91751156 Solucionario Capitulo 23 Paul E Tippens

April 29, 2019 | Author: Ignacio Diaz Muñoz | Category: Proton, Electric Charge, Electrostatics, Quantity, Electromagnetism
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Physics, 6th Edition

Chapter 23. The Electric Force

Chapter 23. The Electric Force Coulomb's Law 23-1. Two balls each having having a charge o 3 C are separated separated by 2! "". #hat #hat is the orce orce o  o  rep$lsion between the"%  F  =

'& ( 1!& ) ⋅ "2 *C2 +'3 ( 1!-6 C+'3 ( 1!-6 C+

;

'2! ( 1!-3 "+2

F = 202 N

23-2. Two point charges o -3 and and   C are 12 12 "" "" apart in a vac$$". #hat is the electrostatic electrostatic orce between the"%  F  =

'& ( 1!& ) ⋅ "2 *C2 +'−3 ( 1!-6 C+' ( 1!-6 C+ '12 ( 1!-3 "+2

;

F = 750 N, attractio

23-3. n alpha particle consists o two two protons 'q 'qe / 1.6 ( 1!-1& C+ and two ne$trons 'no charge+. #hat is the rep$lsive orce between two alpha particles separated by 2 n"% 0α / 2'1.6 ( 1!-1& C+ / 3.2 ( 1!-1& C  F  =

'& ( 1!& ) ⋅ "2 *C2 +'3.2 ( 1!-1& C+'3.2 ( 1!-1& C+ -&

2

'2.!! 2.!! ( 1! "+

F = 2.30 ! "0#"0 N

;

23-. ss$"e that the radi$s o the electrons orbit aro$nd the proton in a hydrogen ato" is appro(i"ately .2 ( 1!-11 ". #hat is the electrostatic orce o attraction%  F  =

'& ( 1!& ) ⋅ "2 *C2 +'1.6 ( 1!-1& C+'− 1.6 ( 1!-1& C+ -11

'.2 ( 1! "+

2

;

F = $.52 ! "0#$ N

23-. #hat is the separation o two -  C charges i the orce orce o rep$lsion between between the" is 2!! )% r  =

kqq   F 

=

'& ( 1!& ) ⋅ "2 *C2 +'− ( 1!-6 C+2 2!! )

6!

;

r = 2%.$ mm

Physics, 6th Edition

Chapter 23. The Electric Force

23-6. Two identical charges separated by 3! "" e(perience a rep$lsive orce o &! ). #hat is the "agnit$de o each charge%

 F

=

kq 2 r2

4 q=

Fr 2 k 

  '&! )+'!.!3! "+2

=

& ( 1!& ) ⋅ "2 *C2

q = &.&! C

4

523-.  1! C charge and a -6  C charge are separated by ! "". #hat is the orce between the". The spheres are placed in contact or a ew "o"ents and then separated again by ! "". #hat is the new orce% 7s it attractive or rep$lsive%  F  =

'& ( 1!& ) ⋅ "2 *C2 +'−6 ( 1!-6 C+'1! ( 1!-6 C+ '! ( 1!-3 "+2

;

F = 33$ N, attractio

1! µC

-6 µC When spheres touch, 6 C of charge are neutralized,

!.! "

leaving  C to be shared by two spheres, or 

 F  =

'& ( 1!& ) ⋅ "2 *C2 +'2 ( 1!-6 C+'2 ( 1!-6 C+ '!.!! "+2

2 µC

2 µC

2 C on each sphere. Now they are again separated.

2 µC

2 µC !.! "

;

F = 5.%2 N, repulsio

523-. Two point charges initially attract each other with a orce o 6!! ). 7 their separation is red$ced to one-third o its original distance, what is the new orce o attraction%

 F



1 r

2

2

4

F1r12

 r12   r     F2 = F1  2  = F 1  1  4   r2   r 2   

= F2 r22 4 2

 3r     F2 = F1  2   = & F 1 4  r 2  

61

F 2 = 5&00 N

r 1 =  r !

