Essential University Physics, 3e (Wolfson) 3e (Wolfson) Chapter 21 Gauss's Law
2 1. 1 Co nc e pt ua lQue s t i o ns 1) If the electric flux through a closed surface is zero, the electric field at points on that surface must be zero. A) True B) False Answer B !ar 1
") The figure shows four #aussian surfaces surrounding a distribution of charges.
$a) %hich #aussian surfaces ha&e an electric flux of 'q 'q(ε through them* $b) %hich #aussian surfaces ha&e no electric flux through them* Answer $a) b $b) c !ar 1
+) %hich of the following statements about #ausss law are correct* $There ma- be more than one correct choice.) A) #ausss law is &alid onl- for s-mmetric charge distributions, such as spheres and c-linders. B) If there is no charge inside of a #aussian surface, the electric field must be zero at points of that surface. ) /nl- charge enclosed within a #aussian surface can produce an electric field at points on that surface. 0) If a #aussian surface is completel- inside an electrostatic conductor, the electric field must alwa-s be zero at all points on that surface. ) The electric flux passing through a #aussian surface depends onl- on the amount of charge inside that surface, not on its size or shape. Answer 0, !ar 1
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5) onsider a spherical #aussian surface of radius R centered at the origin. A charge Q is placed inside the sphere. To maximize the magnitude of the flux of the electric field through the #aussian surface, the charge should be located A) at x 6 , y 6 , z 6 R(". B) at the origin. ) at x 6 R(", y 6 , z 6 . 0) at x 6 , y 6 R(", z 6 . ) The charge can be located an-where, since flux does not depend on the position of the charge as long as it is inside the sphere. Answer !ar 1
7) The graph in the figure shows the electric field strength $not the field lines) as a function of distance from the center for a pair of concentric uniforml- charged spheres. %hich of the following situations could the graph plausibl- represent* $There ma- be more than one correct choice.)
A) a positi&el- charged conducting sphere within another positi&el- charged conducting sphere B) a positi&el- charged conducting sphere within an uncharged conducting sphere ) a solid nonconducting sphere, uniforml- charged throughout its &olume, inside of a positi&elcharged conducting sphere 0) a positi&el- charged nonconducting thin8walled spherical shell inside of a positi&el- charged conducting sphere ) a positi&el- charged nonconducting thin8walled spherical shell inside of another positi&elcharged nonconducting thin8walled spherical shell Answer A, 0 !ar 1
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3) Two long straight parallel lines, 91 and 9", carr- uniform positi&e linear charge densities. The charge densit- on line 9" is twice as great as the charge densit- on line 91. The locus of points where the electric field due to these lines is zero is A) along a line between the lines closer to line 9" than line 91. B) at a point midwa- between the lines. ) along a line perpendicular to lines 91 and 9". 0) along a line between the lines closer to line 91 than line 9". Answer 0 !ar 1
:) At a distance D from a &er- long $essentiall- infinite) uniform line of charge, the electric field strength is 1 ;(. At what distance from the line will the field strength to be " ;(* A) " D B) D ) D( 0) D(" ) D(5 Answer 0 !ar 1
<) A charge Q is uniforml- spread o&er one surface of a &er- large nonconducting s=uare elastic sheet ha&ing sides of length d . At a point P that is 1."7 cm outside the sheet, the magnitude of the electric field due to the sheet is E . If the sheet is now stretched so that its sides ha&e length "d , what is the magnitude of the electric field at P * A) 5 E B) " E ) E 0) E (" ) E (5 Answer !ar 1
>) An uncharged conductor has a hollow ca&it- inside of it. %ithin this ca&it- there is a charge of '1 ? that does not touch the conductor. There are no other charges in the &icinit-. %hich statement about this conductor is true* $There ma- be more than one correct choice.) A) The inner surface of the conductor carries a charge of 81 ? and its outer surface carries no excess charge. B) The inner and outer surfaces of the conductor each contain charges of 87 ?. ) The net electric field within the material of the conductor points awa- from the '1 ? charge. 0) The outer surface of the conductor contains '1 ? of charge and the inner surface contains 81 ?. ) Both surfaces of the conductor carr- no excess charge because the conductor is uncharged. Answer 0 !ar 1
