of magnitude 2.00 ! 104 N/C. Calculate the acceleration ofthe electron. (Ignore gravitation.)

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are (c) ! and (d) qenc if

10 Figure 23-34 shows a closed Gaussiansurface in the shape of a cube of edge length2.00 m. It lies in a region where the nonuni-form electric field is given by

with x in meters.What is the net charge contained by the cube?

11 Figure 23-35 shows a closed Gaussian surface in the shape ofa cube of edge length 2.00 m,with one corner at x1 5.00 m,y1 4.00m.The cube lies in a region where the electric field vector is given by

with y in meters.What is the netcharge contained by the cube?E:" #3.00i # 4.00y2j $ 3.00k N/C,

""

7.00k N/C,$6.00j$4.00)i$(3.00x"E

:

(6.00 $ 3.00y)j] N/C?E:" [#4.00i $

with edge length 1.40 m. What are (a) the net flux ! through thesurface and (b) the net charge qenc enclosed by the surface if

with y in meters? WhatE:"(3.00yj) N/C,

679PROBLEMS

and on the bottom face it is Determine the net charge contained within the cube.

7 A particle of charge 1.8 mC is at the center of a Gaussian cube55 cm on edge.What is the net electric flux through the surface?

8 When a shower is turned on in a closed bathroom, thesplashing of the water on the bare tub can fill the rooms air withnegatively charged ions and produce an electric field in the air asgreat as 1000 N/C. Consider a bathroom with dimensions 2.5 m%3.0 m% 2.0 m. Along the ceiling, floor, and four walls, approximatethe electric field in the air as being directed perpendicular to the sur-face and as having a uniform magnitude of 600 N/C.Also, treat thosesurfaces as forming a closed Gaussian surface around the rooms air.What are (a) the volume charge density r and (b) the number ofexcess elementary charges e per cubic meter in the rooms air?

9 Fig. 23-31 shows a Gaussian surface in the shape of a cubeILW

E:" $ 20k N/C.E

:" #34k N/C,Module 23-1 Electric Flux

1 The square surface shownin Fig. 23-30 measures 3.2 mm oneach side. It is immersed in a uni-form electric field with magnitudeE" 1800 N/C and with field lines atan angle of u" 35 with a normal tothe surface, as shown. Take thatnormal to be directed outward, asthough the surface were one face ofa box. Calculate the electric fluxthrough the surface.2 An electric field given by

pierces aGaussian cube of edge length 2.0 mand positioned as shown in Fig. 23-7.(The magnitude E is in newtons percoulomb and the position x is in me-ters.) What is the electric flux throughthe (a) top face, (b) bottom face, (c) leftface, and (d) back face? (e) What is thenet electric flux through the cube?

3 The cube in Fig. 23-31 has edgelength 1.40 m and is oriented as shownin a region of uniform electric field. Findthe electric flux through the right face ifthe electric field, in newtons per coulomb, is given by (a) (b)

and (c) (d) What is the total flux through thecube for each field?

Module 23-2 Gauss Law4 In Fig. 23-32, a butterfly net isin a uniform electric field of magni-tude E " 3.0 mN/C. The rim, a cir-cle of radius a " 11 cm, is alignedperpendicular to the field. The netcontains no net charge. Find theelectric flux through the netting.

5 In Fig. 23-33, a proton is a dis-tance d/2 directly above the center of a square of side d. What is themagnitude of the electric flux through the square? (Hint:Think of thesquare as one face of a cube with edge d.)

#3.00i $ 4.00k.#2.00j,6.00i,

E:" 4.0i # 3.0(y2 $ 2.0)j

SSM

Tutoring problem available (at instructors discretion) in WileyPLUS and WebAssignSSM Worked-out solution available in Student Solutions Manual

Number of dots indicates level of problem difficulty

Additional information available in The Flying Circus of Physics and at flyingcircusofphysics.com

WWW Worked-out solution is at

ILW Interactive solution is at http://www.wiley.com/college/halliday

Problems

Normal


z

y

x

Figure 23-31 Problems 3,6, and 9.

a


d/2

d

d

+


x

y

z



x

y

z

x1

y1

12 Figure 23-36 shows two non-conducting spherical shells fixed inplace. Shell 1 has uniform surfacecharge density $6.0 mC/m2 on itsouter surface and radius 3.0 cm;shell 2 has uniform surface chargedensity $4.0 mC/m2 on its outersurface and radius 2.0 cm; the shellcenters are separated by L" 10 cm.In unit-vector notation, what is thenet electric field at x " 2.0 cm?