Physics, 6th Edition

Chapter 23. The Electric Force

The esultat Electrostatic Force 23-&.  6! C charge is placed 6! "" to the let o a 2!  C charge. #hat is the res$ltant orce on a -3 C charge placed "idway between the two charges%  F 13

=

'& ( 1!& ) ⋅ "2 *C2 +'6! ( 1!-6 C+'3 ( 1!-6 C+ '3! ( 1!-3 "+2 

 F 1 = 2.1! ( 1!  ), directed to the left   F 23

=

'3! ( 1! "+

2! µC

q1

q

q!  F !

 F 1

'& ( 1!& ) ⋅ "2 *C2 +'2! ( 1!-6 C+'3 ( 1!-6 C+ -3

6! µC

6! "" -3 µC

2

4  F 1 = 2.1! ( 1! ), directed to right.

 F  " = F 1 # F ! = '-2.1! ( 1! )+  '!.!! ( 1! )+4

 F  " = $1.! ( 1! ), let.

23-1!.  point charge o 36  C is placed ! "" to the let o a second point charge o -22  C. #hat orce is e(erted on third charge o 1!  C located at the "idpoint%  F 13

=

'& ( 1!& ) ⋅ "2 *C2 +'36 ( 1!-6 C+'1! ( 1!-6 C+ '! ( 1!-3 "+2

36 µC

! "" 1! µC

q1

q

 F 1 = 2!2 ), directed to the right   F 23

=

'& ( 1!& ) ⋅ "2 *C2 +'22 ( 1!-6 C+'1! ( 1!-6 C+ '! ( 1!-3 "+2

-22 µC  F 1

q!

 F ! 4  F 1 = 123 ), directed to right.

 F  " = F 1 # F ! = 2!2 )  123 )4

 F  " = 326! ), left.

23-11. For Proble" 23-1!, what is the res$ltant orce on a third charge o 12  C placed between the other charges and located 6! "" ro" the 36  C charge%  F 13

 F 23

= =

'& ( 1!& ) ⋅ "2 *C2 +'36 ( 1!-6 C+'12 ( 1!-6 C

36 µC

'6! ( 1!-3 "+2 '& ( 1! ) ⋅ " *C +'22 ( 1! C+'12 ( 1! C+ &

2

2

-6

-3

q1

-6

2

'2! ( 1! "+

4

 %oth to right, so F  " = F 1 # F ! = 1!! )  &! )4

62

! "" q = 12 µC

6! ""  F !

 F = !2! ) , rightward.

-22 µC  F 1

q!

Physics, 6th Edition

Chapter 23. The Electric Force

23-12.  6 C charge is  "" to the right o a -  C charge. #hat is the res$ltant orce on a -2 C charge that is 2! "" to the right o the -  C charge%  F 13

 F 23

= =

'& ( 1!& ) ⋅ "2 *C2 +' ( 1!-6 C+'2 ( 1!-6 C+

'2 ( 1!-3 "+2

q = -2 µC

- µC

'2! ( 1!-3 "+2 '& ( 1!& ) ⋅ "2 *C2 +'2 ( 1!-6 C+'6 ( 1!-6 C+

 ""

q1

2 ""

2! ""

 F 1

 F !

4

 %oth to right, so F  " = F 1 # F ! = 36! )  1. )4

6 µC q!

 F =  ) , rightward 

523-13.  6-C charge is locate 3! c" to the let o a 16- C charge. #hat is the res$ltant orce on a -12 C charge positioned e(actly ! "" below the 16  C charge%  s = '3! ""+2 + '! ""+2

= .3 ""

q1 tan φ

=

! "" 3! ""

4

q!

6 µC 3! "" φ 

φ  = &.!

!

 s  F 13

=

! ""

'& ( 1!& ) ⋅ "2 *C2 +'6 ( 1!-6 C+'12 ( 1!-6 C+ -3

'.3 ( 1! "+

2

 F 1

 F 23

=

'& ( 1! ) ⋅ " *C +'16 ( 1! C+'12 ( 1! C+ 2

2

-6

-3

q = $12 µC

-6

2

'! ( 1! "+

/ 6&1 ), $pward.