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1) @nder electrostatic conditions, the electric field ust outside the surface of an - charged conductor A) is alwa-s parallel to the surface. B) is alwa-s zero because the electric field is zero inside conductors. ) is alwa-s perpendicular to the surface of the conductor. 0) is perpendicular to the surface of the conductor onl- if it is a sphere, a c-linder, or a flat sheet. ) can ha&e nonzero components perpendicular to and parallel to the surface of the conductor. Answer !ar 1
2 1 . 2 Pr o bl e ms 1) A nonuniform electric field is directed along the x8axis at all points in space. This magnitude of the field &aries with x, but not with respect to y or z . The axis of a c-lindrical surface, .< m long and ." m in diameter, is aligned parallel to the x8axis, as shown in the figure. The electric fields E 1 and E ", at the ends of the c-lindrical surface, ha&e magnitudes of 3 ;( and 1 ;( respecti&el-, and are directed as shown. %hat is the net electric flux passing through the c-lindrical surface*
A) 813 ; m"( B) 8+7 ; m"( ) . ; m"( 0) '+7 ; m"( ) '13 ; m"( Answer A !ar 1
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") A cone is resting on a tabletop as shown in the figure with its face horizontal. A uniform electric field of magnitude 577 ;( points &erticall- upward. Cow much electric flux passes through the sloping side surface area of the cone*
Answer 3.+3 ; m"( !ar 1
+) If a rectangular area is rotated in a uniform electric field from the position where the maximum electric flux goes through it to an orientation where onl- half the flux goes through it, what has been the angle of rotation* A) 57D B) "3.3D ) >D 0) +D ) 3D Answer !ar 1
5) The cube of insulating material shown in the figure has one corner at the origin. ach side of the cube has length .< m so the top face of the cube is parallel to the xz 8plane and is at y 6 .< m. It is obser&ed that there is an electric field
that is in the ' y
direction and whose magnitude depends onl- on y. @se #ausss law to calculate the net charge enclosed b- the cube. $ε 6 <.<7 E 181" "(; m")
Answer 1.7 x 1811 !ar 1 7 op-right 2 "13 4earson ducation, Inc.
7) A charge q 6 ". is placed at the origin in a region where there is alread- a uniform electric field
6 $1 ;() . alculate the flux of the net electric field through a #aussian
sphere of radius R 6 1. cm centered at the origin. $ε 6 <.<7 E 181" "(; m") A) 7.7" E 17 ; m"( B) 1.1+ E 17 ; m"( ) "."3 E 17 ; m"( 0) zero Answer !ar 1
3) A charge of 1. × 183 is located inside a sphere, 1."7 cm from its center. %hat is the electric flux through the sphere due to this charge* $ε 6 <.<7 E 181" "(; m") A) .11 ; m"( B) <.> ; m"( ) ."
:) Four dipoles, each consisting of a '18? charge and a 818? charge, are located in the xy8 plane with their centers 1. mm from the origin, as shown. A sphere passes through the dipoles, as shown in the figure. %hat is the electric flux through the sphere due to these dipoles* $ε 6 <.<7 E 181" "(; m")
A) 5.7 E 13 ; m"( B) . ; m"( ) >. E 13 ; m"( 0) 11 E 17 ; m"( Answer A !ar 1
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<) A nonuniform electric field is directed along the x8axis at all points in space. This magnitude of the field &aries with x, but not with respect to y or z . The axis of a c-lindrical surface, .< m long and ." m in diameter, is aligned parallel to the x8axis, as shown in the figure. The electric fields E 1 and E ", at the ends of the c-lindrical surface, ha&e magnitudes of and respecti&el-, and are directed as shown. $ε 6 <.<7 E 181" "(; m") The charge enclosed b- the c-lindrical surface is closest to
A) 81.1 n. B) 1.1 n. ) 8".5 n. 0) 85.< n. ) 5.< n. Answer A !ar 7'
>) Two concentric spheres are shown in the figure. The inner sphere is a solid nonconductor and carries a charge of '7. ? uniforml- distributed o&er its outer surface. The ou ter sphere is a conducting shell that carries a net charge of 8<. ?. ;o other charges are present. The radii shown in the figure ha&e the &alues R1 6 1. cm, R" 6 ". cm, and R+ 6 +. cm. $k 6 1(5πε 6 <.>> E 1> ; m"(") $a) Find the total excess charge on the inner and outer surfaces of the conducting sphere. $b) Find the magnitude and direction of the electric field at the following distances r from the center of the inner sphere $i) r 6 >.7 cm, $ii) r 6 17. cm, $iii) r 6 ":. cm, $i&) r 6 +7. cm.
Answer $a) 87. ? $inner surface), 8+. ? $outer surface) $b) $i) $ii) ". × 13 ;(, radiall- outward $iii) $i&) "." × 17 ;(, radiall- inward !ar 1 : op-right 2 "13 4earson ducation, Inc.
