Gauss's Law27,1 Symmetry

1. An infinite plane of charge is seen edge on. The sign of the charge is not given. Do the electric fieldsshown below have the same symmetry as the charge? If not, why not?

b.

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a,

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d.

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up o.^l Ao*^. srlrn-ctry os th. "horg".

2. Suppose you had a uniformly charged cube. Can you use symmetry aloneto deduce the shape ofthe cube's electric field? Ifso, sketch and describethe field shape. If not, why not?

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27-1

27-2 cHAPrER2T . Gauss'slaw

27.2 The Goncept of Flux

3. The figures shown below are cross sections of three-dimensional closed surfaces. They have a flat top

and bottom surface above and below the plane of the page. However, the electric field is everywhereparallel to the page, so there is no flux through the top or bottom surface. The electric field is uniformover each face of the surface. The field strength, in N/C, is shown.

For each, does the surface enclose a net positive charge, a net negative charge, or no net charge?

b.

Qnet =

--io-

o Qnet =

l0

Qn"t = Qnet = + Qn"t= O

4, The figures shown below are cross sections of three-dimensional closed surfaces. They have a flat topand bottom surface above and below the plane ofthe page, but there is no flux through the top or bot-tom surface. The electric field is perpendicular to and uniform over each face of the surface. The fieldstrength, in N/C, is shown.

Each surface contains no net charge. Draw the missing electric field vector (or write E = d) in th"proper direction. Write the field strength beside it.

b.il( YvFE--O

go2

€tr.l

od

Odo

l,'

Gauss'sLaw . cHAPrsx2l 27-3

27.3 Calculating Electric Flux

5. Draw the area vector 1for eachof these surfaces.

6. How many area vectors are needed to "characterize this closed surface? J

Draw them.

The diameter of the circle equals the edge length of the square.

They are in a uniform electric field. Is the electric flux @1

through the square larger than, smaller than, or equal to the

electric flrx O.z throlsh the circle? Explain,

Becq.\^se- A,t Aa, *nl E,= Et

o*l 'r-a

0,= E,A, Ua=EaAa

Is the electri.c flux 01 through the circle larger than, smaller than, or equal

to the electric flux @2 throughthe hemisphere? Explain.

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7.

B.

27-4 csrprpp,2T . Gauss's Law

9. A uniform electric field is shown below.

Draw and label an edge view of three square surfaces, all the same size, for which

a. The flux is:naxirnurn.

b. The flux is minimum.

c. The flux has half the value of the flux through the square parl of a.

Give the tilt angle of any squares not perpendicular to the field lines.

10. Is the net electric flux through each of the closed surfaces below positive (+), negative ( ), or zero (0)?

+J9/

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"iio ttot or,r*.

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Gauss'slaw . cHAPTBI2T 27-5

27.4 Gauss's Law

27.5 Using Gauss's Law

11. For each of the closed cylinders shown below, are the electric fluxes through the top, the wall, and the

bottom positive (+), negative (-), or zero (0)? Is the net flrix positive, negative, or zero?

a.

@top =

@*ull =

@bot =6=^ ner

@top = -

Owall =

@bot =

@o"t =

@top =

@wall =

@bot =

Onet =

o00O

+

9o

d.

+-

Io

@roo = -J.

@*all = |

I' @bot= T

6=o^ net

@top = -

@wall =-obot = '

Onet =

@top =

@*all =

@bot =

@n"t =

o+-

ogD'

;o

otrq

oA

No

l--+ w;*e/----T----\

tittrtl-+l

@top = OI

Owali = |

obot = O

@net = +

@,op =

Owall =

@bot =

Onet =

o+o+

+-

o_

++

f/li

ffi.-'"@top =

O*u1l =

@bot =

Onet =

13. What is the electric flux through each of these surfaces? Give your answers as multiples of qleu,

@. = *1Go

@" = -%"

@"= 0

27-G cHAPrER2T . Gauss'slaw

12. For this closed cylinder, @top = -15 Nm2/C and @6o, = 5 Nm2/C.What is @*urr?

f wntl = to N^%

$*.J Sr.* + {.*n,r=o

14. What is the electric flux through each of these surfaces?Give your answers as multiples of qle1s.

*^ = *{t

l€^" -Hq/ -@B = 'l€o(D-= O

*o = *tn

/.o@E= o

15. A positively charged balloon expands as it is blown up,increasing in size from the initial to final diameters shown.Do the electric fields at points 1,2, and 3 increase, decrease,or stay the same? Explain your reasoning for each.

