91751156 Solucionario Capitulo 23 Paul E Tippens

Chapter 23. The Electric

Chapter 23. The Electric Force

Force

Coulomb's Law

Coulomb's Law

23-1.

23-1. Two Two balls balls each haeach having a ving a charge charge o 3 o 3 C C are seare separated parated by 2! by 2! "". #"". #hat is hat is the othe orce orce o   rep$lsion between the"%

rep$lsion between the"%

& & 22 22 --66 --66 -3 -3 22 ''& & ( ( 11!! )) "" **CC ++''33 ( ( 11!! CC++''3 3 ( ( 11!! CC++ ''22! ! ( ( 11!! ""++  F   F 

==

⋅⋅

;; F =F =202 N202 N 23-2.

23-2. Two Two point charges point charges o o -3 a-3 and  nd  C C are 1are 12 "2 "" " apart in apart in a vac$$". a vac$$". #hat #hat is is the ethe electrostaticlectrostatic orce between the"%

orce between the"%

& & 22 22 --66 --66 -3 -3 22 ''& & ( ( 11!! )) "" **CC ++'' 3 ( 3 ( 11!! CC++''  ( ( 11!! CC++ ''12 12 ( ( 1!1! ""++  F   F 

==

⋅ ⋅

−−

;; F =F =750 N, attractio750 N, attractio 23-3.

23-3. n alpha particle n alpha particle consists o twconsists o two protons 'o protons 'qqee / 1.6 ( 1! / 1.6 ( 1!-1&-1& C+ and two ne$trons 'no charge+. C+ and two ne$trons 'no charge+.

#hat is the rep$lsive orce between two alpha particles separated by 2 n"% #hat is the rep$lsive orce between two alpha particles separated by 2 n"%

0

0αα / 2'1.6 ( 1! / 2'1.6 ( 1!-1&-1& C+ / 3.2 ( 1! C+ / 3.2 ( 1!-1&-1& C C

&

& 22 22 --11&& --11&&

-& -& 22 ''& & ( ( 11!! )) "" **CC ++''33..22 ( ( 11!! CC++''33..2 2 ( ( 11!! CC++ ''2.2.!! !! ( ( 1!1! ""++  F   F 

==

⋅⋅

;; F =F =2.30 ! "02.30 ! "0#"0#"0 _N N 23-.

23-. ss$"e that the radi$s o the electrons orbit aro$nd the proton in a ss$"e that the radi$s o the electrons orbit aro$nd the proton in a hydrogen ato" ishydrogen ato" is appro(i"ately .2 ( 1!

appro(i"ately .2 ( 1!-11-11 _{".  ". #hat is #hat is the electrostatic orce o the electrostatic orce o attraction%attraction%} &

& 22 22 --11&& --11&&

--1111 22 ''& & ( ( 11!! )) "" **CC ++''11..66 ( ( 11!! CC++'' 11..6 6 ( ( 11!! CC++ ''..2 2 ( ( 11!! ""++  F   F 

==

⋅ ⋅

−−

;; F =F = $.52 ! "0$.52 ! "0#$#$ N N 23-.

23-. #hat #hat is is the the separation o separation o two two - - C C charges i charges i the the orce o orce o rep$lsion betweerep$lsion between the" n the" isis 2!! )% 2!! )% & & 22 22 --66 22  ''& & ( ( 11!! )) "" **CC ++''   ( ( 11!! CC++ 2!! ) 2!! ) kqq kqq r  r   F   F 

⋅ ⋅

−−

=

=

==

;; r =r =2%.$ mm2%.$ mm 6! 6!

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

2 2 2 2 & 2 2 '&! )+'!.!3! "+ 4 & ( 1! ) " *C kq Fr    F q r k 

=

=

=

⋅

4 q = &.&! C

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%

& 2 2 -6 -6

-3 2

'& ( 1! ) " *C +' 6 ( 1! C+'1! ( 1! C+ '! ( 1! "+

 F 

=

⋅

−

; F = 33$ N, attractio

When spheres touch, 6 Cof charge are neutralized, leaving  C to be shared by two spheres, or 

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

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

=

⋅

; 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%

2 2 2 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 4 4 r r   F F r F r F F F  r r r 

     

∝

=

=

_{   }  

=

     

4 r 1 =  r ! 2 2 2 1 1 2 3 & 4 r   F F F  r 



 

=

_

_ 

=



 

F 2 = 5&00 N 61 2

µ

C 2

µ

C -6

µ

C 1!