1) Two concentric conducting spherical shells produce a radiall- outward electric field of magnitude 5>, ;( at a point 5.1 m from the center of the shells. The outer surface of the larger shell has a radius of +.:7 m. If the inner shell contains an excess charge of 87.+ , find the amount of charge on the outer surface of the larger shell. $k 6 1(5πε 6 <.>> E 1> ; m"(") Answer >1.3 ? !ar 1
11) A solid nonconducting sphere of radius R carries a uniform charge densit- throughout its &olume. At a radial distance r 1 6 R(5 from the center, the electric field has a magnitude E . %hat is the magnitude of the electric field at a radial distance r " 6 " R* A) E (5 B) zero ) E (" 0) E ) " E Answer 0 !ar 1
1") A solid nonconducting sphere of radius R carries a charge Q distributed uniforml- throughout its &olume. At a certain distance r 1 $r 1 J R) from the center of the sphere, the electric field has magnitude E . If the same charge Q were distributed uniforml- throughout a sphere of radius " R, the magnitude of the electric field at the same distance r 1 from the center would be e=ual to A) E (<. B) E (". ) " E. 0) < E . ) E . Answer A !ar 1
1+) A spherical, non8conducting shell of inner radius 6 1 cm and outer radius 6 17 cm carries a total charge K 6 17 distributed uniforml- throughout the &olume of the shell. %hat is the magnitude of the electric field at a distance r 6 1" cm from the center of the shell* $k 6 1(5πε 6 <.>> E 1> ; m"(") A) 7.:7 E
;(
B) zero ) ".<: E
;(
0) 7.:7 E
;(
) ".<: E
;(
Answer !ar 7
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15) A non8conducting sphere of radius R 6 :. cm carries a charge Q 6 5. m distributed uniforml- throughout its &olume. At what distance, measured from the center of the sphere, does the electric field reach a &alue e=ual to half its maximum &alue* A) +.7 cm onlB) 5.> cm onl) +.7 cm and >.> cm 0) +.7 cm and 5.> cm ) >.> cm onlAnswer !ar 7
17) lectric charge is uniforml- distributed inside a nonconducting sphere of radius .+ m. The electric field at a point P , which is .7 m from the center of the sphere, is 17, ;( and is directed radiall- outward. At what distance from the center of the sphere does the electric field ha&e the same magnitude as it has at P * A) .11 m B) .1+ m ) .17 m 0) .1: m ) at no other point Answer A !ar 1
13) lectric charge is uniforml- distributed inside a nonconducting sphere of radius .+ m. The electric field at a point P , which is .7 m from the center of the sphere, is 17, ;( and is directed radiall- outward. %hat is the maximum magnitude of the electric field due to this sphere* A) "7, ;( B) +, ;( ) +3, ;( 0) 5", ;( ) 5<, ;( Answer 0 !ar 1
1:) A nonconducting spherical shell of inner radius R1 and outer radius R" contains a uniform &olume charge densit- ρ throughout the shell. @se #ausss law to deri&e an e=uation for the magnitude of the electric field at the following radial distances r from the center of the sphere. Lour answers should be in terms of ρ, R1, R", r , ε , and G. $a) r J R1 $b) R1 J r J R" $c) r M R" Answer $a) E 6
$b) E 6
$c) E 6
!ar 1
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1<) An infinitel- long nonconducting c-linder of radius R 6 ". cm carries a uniform &olume charge densit- of alculate the electric field at distance r 6 1. cm from the axis of the c-linder. $ε 6 <.<7 E 181" "(; m") A) ".7 E 1+ ;( B) 7.1 E 1+ ;( ) zero 0) ". E 1+ ;( ) 1." E 1+ ;( Answer !ar 1
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1>) The cross section of a long coaxial cable is shown in the figure, with radii as g i&en. The linear charge densit- on the inner conductor is and the linear charge densit- on the outer conductor is The inner and outer c-lindrical surfaces are respecti&el- denoted b A, B, C , and D, as shown.