F( / ! 8 F13 cos &.!! / -'2!33 )+ cos &! 4

F( / -1! )

Fy / F23  F13 sin &.!! / 6&1 )  '2!33 )+ sin & !4  F  = '−1! )+2 + '23 )+2

 F ! φ 

 F 1 = 2!33 ), &.!! ) o # &

16 µC

= 26! )4

tanθ  =

23 ) -1! )

Fy / 23 ) 4

/

9es$ltant orce: F9  / 26! ), 66.! ) o # 'or 113.3!+

63

! 66.  ) o #.

Physics, 6th Edition

Chapter 23. The Electric Force

523-1.  charge o 6! nC is located ! "" above a -!-nC charge. #hat is the res$ltant orce on a -!-nC charge located  "" hori;ontally to the right o the -!-nC charge%  s = ' ""+

2

tan φ

6! µC

+ '! ""+ = &1. "" 2

=

! ""  ""

4

q1

φ  = 6!.6

!

 s

! ""  F 13

=

'& ( 1!& ) ⋅ "2 *C2 +'6! ( 1!-6 C+'! ( 1!-6 C+ q!

 F 1 = 26 ), 6!.6! ) o #  F 23

=

 F 1

'&1. ( 1!-3 "+2

-! µC

'& ( 1!& ) ⋅ "2 *C2 +'! ( 1!-6 C+'! ( 1!-6 C+ ' ( 1!-3 "+2

Fy / F13 sin 6!.6!  ! / '26 )+ sin 6!.6! 4 223 ) 632 )

 F !

 ""

φ  q = $! µC

/ & ), rightward .

F( / 8 F13 cos 6!.6!  F23 / - '26 )+ cos 6!.6!  & ) 4

 F  = '632 )+2 + '223 )+2 4 tanθ  =

 F !

F( / 632 )

Fy / 223 )

 4  F  " = &! ),  / 16.3! ) o E.

523-1. Three point charges 01 /   C, 02 / - C, and 03 / 2 C are placed at the corners o  an e0$ilateral triangle, ! "" on each side as described by Fig. 23-1. #hat are the  F !

"agnit$de and direction o the res$ltant orce on the   C charge%  F 21

=

'! ( 1!-3 "+2

=

! ""

'& ( 1!& ) ⋅ "2 *C2 +'2 ( 1!-6 C+' ( 1!-6 C+

 F 1

! "" q / - µC !

F( / '22. )+ cos 6!!  ' )+ cos 6!! / 33. )

F  y = '22. )+ sin 6!! - ' )+ sin 6!! / -1&. )

6

! "" φ 

q 2 µC

'! ( 1!-3 "+2

F 1 = 22. ), 6!! ) o E4

6!! 6!!

F !1 = .! ), 6!! < o E  F 31

q1=  µC

'& ( 1!& ) ⋅ " 2 *C2 +' ( 1!-6 C+' ( 1!-6 C+

φ 

Physics, 6th Edition

Chapter 23. The Electric Force

523-1. 'Cont.+  F  = '33. )+

2

+ '−1&. )+2

9es$ltant electric orce:

=

/ 3&.! )4 tanθ

-1&. ) 33. )

4 θ  = − 3!

!

 F  " = 3&.! ),  / 33!.!!

Challe(e )roblems 23-16. #hat sho$ld be the separation o two -$C charges so that the orce o rep$lsion is  )% r  =

kqq   F 

'& ( 1!& ) ⋅ "2 *C2 +' ( 1!-6 C+2

=

.!! )

;

r = 23.7 cm

23-1. The rep$lsive orce between two pith balls is o$nd to be 6!  ). 7 each pith ball carries a charge o  nC, what is their separation% r =

kqq   F 

=

'& ( 1!& ) ⋅ "2 *C2 +' ( 1!-& C+2 6! ( 1!-6 )

;

r = *$.0 mm

23-1. Two identical $n=nown charges e(perience a "$t$al rep$lsive orce o  ) when separated by 6! "". #hat is the "agnit$de o each charge%