Shell1 Shell

2

y

x

L


6 At each point on the surface of the cube shown in Fig. 23-31,the electric field is parallel to the z axis. The length of each edgeof the cube is 3.0 m. On the top face of the cube the field is

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680 CHAPTER 23 GAUSS LAW

y

x

z

z1

z2

z1

z2

x2

y2

x2x1Figure 23-38 Problem 16.

A

B

(

105

N

m2 /

C)

(a) (b)

0

r

s

s

2 s


Es

0

Es

E (1

03 N

/C)

1 2 3

r (cm)

4 5 6

(a)

(b)


13 The electric field in a certain region of Earths atmo-sphere is directed vertically down. At an altitude of 300 m the fieldhas magnitude 60.0 N/C; at an altitude of 200 m, the magnitude is100 N/C. Find the net amount of charge contained in a cube 100 mon edge, with horizontal faces at altitudes of 200 and 300 m.

14 Flux and nonconducting shells. A charged particle is sus-pended at the center of two concentric spherical shells that arevery thin and made of nonconducting material. Figure 23-37ashows a cross section. Figure 23-37b gives the net flux ! through aGaussian sphere centered on the particle, as a function of the ra-dius r of the sphere. The scale of the vertical axis is set by !s"5.0# 105 N ?m2/C. (a) What is the charge of the central particle?What are the net charges of (b) shell A and (c) shell B?

SSM

15 A particle of charge $q is placed at one corner of a Gaussiancube. What multiple of q/0 gives the flux through (a) each cube faceforming that corner and (b) each of the other cube faces?

16 The box-like Gaussian surface shown in Fig. 23-38 en-closes a net charge of 24.00 C and lies in an electric field givenby with x and z in me-ters and b a constant.The bottom face is in the xz plane; the top faceis in the horizontal plane passing through y2" 1.00 m. For x1"1.00 m, x2" 4.00 m, z1" 1.00 m, and z2" 3.00 m, what is b?

E:" [(10.0 $ 2.00x)i % 3.00j $ bzk] N/C,

$

20 Flux and conducting shells. A charged particle is held at thecenter of two concentric conducting spherical shells. Figure 23-39ashows a cross section. Figure 23-39b gives the net flux ! through aGaussian sphere centered on the particle, as a function of the radius rof the sphere. The scale of the vertical axis is set by !s" 5.0#105 N ?m2/C. What are (a) the charge of the central particle and thenet charges of (b) shell A and (c) shell B?

21 An isolated conductor has net charge $10# 10%6 C and a cav-ity with a particle of charge q"$3.0# 10%6 C.What is the charge on(a) the cavity wall and (b) the outer surface?

Module 23-4 Applying Gauss Law: Cylindrical Symmetry22 An electron is released 9.0 cm from a very long nonconduct-ing rod with a uniform 6.0 mC/m. What is the magnitude of theelectrons initial acceleration?

23 (a) The drum of a photocopying machine has a length of 42 cmand a diameter of 12 cm.The electric field just above the drums sur-face is 2.3 # 105 N/C. What is the total charge on the drum? (b) Themanufacturer wishes to produce a desktop version of the machine.This requires reducing the drum length to 28 cm and the diameter to8.0 cm. The electric field at the drum surface must not change. Whatmust be the charge on this new drum?


s

s

0

(1

05 N

m

2 /C

)

r

AB

(a) (b)

26 Figure 23-41a shows a narrow charged solid cylinder that iscoaxial with a larger charged cylindrical shell. Both are noncon-

Module 23-3 A Charged Isolated Conductor17 A uniformly charged conducting sphere of 1.2 m diam-eter has surface charge density 8.1 mC/m2. Find (a) the net chargeon the sphere and (b) the total electric flux leaving the surface.

18 The electric field just above the surface of the charged con-ducting drum of a photocopying machine has a magnitude E of2.3 # 10 5 N/C. What is the surface charge density on the drum?

19 Space vehicles traveling through Earths radiation belts canintercept a significant number of electrons. The resulting chargebuildup can damage electronic components and disrupt operations.Suppose a spherical metal satellite 1.3 m in diameter accumulates2.4 mC of charge in one orbital revolution. (a) Find the resulting sur-face charge density. (b) Calculate the magnitude of the electric fieldjust outside the surface of the satellite, due to the surface charge.