µ

C !.! " 2

µ

C 2

µ

C !.! "

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%

& 2 2 -6 -6

13 -3 2

'& ( 1! ) " *C +'6! ( 1! C+'3 ( 1! C+ '3! ( 1! "+

 F 

=

⋅

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

& 2 2 -6 -6 23 -3 2 '& ( 1! ) " *C +'2! ( 1! C+'3 ( 1! C+ '3! ( 1! "+  F 

=

⋅

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%

& 2 2 -6 -6

13 -3 2

'& ( 1! ) " *C +'36 ( 1! C+'1! ( 1! C+ '! ( 1! "+

 F 

=

⋅

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

& 2 2 -6 -6 23 -3 2 '& ( 1! ) " *C +'22 ( 1! C+'1! ( 1! C+ '! ( 1! "+  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%

& 2 2 -6 -6 13 -3 2 '& ( 1! ) " *C +'36 ( 1! C+'12 ( 1! C '6! ( 1! "+  F 

=

⋅

& 2 2 -6 -6 23 -3 2 '& ( 1! ) " *C +'22 ( 1! C+'12 ( 1! C+ '2! ( 1! "+  F 

=

⋅

4

 %oth to right, so F  " = F 1# F ! = 1!! )  &! )4  F = !2! ) , rightward.

62  F  1  F ! q  q_! q 1 -3

µ

C 2!

µ

C 6!

µ

C 6! ""  F ₁  F  ! q  q_! q 1 1!

µ

C -22

µ

C 36

µ

C ! "" q = 12

µ

C -22

µ

C 6! ""  _F  1  F  ! q_! q 1 36

µ

C ! ""

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%

& 2 2 -6 -6 13 -3 2 '& ( 1! ) " *C +' ( 1! C+'2 ( 1! C+ '2! ( 1! "+  F 

=

⋅

& 2 2 -6 -6 23 -3 2 '& ( 1! ) " *C +'2 ( 1! C+'6 ( 1! C+ '2 ( 1! "+  F 

=

⋅

4

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

2 2 '3! ""+ '! ""+ .3 ""  s

 =

+

=

! ! "" tan 4 &.! 3! "" φ

=

φ 

=

& 2 2 -6 -6 13 -3 2 '& ( 1! ) " *C +'6 ( 1! C+'12 ( 1! C+ '.3 ( 1! "+  F 

=

⋅

 F 1 = 2!33 ), &.!! ) o # & 2 2 -6 -6 23 -3 2 '& ( 1! ) " *C +'16 ( 1! C+'12 ( 1! C+ '! ( 1! "+  F 

=

⋅

/ 6&1 ), $pward.

F( / ! 8 F13 cos &.!! / -'2!33 )+ cos &! 4 F( / -1! )

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

2 2 23 )

' 1! )+ '23 )+ 26! )4 tan

-1! )

 F 

=

−

+

=

θ 

=

4 / 66.! _{) o #.}

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

63 q = -2

µ

C 6

µ

C 2! ""  F  1  F _! q ! q 1 -

µ

C  "" 2 "" φ  φ   s ! "" q_ = $12

µ

C 16

µ

C 3! ""  F  1  F _! q_! q 1 6

µ

C

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%

2 2 ' ""+ '! ""+ &1. ""  s

 =

+

=

! ! "" tan 4 6!.6  "" φ

=

φ 

=

& 2 2 -6 -6 13 -3 2 '& ( 1! ) " *C +'6! ( 1! C+'! ( 1! C+ '&1. ( 1! "+  F 

=

⋅

 F 1 = 26 ), 6!.6! ) o # & 2 2 -6 -6 23 -3 2 '& ( 1! ) " *C +'! ( 1! C+'! ( 1! C+ ' ( 1! "+  F 

=

⋅

/ & ), rightward .

F( / 8 F13 cos 6!.6!  F23/ - '26 )+ cos 6!.6! & ) 4 F( / 632 )

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

2 2 223 )

'632 )+ '223 )+ 4 tan

632 )

 F 

=

+

θ 

=

 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$ilateral triangle, ! "" on each side as described by Fig. 23-1. #hat are the "agnit$de and direction o the res$ltant orce on the  C charge%

& 2 2 -6 -6 21 -3 2 '& ( 1! ) " *C +' ( 1! C+' ( 1! C+ '! ( 1! "+  F 

=

⋅

F !1 = .! ), 6!! < o E & 2 2 -6 -6 31 -3 2 '& ( 1! ) " *C +'2 ( 1! C+' ( 1! C+ '! ( 1! "+  F 

=

⋅

F 1 = 22. ), 6!! ) o E4 F( / '22. )+ cos 6!! ' )+ cos 6!! / 33. )

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

6  F  ! φ   s ! "" q_ = $!

µ

C -!

µ

C  ""  F  1  F _! q ! q 1 6!