 F

=

kq 2 r

2

4 q=

Fr 2 k 

  ' )+'!.!6! "+2

=

& ( 1! ) ⋅ " *C &

2

2

4

q = .3 C

23-1&. >ne ob?ect contains an e(cess o  ( 1!1 electrons and another has a deiciency o  ( 1!1 electrons. #hat is the orce each e(erts on the other i the ob?ects are 3! "" apart% 7s it attraction or rep$lsion% ' 1e / 1.6 ( 1!-1& C, e&cess = $, deficiency = #. + q1 = ' ( 1!1 e+'1.6 ( 1!-1& C*e+ / -! C4 q! = ' ( 1!1 e+'1.6 ( 1!-1& C*e+ / 6 C  F  =

'& ( 1!& ) ⋅ "2 *C2 +'! ( 1!-6 C+'6 ( 1!-6 C+ -3

2

'3! ( 1! "+

6

4

 F = .12 ( 1! ), attraction

Physics, 6th Edition

Chapter 23. The Electric Force

23-2!. 7 it were possible to p$t 1 C o charge on each o two spheres separated by a distance o 1 ", what wo$ld be the rep$lsive orce in newtons. '& ( 1!& ) ⋅ "2 *C2 +'1 C+'1 C+

 F  =

'1 "+

2

 F = & ( 1!& ) @

4

'he coulo(b is a very large unit for electrostatics applications.

23-21. Aow "any electrons "$st be placed on each o two spheres separated by  "" in order to  prod$ce a rep$lsive orce o one !! )%

=

 F

kq 2 r

2

4 q=

Fr 2 k 

  '!! )+'!.!! "+2

=

& ( 1! ) ⋅ " *C &

1e     4 -1&  1.6 ( 1! C  

q = 3 ( 1!-& C 

2

2

4

q = 3 nC

q = .2 ( 1!12 electrons

23-22.  8!-nC charge is placed ! "" to the let o a 6-nC charge. #hat is the res$ltant orce on a 812-nC charge place  "" to the right o the 6-nC charge%  F 1 =

 F 2

=

'& ( 1!& ) ⋅ "2 *C2 +'! ( 1!-& C+'12 ( 1!-& C+ -3

' ( 1! "+

2

'& ( 1! ) ⋅ " *C +'6 ( 1! C+'12 ( 1! C+ &

2

2

-&

6 nC -! nC

' ( 1!-3 "+2

 ""

! "" q!

q1

-&

q = $12 nC

 F 1

 F !

4

 %oth to right, so F  " = F 1 # F ! = 1. ") 8 1!.1 ")4

 F = $.2 ") , leftward 

23-23.  -C charge is placed 6 c" to the right o a 2- C charge. #hat is the res$ltant orce on a 8& nC charge placed 2 c" to the let o the 2- C charge%  F 1 =

'& ( 1!& ) ⋅ "2 *C2 +'& ( 1!-& C+'2 ( 1!-6 C+

-& nC

'2 ( 1!-2 "+2 2 c"

2 µC q1

F1 / ! "), to right   F !

66

6 c"  F 1

 µC q!

Physics, 6th Edition

Chapter 23. The Electric Force

 F 2

=

'& ( 1!& ) ⋅ "2 *C2 +' ( 1!-6 C+'& ( 1!-& C+ -2

' ( 1! "+  "esultant force)

2

4

 F ! / 63.3 ") , to right 

F  " = ! ")  63.3 ")4

 F  " = 6 ")

23-2. n e0$al n$"ber o electrons is placed on two "etal spheres 3.! c" apart in air. Aow "any electrons are on each sphere i the res$ltant orce is !! )%

=

 F

kq 2 r

2

Fr 2

4 q=

  '!! )+'!.!3 "+2

=



& ( 1! ) ⋅ " *C &

2

1e     4 -1&  1.6 ( 1! C  

q = 21.2 ( 1! C  -6

2

q = 21.2 C

4

q = 1.33 ( 1!1 electrons

23-2.  -nC charge is placed on a -g sphere that is ree to "ove.  i(ed 1!- C point charge is  c" away. #hat is the initial acceleration o the - C charge%  F 2