SSM

R

++

++

++

++

++

++

++

+

++

++ ++ ++

+


24 Figure 23-40 shows a section of along, thin-walled metal tube of radiusR" 3.00 cm, with a charge per unitlength of l" 2.00# 10%8 C/m. Whatis the magnitude E of the electric fieldat radial distance (a) r"R/2.00 and(b) r" 2.00R? (c) Graph E versus rfor the range r" 0 to 2.00R.

25 An infinite line of chargeSSMproduces a field of magnitude 4.5#104 N/C at distance 2.0 m. Find thelinear charge density.

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681PROBLEMS

35 Figure 23-46a shows three plastic sheets that are large, paral-lel, and uniformly charged. Figure 23-46b gives the component of thenet electric field along an x axis through the sheets. The scale of thevertical axis is set by Es!6.0"105 N/C.What is the ratio of the chargedensity on sheet 3 to that on sheet 2?

ducting and thin and have uniform surface charge densities ontheir outer surfaces. Figure 23-41b gives the radial component E ofthe electric field versus radial distance r from the common axis,and Es! 3.0"103 N/C.What is the shells linear charge density?

27 A long, straight wire has fixed negative charge with a lin-ear charge density of magnitude 3.6 nC/m. The wire is to be en-closed by a coaxial, thin-walled nonconducting cylindrical shell ofradius 1.5 cm.The shell is to have positive charge on its outside sur-face with a surface charge density s that makes the net externalelectric field zero. Calculate s.

28 A charge of uniform linear density 2.0 nC/m is distributedalong a long, thin, nonconducting rod.The rod is coaxial with a longconducting cylindrical shell (inner radius! 5.0 cm, outer radius!10 cm). The net charge on the shell is zero. (a) What is the magni-tude of the electric field 15 cm from the axis of the shell? What isthe surface charge density on the (b) inner and (c) outer surface ofthe shell?

29 Figure 23-42 isWWWSSM

the plates have excess surface charge densities of opposite signs andmagnitude 7.00" 10#22 C/m2. In unit-vector notation, what is theelectric field at points (a) to the left of the plates, (b) to the right ofthem, and (c) between them?

34 In Fig. 23-45, a small circular hole of radius R ! 1.80 cm hasbeen cut in the middle of an infinite, flat, nonconducting surfacethat has uniform charge density s! 4.50 pC/m2. A z axis, with itsorigin at the holes center, is perpendicular to the surface. In unit-vector notation, what is the electric field at point P at z ! 2.56 cm?(Hint: See Eq. 22-26 and use superposition.)

36 Figure 23-47 shows cross sec-tions through two large, parallel, non-conducting sheets with identical distri-butions of positive charge with surfacecharge density s! 1.77" 10#22 C/m2.In unit-vector notation, what is atpoints (a) above the sheets, (b) be-tween them, and (c) below them?

37 A square metal plate of edge length 8.0 cm andWWWSSM

E:

a section of a conducting rod of ra-dius R1 ! 1.30 mm and length L !11.00 m inside a thin-walled coax-ial conducting cylindrical shell ofradius R2! 10.0R1 and the (same)length L.The net charge on the rodis Q1 !$3.40 " 10#12 C; that onthe shell is Q2 !#2.00Q1. Whatare the (a) magnitude E and (b) di-rection (radially inward or out-ward) of the electric field at radial

R1R2

Q1

Q 2


distance r ! 2.00R2? What are (c) E and (d) the direction at r !5.00R1? What is the charge on the (e) interior and (f) exterior sur-face of the shell?

30 In Fig. 23-43, short sections oftwo very long parallel lines ofcharge are shown, fixed in place,separated by L! 8.0 cm. The uni-form linear charge densities are$6.0 mC/m for line 1 and #2.0mC/m for line 2. Where along the xaxis shown is the net electric fieldfrom the two lines zero?

31 Two long, charged,thin-walled, concentric cylindrical shells have radii of 3.0 and6.0 cm. The charge per unit length is 5.0 " 10#6 C/m on the innershell and #7.0 " 10#6 C/m on the outer shell. What are the (a)magnitude E and (b) direction (radially inward or outward) of theelectric field at radial distance r ! 4.0 cm? What are (c) E and(d) the direction at r! 8.0 cm?