µ

C 6!! 6!! ! "" φ  ! "" q !/ -

µ

C 2

µ

C ! ""  F  1  F _! φ  q_ q 1= 

µ

C

523-1. 'Cont.+ '33. )+2 ' 1&. )+ / 3&.! )4 tan2 -1&. )4 3!! 33. )

 F 

=

+

−

θ

=

θ 

=

−

9es$ltant electric orce:  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  )%

& 2 2 -6 2  '& ( 1! ) " *C +' ( 1! C+ .!! ) kqq r   F 

⋅

=

=

; 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%

& 2 2 -& 2 -6  '& ( 1! ) " *C +' ( 1! C+ 6! ( 1! ) kqq r   F 

⋅

=

=

; 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%

2 2 2 2 & 2 2 ' )+'!.!6! "+ 4 & ( 1! ) " *C kq Fr    F q r k 

=

=

=

⋅

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

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

=

⋅

4  F = .12 ( 1! ), attraction 6

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.

& 2 2

2

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

 F 

=

⋅

4  F = & ( 1!& _{) @}

'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 !! )%

2 2 2 2 & 2 2 '!! )+'!.!! "+ 4 & ( 1! ) " *C kq Fr    F q r k 

=

=

=

⋅

4 q = 3 nC -& -1& 1 e 3 ( 1! C 1.6 ( 1! C q

=



_

 

_ 



 

4 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%

& 2 2 -& -&

1 -3 2

'& ( 1! ) " *C +'! ( 1! C+'12 ( 1! C+ ' ( 1! "+

 F 

=

⋅

& 2 2 -& -&

2 -3 2

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

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

=

⋅

F1 / ! "), to right  66 6 nC ! ""  F  1  F  ! q ! q 1 -! nC  "" q = $12 nC 

µ

C 2 c" -& nC 6 c"  F  1  F  ! q ! q 1 2

µ

C

& 2 2 -6 -& 2 -2 2 '& ( 1! ) " *C +' ( 1! C+'& ( 1! C+ ' ( 1! "+  F 

=

⋅

4  F ! / 63.3 ") , to right   "esultant force) 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 !! )%

2 2 2 2 & 2 2 '!! )+'!.!3 "+ 4 & ( 1! ) " *C kq Fr    F q r k 

=

=

=

⋅

4 q = 21.2 C -6 -1& 1 e 21.2 ( 1! C 1.6 ( 1! C q

=



_

 

_ 



 

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%

& 2 2 -& -6 2 -2 2 '& ( 1! ) " *C +' ( 1! C+'1! ( 1! C+ ' ( 1! "+  F 

=

⋅

4  F ! / 22 ") !.22 ) !.!! =g  F  a (

= =

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 4 .2 6! "" φ

=

φ 

=

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

=

⋅

F ! = 2!.! ), .2!< o E & 2 2 -6 -6 1 -3 2 '& ( 1! ) " *C +'2 ( 1! C+' ( 1! C+ '6! ( 1! "+  F 

=

⋅

4  F 1 = 2!.! ), .2! < o # 6 φ   F ₁ φ  ! "" φ  6! "" q_!/ -

µ

C -

µ

C 6! ""  F  ! φ  q  q 1= 2

µ

C

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% & 2 2 -6 -6 1 -3 2 '& ( 1! ) " *C +'6! ( 1! C+'2 ( 1! C+ '3! ( 1! "+  F 

=

⋅

 F 1 = 1 =), directed to the right 

& 2 2 -6 -6 2 -3 2 '& ( 1! ) " *C +'6! ( 1! C+'16 ( 1! C+ '! ( 1! "+  F 

=

⋅

4  F 1 = 3.6 =), directed to left.  F  "= F 1 # F ! = 1 =)  3.6 =)+4  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%

& 2 2 -6 -6 2 '& ( 1! ) " *C +'2! ( 1! C+'! ( 1! C+ '!.6!! "+  F 

=

⋅

4  F / 2.! ) - 2.! ) 2 ( 1! =g  F  a (

= =

4 a = 1.2 ( 1!6 _"*s

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%

3 1 3 2 1 24 2 2 '6 + kq q kq q  F F   & &

=

=

−

6 ! "" q = 6!

µ

C 16

µ

C 3! ""  F  1  F _! q_! q 1 2

µ

C ! "" q  q₁ 6 c" - &  & 

µ

C 

µ

C  F 1 = F ! q !

2 2 1 2 1 1 2 2 2 2 4 4 ( '6 + '6 + q q q  F F &  & & q

  

=

=

=

_{ } 

−

−

_{ } 

'ake square root of both sides) 1

2  C '6 - (+ / '6 - (+4 ( / !.!'6 - (+  C q  & q  µ   µ 

=

*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%

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

3 1 3 2 1 24 2 2 '! + kq q kq q  F F   & &

=

=

+

2 2 1 2 1 1 2 2 2 2 4 4 ( '! + '! + q q q  F F &  & & q

  

=

=

=

_{ } 

+

+

_{ } 

'ake square root of both sides) 1

2  nC '! (+ / '! (+4 ( / !.6'! (+ 1 nC q  & q

=

+

+

*olving for &, we obtain) & = 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%