=

'& ( 1!& ) ⋅ "2 *C2 +' ( 1!-& C+'1! ( 1!-6 C+ ' ( 1!-2 "+2 a=

 F  (

!.22 )

=

!.!! =g

4

 F ! / 22 ")

a = 6.2 "*s2

523-26. #hat is the res$ltant orce on a 2  C charge an e0$al distance o 6! "" ro" each o  two 8-C charges that are ! "" apart in air% tan φ

 F 2

=

=

! "" 6! ""

φ  = .2

4

φ  6! ""

'& ( 1!& ) ⋅ "2 *C2 +' ( 1!-6 C+'2 ( 1!-6 C+ -3

2

'6! ( 1! "+

'6! ( 1! "+

φ  6! ""

q

φ 

- µC

q! / - µC ! ""

'& ( 1!& ) ⋅ "2 *C2 +'2 ( 1!-6 C+' ( 1!-6 C+ -3

φ   F  !

 F 1

!

F ! = 2!.! ), .2! < o E  F 1 =

q1= 2 µC

2

6

4

 F 1 = 2!.! ), .2! < o #

Physics, 6th Edition

Chapter 23. The Electric Force 523-26. 'Cont.+

F( / '2! )+ cos .2!  '2! )+ cos .2! / 13.33 ) - 13.33 )4 F( / ! F y / '2! )+ sin .2!  '2! )+ sin .2! / 1.& )  1.& )4

Fy / -2&. )

 "esultant force) F  " = 2&. ), downward 

523-2. Two charges o 2 and 16  C are ! "" apart.  third charge o 6!  C is placed  between the other charges 3! "" ro" the 2  C charge. Find the res$ltant orce on the third charge%  F 1 =

2

2

-6

-6

'3! ( 1!-3 "+2

=

3! ""

q1

16 µC q!

! ""

 F 1

 F !

 F 1 = 1 =), directed to the right   F 2

! ""

q = 6! µC

2 µC

'& ( 1! ) ⋅ " *C +'6! ( 1! C+'2 ( 1! C+ &

'& ( 1!& ) ⋅ "2 *C2 +'6! ( 1!-6 C+'16 ( 1!-6 C+ '! ( 1!-3 "+2  F  " = F 1 # F ! = 1 =)  3.6 =)+4

4  F 1 = 3.6 =), directed to left.

 F  " = 11. =), right.

523-2.  !.!2-g pith ball is s$spended reely. The ball is given a charge o 2!  C and placed !.6 " ro" a charge o !  C. #hat will be the initial acceleration o the pith ball%  F  =

'& ( 1!& ) ⋅ "2 *C2 +'2! ( 1!-6 C+'! ( 1!-6 C+ '!.6!! "+2 a=

 F  (

=

2.! ) -

2 ( 1! =g

4

 F / 2.! )

a = 1.2 ( 1!6 "*s

4

523-2&.   C charge is located 6 c" ro" an   C charge. t what point on a line ?oining the two charges will the res$ltant orce be ;ero%

 F 1 = F !

 µC  F1

= F 2 4

kq3 q1  &

2

=

kq3 q2 '6 − &+2

q!

6

 &

6 c" - & q

 µC q1

Physics, 6th Edition

Chapter 23. The Electric Force

 F1

= F2 4

q1

=

 & 2

q2 '6 − &+2 q1

'ake square root of both sides)  & =

q2

4

(2

 q   =  1  '6 − &+2  q2     µ C

'6 - (+ /

  µ C

'6 - (+4

( / !.!'6 - (+

*olving for &, we obtain) & = 2.& c"  fro( -C charge

523-3!.  charge o  nC is placed ! "" to the let o a 81 nC charge. #here sho$ld a third charge be placed i it is to e(perience a ;ero res$ltant orce%  F 1 = F !

+onsidering the sign of the charges and  their (agnitudes, the charge (ust be to the left of the  n+ charge as shown.  F1

= F 2 4

kq3 q1  &

2

=

kq3 q2 '! + &+

 &

q

= F2 4

 F1

2

q1

'ake square root of both sides)  & =

q2

*olving for &, we obtain)

q1  &

2

=

'! + (+ /

q2 '! + &+  nC 1 nC

2

-1 nC

 nC

(2

4

! ""

q1

q!

 q   =  1  '! + &+2  q2  

'! + (+4

( / !.6'! (+

& = 12 "" left of  nC charge.