32 A long, nonconducting, solid cylinder of radius 4.0 cm has anonuniform volume charge density r that is a function of radial dis-tance r from the cylinder axis: r !Ar 2. For A ! 2.5 mC/m5, what is themagnitude of the electric field at(a) r! 3.0 cm and (b) r! 5.0 cm?

Module 23-5 Applying GaussLaw: Planar Symmetry33 In Fig. 23-44, two large, thinmetal plates are parallel and closeto each other. On their inner faces,

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Line 1 Line 2

x

y

L/2 L/2

+++++++++

x


+ + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + Pz

R


(a)

(b)x

x

Es

0

E (1

05 N

/C)

1 2 3


+++++++++++

+++++++++++

y

x


negligible thickness has a total charge of 6.0" 10#6 C. (a) Estimatethe magnitude E of the electric field just off the center of the plate (at,say, a distance of 0.50 mm from the center) by assuming that thecharge is spread uniformly over the two faces of the plate. (b)EstimateE at a distance of 30 m (large relative to the plate size) by as-suming that the plate is a charged particle.

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E (1

07 N

/C)

Es

0 1 2 r (cm)

3 4 5


a b

c


38 In Fig. 23-48a, an electron is shot directly away from a uni- Module 23-6 Applying Gauss Law: Spherical Symmetry44 Figure 23-52 gives the magni-tude of the electric field inside andoutside a sphere with a positive chargedistributed uniformly throughout itsvolume.The scale of the vertical axis isset by Es!5.0"107 N/C. What is thecharge on the sphere?45 Two charged concentric spher-ical shells have radii 10.0 cm and 15.0 cm. The charge on the innershell is 4.00 " 10#8 C, and that on the outer shell is 2.00" 10#8 C.Find the electric field (a) at r ! 12.0 cm and (b) at r! 20.0 cm.46 Assume that a ball of charged particles has a uniformlydistributed negative charge density except for a narrow radialtunnel through its center, from the surface on one side to thesurface on the opposite side. Also assume that we can position aproton anywhere along the tunnel or outside the ball. Let FR bethe magnitude of the electrostatic force on the proton when it islocated at the balls surface, at radius R. As a multiple of R, howfar from the surface is there a point where the force magnitude is0.50FR if we move the proton (a) away from the ball and (b) intothe tunnel?47 An unknown charge sits on a conducting solid sphere ofradius 10 cm. If the electric field 15 cm from the center of thesphere has the magnitude 3.0 " 103 N/C and is directed radially in-ward, what is the net charge on the sphere?48 A charged particle is held at the center of a sphericalshell. Figure 23-53 gives the magnitude E of the electric field ver-sus radial distance r. The scale of the vertical axis is set by Es!10.0 " 107 N/C. Approximately, what is the net charge on theshell?

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Both are fixed in place. If d! 0.200 m,at what (a) positive and (b) negativecoordinate on the x axis (other than in-finity) is the net electric field ofE

:net

3 6

t (ps)

9

12

vs

0

vs

v (1

05 m

/s)

+ + + + + + + + e

(a) (b)

49 In Fig. 23-54, a solid sphere ofradius a! 2.00 cm is concentric with aspherical conducting shell of inner ra-dius b! 2.00a and outer radius c!2.40a. The sphere has a net uniformcharge q1!$5.00 fC; the shell has anet charge q2!#q1.What is the mag-nitude of the electric field at radialdistances (a) r! 0, (b) r! a/2.00, (c) r! a, (d) r! 1.50a, (e) r ! 2.30a, and(f) r! 3.50a? What is the net chargeon the (g) inner and (h) outer surfaceof the shell?


formly charged plastic sheet, at speed vs! 2.0" 105 m/s.The sheet isnonconducting, flat, and very large. Figure 23-48b gives the electronsvertical velocity component v versus time t until the return to thelaunch point.What is the sheets surface charge density?

39 In Fig. 23-49, a small, nonconductingSSM +++++++++++++ m, q


ball of mass m! 1.0 mg and charge q ! 2.0 "10#8 C (distributed uniformly through its vol-ume) hangs from an insulating thread that makesan angle u ! 30 with a vertical, uniformlycharged nonconducting sheet (shown in cross sec-tion). Considering the gravitational force on theball and assuming the sheet extends far verticallyand into and out of the page, calculate the surfacecharge density s of the sheet.