3 1 3 2 1 24 2 2 '! + kq q kq q  F F   & &

=

=

−

2 2 1 2 1 1 2 2 2 2 4 4 ( '! + '! + q q q  F F &  & & q

  

=

=

=

_{ } 

−

−

_{ } 

'ake square root of both sides) 1

2 & C '! - (+ / '! - (+4 ( / !.!'! - (+ 16 C q  & q  µ   µ  =

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

6& q  q! ! ""  & -1 nC  nC  F  1 = F ! q 1 ! "" - ( q 1 q!  & 16

µ

C &

µ

C  F 1 = F ! q_

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% ! 2 "" cos 4 / 1. ! "" φ

=

φ  '  y = ' sin φ   = (g  -3 2 ! '3 ( 1! =g+'&. "*s + sin sin 1. (g  '  φ 

=

=

4 ' = 3!.& ")

'  & = ' cos 1.! / '3!.& ")+ cos 1.!4 '  & = &.6 ")4 Σ  F  & / !:  F = '  &

2 2 -3 -3 2 2 & 2 2 '&.6 ( 1! )+'! ( 1! "+ &.6 ")4 & ( 1! ) " *C kq Fr    F q r k  = = = = ⋅ - 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%

& 2 2 -6 -6

2

'& ( 1! ) " *C +'12 ( 1! C+'! ( 1! C+ '!.!! "+

 F 

=

⋅

; F =%75 N, attractio

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

leaving ! + to be shared by two spheres, or 

#1 C on each sphere. Now they are again separated)

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

=

⋅

4 F =27% N, repulsio !  F   F  (g  _{2 ""} (g 

φ

T _{! ""} 1

µ

C 1

µ

C !

µ

C -12

µ

C !.! " 1

µ

C 1

µ

C !.! "

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 q! = ! C 8q1 2 1 2 1 1 1 1 2 2 2 '! C - + '! + 4 kq q kq q kq + kq  F F  r r r   µ µ 

−

=

=

=

4 2 2 1 1 '! C+  Fr  q q k 

=

 µ 

−

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

⋅

=

 µ  q

−

q ; q1 2 _{–-$0 ! "0}#% _Cq 1+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". <how that the res$ltant orce on each charge has a "agnit$de e0$al to 1&1 ). #hat is the direction o the orce% #hat will change i the charges are each q / -2! C% ' ike charges repel, so sign doesn2t (atter .+

2 2 '6 c"+ '6 c"+ . c"  "

 =

+

=

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

=

⋅

& 2 2 -6 -6 2 -2 2 '& ( 1! ) " *C +'2! ( 1! C+'2! ( 1! C+ '. ( 1! "+  F 

=

⋅

 F 1 = 1!!! )4  F ! = !! )4  For a square, the angle

φ

 / !

 F  & = '1!!! )+  '!! )+ cos !  ! / 13 )4  F  y = 1!!! )  '!! )+ sin ! / 13 )

1

φ

q 1 ! "" q ! q _q q 6 c" _{6 c"} 6 c" 6 c"  F _!  F  1  F  1  "

φ

2 2

'13 )+ '13 )+ 4

 F 

 =

+

F = 1&1 ), ! away fro( center.

523-36. Two charges q1 and q! are separated by a distance r. They e(perience a orce F  at this

distance. 7 the initial separation is decreased by only ! "", the orce between the two charges is do$bled. #hat was the initial separation%

1 2 1 2 2 2 2 ' ! ""+ kq q kq q  & = &− 4 2 2 2 1 ' ! ""+  &

=

&

−

 &! _{/ 2'( 8 ! ""+}2 _{Take square root of both sides)}

 & / 1.1'( 8 ! ""+4 ( / 13 ""

523-3. Two -g pith balls are s$spended ro" sil= threads 6! c" long and attached to a co""on  point. #hen the spheres are given e0$al a"o$nts o negative charge, the balls co"e to

rest 3! c" apart. Calc$late the "agnit$de o the charge on each pith ball.

! !.1 " cos 4 / . !.6! " φ

=

φ  '  y = ' sin φ   = (g  -3 2 ! ' ( 1! =g+'&. "*s + sin sin . (g  '  φ 

=

=

4 ' = 1.! ")

'  & = ' cos 1.! / '1.! ")+ cos .!4 '  & = 2!.2 ")4 Σ  F  & / !:  F = '  &

2 2 - 2 2 & 2 2 '2.!2 ( 1! )+'!.3! "+ 2!.2 ")4 & ( 1! ) " *C kq Fr    F q r k 

=

=

=

=

⋅

- q = $! nC 2  F   F  (g  _{!.1 "} (g 

φ

T _{!.6 "} !F   F   F  q₁ ( 8 ! "" q_! q₁ ( q_! !F