523-31.  16-C charge is ! "" to the right o a &  C. #here sho$ld a third charge be  placed so that the res$ltant orce is ;ero%  F1 = F 2 4

 F1

kq3 q1  & 2

= F2 4

'ake square root of both sides)

= q1  &

2

kq3 q2

q1

'! − &+2

=

 &

=

q2 '! − &+ q1 q2

2

 F 1 = F !

& µC

4

(2

'! - (+ /

 &

q

! "" - (

q!

 q   =  1  '! − &+2  q2   &  µ C 16  µ C

'! - (+4

( / !.!'! - (+

*olving for &, we obtain) & = 3.3 ""  fro( &-C charge

6&

16 µC

Physics, 6th Edition

Chapter 23. The Electric Force

523-32. Two 3-g spheres are s$spended ro" a co""on point with two ! "" light sil= threads o negligible "ass. #hat charge "$st be placed on each sphere i their inal positions are ! "" apart%

T

2 ""

cos φ

=

'  =

(g 

! ""

sin φ 

=

!

4

φ  / 1.

!

sin 1.

(g 

4

 F

=

r2

=

&.6 ")4

q

Fr 2 =



'  & = &.6 ")4

 F  & / !: Σ 

  '&.6 ( 1!-3 )+'! ( 1! -3"+ 2 =

& ( 1!& ) " 2 *C 2

-

 F 

2 ""

' = 3!.& ")

'  & = ' cos 1.! / '3!.& ")+ cos 1.!4 kq 2

φ

 F 

'  y = ' sin φ   = (g 

'3 ( 1!-3 =g+'&. "*s2 +

! ""

(g 

 F = '  &

q = 1. nC



Critical Thi+i( uestios 523-33.  s"all "etal sphere is given a charge o !  C, and a second sphere  c" away is given a charge o 812  C. #hat is the orce o attraction between the"% 7 the two spheres are allowed to to$ch and are then again placed  c" apart, what new electric orce e(ists between the"% 7s it attraction or rep$lsion%  F  =

'& ( 1!& ) ⋅ "2 *C2 +'12 ( 1!-6 C+'! ( 1!-6 C+ '!.!! "+2

;

F = %75 N, attractio

When spheres touch,  + of charge are neutralized,

-12 µC

! µC !.! "

leaving ! + to be shared by two spheres, or  #1 C on each sphere. Now they are again separated)

1 µC

1 µC 1 µC

1 µC  F  =

'& ( 1! ) ⋅ " *C +'1 ( 1! C+'1 ( 1! C+ &

2

2

-6

'!.!! "+

!.! "

-6

2

F = 27% N, repulsio

!

4

Physics, 6th Edition

Chapter 23. The Electric Force

523-3. The total charge on two "etal spheres ! "" apart is !  C. 7 they repel each other  with a orce o !! ), what is the charge on each sphere% q1 # q! = ! C4  F

=

kq1q2 r

2

=

kq1 '! µ C - q1 + r

2

'!! )+'! ( 1!-3 "+2 '& ( 1! ) ⋅ " *C + &

2

2

F  =

kq1 '!µ + + − kq12 2



= '!  µ C+q1 − q12 ;

q!

q1

q! = ! C 8 q1 4

! ""

 Fr 2

4



= '!  µ C+q1 − q12

q12 – -$0 ! "0#% Cq1+ 222 ! "0#"2 C2 / 0

 Solve the quadratic equation with a / " , b = # $0 ! "0#% , and c = 222 ! "0#"2

q1 / .1

C

and q1 / 2.& C

 Now q! = ! C / q1 yields the following ) q! = !.0 C and q! = .1 C 'hus, one charge is .1 C and the other is 2.&  C

523-3. Fo$r s"all spheres are each given charges o 0 / 2!  C and placed at the corners o a s0$are with sides o length 6 c".
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