The radial component of the electric
field at a point that +5 mm from the axis is closest to
A) 813, ;(. B) '13, ;(. ) 8+:, ;(. 0) '+:, ;(. ) zero Answer A !ar 7'
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") The cross section of a long coaxial cable is shown in the figure, with radii as g i&en. The linear charge densit- on the inner conductor is and the linear charge densit- on the outer conductor is The inner and outer c-lindrical surfaces are respecti&el- denoted b A, B, C , and D, as shown. point that is
The magnitude of the electric field at a
from the axis is closest to
A) 1:, ;(. B) 17, ;(. ) 1+, ;(. 0) 11, ;(. ) >3 ;(. Answer A !ar 7'
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"1) harge is distributed uniforml- throughout a large insulating c-linder of radius R. The charge per unit length in the c-lindrical &olume is λ. $a) @se #ausss law to find the magnitude of the electric field at a distance r from the central axis of the c-linder for r J R. Lour answer should be in terms of r , R, λ, ε , and G. $b) hecH the reasonableness of -our answer b- e&aluating it at the surface of the c-linder. Answer $a) 6 $b) At r = R, E 6 λ/ "πε R, which is reasonable. !ar 1
"") A &er- large sheet of a conductor carries a uniform charge densit- of 5. p(mm" on its surfaces. %hat is the electric field strength +. mm outside the surface of the conductor* $ε 6 <.<7 E 181" "(; m") A) 5.7" E 17 ;( B) "."3 E 17 ;( ) >.5 E 17 ;( 0) .57" ;( ) .""3 ;( Answer A !ar 1
"+) A huge $essentiall- infinite) horizontal noncondu cting sheet 1. cm thicH has charge uniforml- spread o&er both faces. The upper face carries '>7. n(m" while the lower face carries 8"7. n( m". %hat is the magnitude of the electric field at a point within the sheet ". cm below the upper face* $ε 6 <.<7 E 181" "(; m") A) . ;( B) +.>7 E 1+ ;( ) 3.:< E 1+ ;( 0) :.>1 E 1+ ;( ) 1.+3 E 15 ;( Answer !ar 1
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"5) As shown in the figure, a s=uare insulating slab 7. mm thicH measuring ". m E ". m has a charge of <. E 1811 distributed uniforml- throughout its &olume. @se #aussNs law to determine the electric field at point P , which is located within the slab beneath its center, 1. mm from one of the faces. $ε 6 <.<7 E 181" "(; m")
A) .3< ;( B) 15 ;( ) "+ ;( 0) +5 ;( ) 7: ;( Answer A !ar 1
"7) Two extremel- large nonconducting horizontal sheets each carr- uniform charge densit- on the surfaces facing each other. The upper sheet carries '7. ?(m". The electric field midwa between the sheets is 5."7 E 17 ;( pointing downward. %hat is the surface charge densit- on the lower sheet* $ε 6 <.<7 E 181" "(; m") Answer 8".7" ?(m" !ar 1
"3) onsider two closel- spaced and oppositel- charged parallel metal plates. The plates are s=uare with sides of length L and carr- charges Q and 8Q on their facing surfaces. %hat is the magnitude of the electric field in the region between the plates* A) 6
B) 6 ) 6 0) 6
) 6 Answer A !ar 1
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":) A neutral hollow spherical conducting shell of inner radius 1 . cm and outer radius +. cm has a '".8? point charge placed at its center. Find the surface charge densit$a) on the inner surface of the shell. $b) on the outer surface of the shell. Answer $a) 817> ?(m" $b) '1:: ?(m" !ar 1
"<) A hollow conducting spherical shell has radii of .< m and 1." m, as shown in the figure. The sphere carries an excess charge of 87 n. A point charge of '+ n is present at the center. The surface charge densit- on the inner spherical surface is closest to
A) zero. B) '5. × 18< (m". ) '3. × 18< ( m". 0) 85. × 18< ( m". ) 83. × 18< ( m". Answer 0 !ar 1
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">) A hollow conducting spherical shell has radii of .< m and 1." m, as shown in the figure. The sphere carries a net excess charge of 87 n. A point charge of '+ n is present at the center. $k 6 1(5πε 6 <.>> E 1> ; m"() The radial component of the electric field at a point that is .3 m from the center is closest to
A) zero. B) '7 ;(. ) ':7 ;(. 0) 87 ;(. ) 8:7 ;(. Answer !ar 1
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+) A hollow conducting spherical shell has radii of .< m and 1." m, as shown in the figure. The sphere carries a net excess charge of 87 n. A point charge of '+ n is present at the center. $k 6 1(5πε 6 <.>> E 1> ; m"() The radial component of the electric field at a point that is 1.7 m from the center is closest to
A) '1" ;(. B) '" ;(. ) 8< ;(. 0) 813 ;(. ) 8" ;(. Answer !ar 1
+1) An irregular conductor carries a surface charge densit- of 83.:7 ?(m" at and in the &icinitof a point P on the surface. An electron is released ust abo&e P outside the conductor. %hat are the magnitude and direction of its acceleration the instant after it is released* $ε 6 <.<7 E 181" "(; m", e 6 1.3 E 181> , mel 6 >.11 E 18+1 Hg) Answer 1.+5 E 11: m(s", awa- from P !ar 1
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