40 Figure 23-50 shows a very large nonconduct-ing sheet that has a uniform surface charge densityof s!#2.00 mC/m2; it also shows a particle ofcharge Q! 6.00 mC, at distance d from the sheet.

y

xQ

d


the sheet and particle zero? (c) If d!0.800 m, at what coordinate on the xaxis is

41 An electron is shot directlytoward the center of a large metal

E:net ! 0?

plate that has surface charge density #2.0" 10#6 C/m2. If the initialkinetic energy of the electron is 1.60" 10#17 J and if the electron is tostop (due to electrostatic repulsion from the plate) just as it reachesthe plate,how far from the plate must the launch point be?

42 Two large metal plates of area 1.0 m2 face each other, 5.0cm apart, with equal charge magnitudes but opposite signs.The field magnitude E between them (neglect fringing) is 55 N/C.Find .

43 Figure 23-51 shows a cross sec-tion through a very large nonconductingslab of thickness d ! 9.40 mm and uni-form volume charge density r ! 5.80fC/m3. The origin of an x axis is at theslabs center. What is the magnitude ofthe slabs electric field at an x coordi-nate of (a) 0, (b) 2.00 mm, (c) 4.70 mm,and (d) 26.0 mm?

!q !

!q !

d/2

d

x0


E (1

07 N

/C) Es

20r (cm)

4


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683PROBLEMS

50 Figure 23-55 shows two non-conducting spherical shells fixed inplace on an x axis. Shell 1 has uniformsurface charge density !4.0 mC/m2

on its outer surface and radius 0.50cm, and shell 2 has uniform surfacecharge density "2.0 mC/m2 on itsouter surface and radius 2.0 cm; thecenters are separated by L# 6.0 cm.

57 A thin-walled metal spherical shell has radius 25.0 cm andcharge 2.00$ 10"7 C. Find E for a point (a) inside the shell, (b)just outside it, and (c) 3.00 m from the center.

58 A uniform surface charge of density 8.0 nC/m2 is distributed overthe entire xy plane. What is the electric flux through a sphericalGaussian surface centered on the origin and having a radius of 5.0 cm?

59 Charge of uniform volume density r # 1.2 nC/m3 fills an infi-nite slab between x #"5.0 cm and x #!5.0 cm. What is the mag-nitude of the electric field at any point with the coordinate (a) x #4.0 cm and (b) x # 6.0 cm?

60 The chocolate crumb mystery. Explosions ignited byelectrostatic discharges (sparks) constitute a serious danger in fa-cilities handling grain or powder. Such an explosion occurred inchocolate crumb powder at a biscuit factory in the 1970s. Workersusually emptied newly delivered sacks of the powder into a loadingbin, from which it was blown through electrically grounded plasticpipes to a silo for storage. Somewhere along this route, two condi-tions for an explosion were met: (1) The magnitude of an electricfield became 3.0 $ 106 N/C or greater, so that electrical break-down and thus sparking could occur. (2) The energy of a spark was150 mJ or greater so that it could ignite the powder explosively. Letus check for the first condition in the powder flow through theplastic pipes.

Suppose a stream of negatively charged powder was blownthrough a cylindrical pipe of radius R # 5.0 cm. Assume that thepowder and its charge were spread uniformly through the pipewith a volume charge density r. (a) Using Gauss law, find an ex-pression for the magnitude of the electric field in the pipe as afunction of radial distance r from the pipe center. (b) Does E in-crease or decrease with increasing r? (c) Is directed radially in-ward or outward? (d) For r # 1.1 $ 10"3 C/m3 (a typical value atthe factory), find the maximum E and determine where that maxi-mum field occurs. (e) Could sparking occur, and if so, where? (Thestory continues with Problem 70 in Chapter 24.)

61 A thin-walled metal spherical shell of radius a has a chargeqa. Concentric with it is a thin-walled metal spherical shell of radiusb% a and charge qb. Find the electric field at points a distance r fromthe common center, where (a) r& a, (b) a& r& b, and (c) r% b.(d) Discuss the criterion you would use to determine how the chargesare distributed on the inner and outer surfaces of the shells.

62 A particle of charge q# 1.0$ 10"7 C is at the center of aspherical cavity of radius 3.0 cm in a chunk of metal. Find the electricfield (a) 1.5 cm from the cavity center and (b) anyplace in the metal.

