# 91751156 Solucionario Capitulo 23 Paul E Tippens

13

(1)

## Force

### 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!

(2)

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

C 2

C -6

C 1!

C !.! " 2

C 2

## µ

C !.! "

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### 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

C 2!

C 6!

## µ

C 6! ""  F 1  F  ! q  q! q 1 1!

C -22

C 36

C ! "" q = 12

C -22

## µ

C 6! ""  F  1  F  ! q! q 1 36

## µ

C ! ""

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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

C 6

## µ

C 2! ""  F  1  F ! q ! q 1 -

## µ

C  "" 2 "" φ  φ   s ! "" q = \$12

C 16

## µ

C 3! ""  F  1  F ! q! q 1 6

## µ

C

(5)

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 = \$!

C -!

## µ

C  ""  F  1  F ! q ! q 1 6!

## µ

C 6!! 6!! ! "" φ  ! "" q !/ -

C 2

## µ

C ! ""  F  1  F ! φ  q q 1= 

## µ

C

(6)

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

(7)

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

(8)

& 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!/ -

C -

## µ

C 6! ""  F  ! φ  q  q 1= 2

## µ

C

(9)

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!

C 16

## µ

C 3! ""  F  1  F ! q! q 1 2

## µ

C ! "" q  q1 6 c" - &  & 

C 

## µ

C  F 1 = F ! q !

(10)

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

C &

## µ

C  F 1 = F ! q

(11)

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

T ! "" 1

C 1

C !

C -12

C !.! " 1

C 1

## µ

C !.! "

(12)

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

(13)

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

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