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 electrons 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$ced to one-third o its original distance, what is the new orce o attraction%
F
∝
1 r
2
2
4
F1r12
r12 r F2 = F1 2 = F 1 1 4 r2 r 2
= F2 r22 4 2
3r F2 = F1 2 = & F 1 4 r 2
61
F 2 = 5&00 N
r 1 = r !
Physics, 6th Edition
Chapter 23. The Electric Force
The esultat Electrostatic Force 23-&. 6! C charge is placed 6! "" to the let o a 2! C charge. #hat is the res$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 " = $1.! ( 1! ), let.
23-1!. point charge o 36 C is placed ! "" to the let o a second point charge o -22 C. #hat orce is e(erted on third charge o 1! C located at the "idpoint% F 13
=
'& ( 1!& ) ⋅ "2 *C2 +'36 ( 1!-6 C+'1! ( 1!-6 C+ '! ( 1!-3 "+2
36 µC
! "" 1! µC
q1
q
F 1 = 2!2 ), directed to the right F 23
=
'& ( 1!& ) ⋅ "2 *C2 +'22 ( 1!-6 C+'1! ( 1!-6 C+ '! ( 1!-3 "+2
-22 µC F 1
q!
F ! 4 F 1 = 123 ), directed to right.
F " = F 1 # F ! = 2!2 ) 123 )4
F " = 326! ), left.
23-11. For Proble" 23-1!, what is the res$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$ltant orce on a -2 C charge that is 2! "" to the right o the - C charge% F 13
F 23
= =
'& ( 1!& ) ⋅ "2 *C2 +' ( 1!-6 C+'2 ( 1!-6 C+
'2 ( 1!-3 "+2
q = -2 µC
- µC
'2! ( 1!-3 "+2 '& ( 1!& ) ⋅ "2 *C2 +'2 ( 1!-6 C+'6 ( 1!-6 C+
""
q1
2 ""
2! ""
F 1
F !
4
%oth to right, so F " = F 1 # F ! = 36! ) 1. )4
6 µC q!
F = ) , rightward
523-13. 6-C charge is locate 3! c" to the let o a 16- C charge. #hat is the res$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 ), $pward.
F( / ! 8 F13 cos &.!! / -'2!33 )+ cos &! 4
F( / -1! )
Fy / F23 F13 sin &.!! / 6&1 ) '2!33 )+ sin & !4 F = '−1! )+2 + '23 )+2
F ! φ
F 1 = 2!33 ), &.!! ) o # &
16 µC
= 26! )4
tanθ =
23 ) -1! )
Fy / 23 ) 4
/
9es$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 = 26 ), 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! ! / '26 )+ sin 6!.6! 4 223 ) 632 )
F !
""
φ q = $! µC
/ & ), rightward .
F( / 8 F13 cos 6!.6! F23 / - '26 )+ cos 6!.6! & ) 4
F = '632 )+2 + '223 )+2 4 tanθ =
F !
F( / 632 )
Fy / 223 )
4 F " = &! ), / 16.3! ) o E.
523-1. Three point charges 01 / C, 02 / - C, and 03 / 2 C are placed at the corners o an e0$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$de o each charge%
F
=
kq 2 r
2
4 q=
Fr 2 k
' )+'!.!6! "+2
=
& ( 1! ) ⋅ " *C &
2
2
4
q = .3 C
23-1&. >ne ob?ect contains an e(cess o ( 1!1 electrons and another has a deiciency o ( 1!1 electrons. #hat is the orce each e(erts on the other i the ob?ects are 3! "" apart% 7s it attraction or rep$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$lsive orce o one !! )%
=
F
kq 2 r
2
4 q=
Fr 2 k
'!! )+'!.!! "+2
=
& ( 1! ) ⋅ " *C &
1e 4 -1& 1.6 ( 1! C
q = 3 ( 1!-& C
2
2
4
q = 3 nC
q = .2 ( 1!12 electrons
23-22. 8!-nC charge is placed ! "" to the let o a 6-nC charge. #hat is the res$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 let 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$ltant orce is !! )%
=
F
kq 2 r
2
Fr 2
4 q=
'!! )+'!.!3 "+2
=
k
& ( 1! ) ⋅ " *C &
2
1e 4 -1& 1.6 ( 1! C
q = 21.2 ( 1! C -6
2
q = 21.2 C
4
q = 1.33 ( 1!1 electrons
23-2. -nC charge is placed on a -g sphere that is ree to "ove. i(ed 1!- C point charge is c" away. #hat is the initial acceleration o the - C charge% F 2
=
'& ( 1!& ) ⋅ "2 *C2 +' ( 1!-& C+'1! ( 1!-6 C+ ' ( 1!-2 "+2 a=
F (
!.22 )
=
!.!! =g
4
F ! / 22 ")
a = 6.2 "*s2
523-26. #hat is the res$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 let 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 =
k
' & = &.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( uestios 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
r
= '! µ C+q1 − q12 ;
q!
q1
q! = ! C 8 q1 4
! ""
Fr 2
4
k
= '! µ C+q1 − q12
q12 – -$0 ! "0#% Cq1+ 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$are with sides o length 6 c".
q
6 c"
6 c"
"
F !
F 1
φ F 1
'& ( 1!& ) ⋅ "2 *C2 +'2! ( 1!-6 C+'2! ( 1!-6 C+ '6 ( 1!-2 "+2 '& ( 1!& ) ⋅ "2 *C2 +'2! ( 1!-6 C+'2! ( 1!-6 C+ -2
q
2
'. ( 1! "+
6 c"
φ q 6 c"
F 1 = 1!!! )4
F ! = !! )4 For a square, the angle φ / !
F & = '1!!! )+ '!! )+ cos ! ! / 13 )4 F y = 1!!! ) '!! )+ sin ! / 13 )
1
Physics, 6th Edition
Chapter 23. The Electric Force F =
'13 )+
2
+ '13 )+2 4
F = 1&1 ), !
away fro( center.
523-36. Two charges q1 and q! are separated by a distance r. They e(perience a orce F at this distance. 7 the initial separation is decreased by only ! "", the orce between the two charges is do$bled. #hat was the initial separation% 2kq1q2 &
2
=
kq1q2 ' & ! ""+ −
2
4
2 &
2
=
1 ' & − ! ""+
F
F
(
q1
q!
2
( 8 ! "" !
& / 2'( 8 ! ""+
2
Take square root of both sides)
& / 1.1'( 8 ! ""+4
q1
!F q!
!F
( / 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. cos φ
=
!.1 " !.6! "
4
!
φ / .
' y = ' sin φ = (g
T
!.6 "
φ
F
' =
(g sin φ
=
' ( 1!-3 =g+'&. "*s2 + sin .!
4
(g
' & = ' cos 1.! / '1.! ")+ cos .!4
F
=
kq 2 r2
= 2!.2 ")4
q=
' = 1.! ")
Fr 2 k
=
' & = 2!.2 ")4
F & / !: Σ
'2.!2 ( 1!- )+'!.3! "+2 & ( 1!& ) ⋅ "2 *C2
2
!.1 "
-
F
(g
F = ' &
q = $! nC