63 A proton at speed v# 3.00$ 105 m/s orbits at radius r# 1.00 cmoutside a charged sphere. Find the spheres charge.

64 Equation 23-11 (E# s/0) gives the electric field at points near acharged conducting surface. Apply this equation to a conductingsphere of radius r and charge q, and show that the electric field outsidethe sphere is the same as the field of a charged particle located at thecenter of the sphere.

65 Charge Q is uniformly distributed in a sphere of radius R. (a)What fraction of the charge is contained within the radius r# R/2.00? (b) What is the ratio of the electric field magnitude at r# R/2.00 to that on the surface of the sphere?

66 A charged particle causes an electric flux of "750 N ?m2/C topass through a spherical Gaussian surface of 10.0 cm radius cen-tered on the charge. (a) If the radius of the Gaussian surface were

SSM

E:

E:

Other than at x # ', where on the xaxis is the net electric field equal tozero?51 In Fig. 23-56, anonconducting spherical shell of in-ner radius a# 2.00 cm and outer ra-dius b# 2.40 cm has (within its thick-ness) a positive volume chargedensity r#A/r, where A is a constantand r is the distance from the centerof the shell. In addition, a small ball ofcharge q# 45.0 fC is located at thatcenter. What value should A have if

WWWSSM

x

Shell1

Shell2

y

L


ba

q +


a

b

++++++

++++++++

++++

++++

++

++

+++

+++

+++

+++

+++

++

++

++++

++++

++++++++

++++++


of radius R # 5.60 cm varies with radial distance r as given by r #(14.1 pC/m3)r/R. (a) What is the spheres total charge? What is thefield magnitude E at (b) r # 0, (c) r # R/2.00, and (d) r # R? (e)

the electric field in the shell (a( r(b) is to be uniform?

52 Figure 23-57 shows a spheri-cal shell with uniform volume chargedensity r# 1.84 nC/m3, inner radiusa# 10.0 cm, and outer radius b#2.00a. What is the magnitude of theelectric field at radial distances (a) r#0; (b) r# a/2.00, (c) r# a, (d) r#1.50a, (e) r# b,and (f) r# 3.00b?

53 The volume charge den-sity of a solid nonconducting sphere

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R R

P1 2


Graph E versus r.

54 Figure 23-58 shows, in crosssection, two solid spheres with uni-formly distributed charge through-out their volumes. Each has radiusR. Point P lies on a line connectingthe centers of the spheres, at radialdistance R/2.00 from the center of sphere 1. If the net electric fieldat point P is zero, what is the ratio q2/q1 of the total charges?

55 A charge distribution that is spherically symmetric but notuniform radially produces an electric field of magnitude E # Kr 4,directed radially outward from the center of the sphere. Here r isthe radial distance from that center, and K is a constant. What isthe volume density r of the charge distribution?

Additional Problems56 The electric field in a particular space is N/C,with x in meters. Consider a cylindrical Gaussian surface of radius20 cm that is coaxial with the x axis. One end of the cylinder is at x # 0. (a) What is the magnitude of the electric flux through theother end of the cylinder at x # 2.0 m? (b) What net charge is en-closed within the cylinder?

E:# (x ! 2)i

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doubled, how much flux would pass through the surface? (b) Whatis the charge of the particle?

67 The electric field at point P just outside the outer sur-face of a hollow spherical conductor of inner radius 10 cm andouter radius 20 cm has magnitude 450 N/C and is directed outward.When a particle of unknown charge Q is introduced into the centerof the sphere, the electric field at P is still directed outward but isnow 180 N/C. (a) What was the net charge enclosed by the outersurface before Q was introduced? (b) What is charge Q? After Q isintroduced, what is the charge on the (c) inner and (d) outer sur-face of the conductor?

68 The net electric flux through each face of a die (singular ofdice) has a magnitude in units of 103 N ?m2/C that is exactly equalto the number of spots N on the face (1 through 6). The flux is in-ward for N odd and outward for N even. What is the net charge in-side the die?

69 Figure 23-59 shows, incross section, three infinitelylarge nonconducting sheets onwhich charge is uniformlyspread. The surface chargedensities are s1!"2.00mC/m2, s2! "4.00 mC/m2,and s3!#5.00mC/m2, anddistance L! 1.50 cm. In unit-vector notation,what is the netelectric field at point P?

70 Charge of uniform vol-ume density r ! 3.2 mC/m3 fills a nonconducting solid sphere ofradius 5.0 cm. What is the magnitude of the electric field (a) 3.5cm and (b) 8.0 cm from the spheres center?