Point l: S{a1s lho r^-.. A spl.,c,rico.\ Go..rrssiq.n

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"*n"tostd, bv q'Go.$ssio.n

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Gauss'sLaw . cnaptex2T 27-7

I 6. Three charges, all the same charge q, are surrounded by three spheres of equal radii.

a. Rank in order, from largest to smallest, the fluxes (D1 , (D2, and O, through the spheres.

order: E, = Q O= 0.Explanatron:

Thc. flux is equo\ to +!"diviJ.l by eo. 'Eoch Srr't'tqcc'

Tolr",[ ctrq,noc .r."liosol bv *ho surfrcee...tosoS dl" So.ync- q,*o,^1 of .hc5e.

b. Rank in order, from largest to smallest, the electric field strengths E1, E2,and E3 on the surfaces ofthe spheres.

order: E, = Ea= EaExplanation:

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17, Two spheres of different diameters surround equal charges. Three students are discussing the situation.

Student l: The flux through spheres A and B are equal because they enclose equal charges.

Student 2: But the electric field on sphere B is weaker than the electric field on sphere A. The fluxdepends on the electric field strength, so the flux through A is larger than the flux through B.Sfudent 3: I thought we leamed that flux was about surface area. Sphere B is larger than sphere A,so I think the flux through B is larger than the flux through A.Which of these students, if any, do you agree with? Explain.

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thcis

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27-g cHAPrER2T' Gauss's Law

18. A sphere and an ellipsoid suround equal charges. Four students are discussing the sifuation.

Sfudent 1: The fluxes through A and B are equal because the average radius is the same.

Student 2:Iagree that the fluxes are equal, but it's because they enclose equal charges.

Studelt 3: The electrj,c field js not perpendic,nlar to the sur,face for B, and,that makes the flux through

B less than the flux through A.

Student 4: I don't think that Gauss's iaw even appiies to a situation like B, so we can't compare the

fluxes through A and B.

Which of these students, if any, do you agree with? Explain.

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by *h" Squtc- fq.tor so +h".+ the {l.^xes arL e1uot.

19. Two parallel, infinite planes of charge have charge densities?gs ztt and-r7. A Gaussian cylinder with cross sectionl extends??1 distance Z to either side.

a. Is E perpendicular or parallel to the surface at the:

Top I Boftom wall ll

c. By inspection, write the electric fluxes through the three surfaces in tetms of E1oo, E,ooy, E*u11, L, L6,

b. Is the electric field E oo emerging from the top surface

stronger than, weaker than, or equal in strength to the fieldE6o, emerging from the bottom? Explain.

E1,^",\. E*oo= * - A = t up

Es51*o,,r= * t=tJo*^andA. (You rnay not need all of these.)

@top= E*.,A obot= Ero*A @watl = O

d. How much charge is enclosed within the cylinder? Write Qn in tetms of r7, L, L11, and A.

Qin= tel-1)A= ?',lA

e. By combining your answers from parts b, c, and d, use Gauss's law to determine the electric field

strength above the top plane. Show your work.

6 = F ff= Q,. = 1A-*o? **op €o €o

Tl,F-o

a

EF.l

o

o

d

o

t-1-r,op

Gauss'sLaw . cHaprqp.2T 27-9

27.6 Conductors in Electrostatic Equilibrium

20. A smal1 metal sphere hangs by a thread within a larger, hollowconducting sphere. A charged rod is used to transfer positivecharge to the outer surface ofthe hollow sphere.

a. Suppose the thread is an insulator. After the charged rod touchesthe outer sphere and is removed, are the following surfacespositive, negative, or not charged?

The smali sphere:

The inner surface ofthe hollow sphere:

The outer surface ofthe hollow sphere:

b. Suppose the thread is a conductor. After the charged rod touches the outer sphere and is removed,are the following surfaces positive, negative, or not charged?

The small sphere:

The inner surface ofthe hollow sphere:

The outer surface ofthe hoilow sphere:

21. A small metal sphere hangs by an insulating thread within a larger,hollow conducting sphere. A conducting wire extends from thesmall sphere through, but not touching, a small hole in the hollowsphere. A charged rod is used to transfer positive charge to thewire. After the charged rod has touched the wire and beenremoved, are the following surfaces positive, negative, or notcharged?

The small sphere:

The imrer surface of the hollow sphere:

The outer surface ofthe hollow sphere:

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ch",i3ed

Posi+ive

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posi+ive

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o

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o

ONo

22. A -10 nC point charge is inside a hole in a conductor. The conductor has

no net charge.

a. What is the total charge on the inside surface of the conductor?

+ 10 " C

b. What is the total charge on the outside surface of the conductor?

-tO",C

27-10 cHAPTER2T . Gauss'sLaw

23. A -10 nC point charge is inside a hole in a conductor. The

conductor has a net charge of+10 nC,

a. What is the total charge on the inside surface of the conductor?

+10^C

b. What is the total charge on the outside surface of the conductor?

An insulating thread is used to lower a positively charged metal ball into a metal container. Initially,the container has no net charge. Use plus and minus signs to show the charge distribution on the ball atthe times shown in the figure. (The ball's charge is already shown in the first frame.)

24.

Ball hasn't touched Ball has touched Ball has been withdrawn

o

o

oo.

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