71 A Gaussian surface in the form of a hemisphere of radius R !5.68 cm lies in a uniform electric field of magnitude E ! 2.50 N/C.The surface encloses no net charge. At the (flat) base of the sur-face, the field is perpendicular to the surface and directed into thesurface. What is the flux through (a) the base and (b) the curvedportion of the surface?

72 What net charge is enclosed by the Gaussian cube ofProblem 2?

73 A nonconducting solid sphere has a uni-form volume charge density r. Let be thevector from the center of the sphere to a gen-eral point P within the sphere. (a) Show thatthe electric field at P is given by (Note that the result is independent of the ra-dius of the sphere.) (b) A spherical cavity ishollowed out of the sphere, as shown in Fig. 23-60. Using superposition concepts, show thatthe electric field at all points within the cavityis uniform and equal to where is the position vectorfrom the center of the sphere to the center of the cavity.

74 A uniform charge density of 500 nC/m3 is distributed through-out a spherical volume of radius 6.00 cm. Consider a cubicalGaussian surface with its center at the center of the sphere. What isthe electric flux through this cubical surface if its edge length is(a) 4.00 cm and (b) 14.0 cm?

75 Figure 23-61 shows a Geiger counter, a device used to detectionizing radiation, which causes ionization of atoms. A thin, posi-

a:E:! ra:/30,

E:! rr:/30.

r:

SSM

tively charged central wire is sur-rounded by a concentric, circular, con-ducting cylindrical shell with an equalnegative charge, creating a strong ra-dial electric field. The shell contains alow-pressure inert gas.A particle of ra-diation entering the device throughthe shell wall ionizes a few of the gasatoms. The resulting free electrons (e)are drawn to the positive wire.However, the electric field is so intensethat, between collisions with gasatoms, the free electrons gain energysufficient to ionize these atoms also.More free electrons are thereby cre-ated, and the process is repeated untilthe electrons reach the wire. The re-sulting avalanche of electrons is col-lected by the wire, generating a signal that is used to record thepassage of the original particle of radiation. Suppose that the ra-dius of the central wire is 25 mm, the inner radius of the shell 1.4cm, and the length of the shell 16 cm. If the electric field at theshells inner wall is 2.9$ 104 N/C, what is the total positive chargeon the central wire?

76 Charge is distributed uniformly throughout the volume of an in-finitely long solid cylinder of radius R. (a) Show that, at a distance r%R from the cylinder axis,

where r is the volume charge density. (b) Write an expression for Ewhen r& R.

77 A spherical conducting shell has a charge of #14 mC onits outer surface and a charged particle in its hollow. If the netcharge on the shell is #10 mC, what is the charge (a) on the innersurface of the shell and (b) of the particle?

78 A charge of 6.00 pC is spread uniformly throughout the volumeof a sphere of radius r! 4.00 cm.What is the magnitude of the electricfield at a radial distance of (a) 6.00 cm and (b) 3.00 cm?

79 Water in an irrigation ditch of width w! 3.22 m and depth d!1.04 m flows with a speed of 0.207 m/s. The mass flux of the flowingwater through an imaginary surface is the product of the watersdensity (1000 kg/m3) and its volume flux through that surface. Findthe mass flux through the following imaginary surfaces: (a) a sur-face of area wd, entirely in the water, perpendicular to the flow;(b) a surface with area 3wd/2, of which wd is in the water, perpendi-cular to the flow; (c) a surface of area wd/2, entirely in the water, per-pendicular to the flow; (d) a surface of area wd, half in the water andhalf out, perpendicular to the flow; (e) a surface of area wd, entirely inthe water,with its normal 34.0 from the direction of flow.

80 Charge of uniform surface density 8.00 nC/m2 is distributedover an entire xy plane; charge of uniform surface density 3.00 nC/m2

is distributed over the parallel plane defined by z! 2.00 m.Determine the magnitude of the electric field at any point having a zcoordinate of (a) 1.00 m and (b) 3.00 m.

81 A spherical ball of charged particles has a uniform chargedensity. In terms of the balls radius R, at what radial distances(a) inside and (b) outside the ball is the magnitude of the ballselectric field equal to of the maximum magnitude of that field?14

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,

2L

L/2

L

P

3

2

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y

x


a


+ +

+ +

+ +

+ +

+ +

+ +

SignalParticle

Chargedcylindrical shell

e

Chargedwire


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