-
8/12/2019 Cap21 Prob
1/92
!
Chapter 21
The Electric Field 1: Discrete Charge Distributions
Conceptual Problems
! " #$%&'() *+& ',-.,)&/ ,0 *(,-) 123'2 *+&
',-.,)&/ ,0 '2*+4&/
.*+(3'5&) 6.+,(,7) *7/ &5&'(+,7)89 2,1&:&+;
1& +*+&5< ,$)&+:& (2& &00&'() ,0
(2&
&5&'(+,)(*(3' 0,+'&= >?.5*37 12< 1& /, 7,(
,$)&+:& (2&)& &00&'()=
Determine the Concept @2& 7&( '2*+4& ,7 5*+4&
,$%&'() 3) *51*
-
8/12/2019 Cap21 Prob
2/92
L2*.(&+ I!I
Determine the Concept
6a8 L,77&'( (2& -&(*5 ).2&+& (, 4+,C7/9
$+374 (2& 37)C5*(374 +,/ 7&*+ (2& -&(*5
).2&+& *7/ /3)',77&'( (2& ).2&+& 0+,-
4+,C7/9 (2&7 +&-,:& (2& 37)C5*(374 +,/=
@2& ).2&+& 1355 $& 7&4*(3:&5<
'2*+4&/=
6b8 U+374 (2& 37)C5*(374 +,/ 37 ',7(*'( 13(2 (2&
-&(*5 ).2&+&9 ),-& ,0 (2&
.,)3(3:&'2*+4& ,7 (2& +,/ 1355 $&
(+*7)0&++&/ (, (2& -&(*5 ).2&+&=
5 6a8 @1, .,37( .*+(3'5&) (2*( 2*:& '2*+4&) ,0 GVq
*7/ HWq *+& )&.*+*(&/
$< /3)(*7'& d= X)& 03&5/ 537&) (, /+*1 *
:3)C*53A*(3,7 ,0 (2& &5&'(+3' 03&5/ 37 (2&
7&342$,+2,,/ ,0 (23) )
-
8/12/2019 Cap21 Prob
3/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) W
7 D )3-.5& /&-,7)(+*(3,7 ,0 &5&'(+,)(*(3'
*((+*'(3,7 '*7 $& /,7& )3-.55&'(+3' Y3&5/ !Z
P3)'+&(& L2*+4& P3)(+3$C(3,7) a
Picture the Problem 6a8 E& '*7 C)&
(2& 4&,-&(+< ,0 *7 &QC35*(&+*5
(+3*745&(, &?.+&))x!*7/xI37 (&+-) ,0 (2&
)3/&
a ,0 (2& (+3*745&= 6b8 @2& &5&'(+3'
03&5/
,7 (2& x *?3) * /3)(*7'& x 0+,- (2&
,+3437 ,7 (2& 37(&+:*5 \ x` xI 3) (2&
)C.&+.,)3(3,7 ,0 (2& &5&'(+3' 03&5/)
/C&(, (2& .,37( '2*+4&) qI; qW; *7/ qV5,'*(&/ *(
(2& :&+(3'&) ,0 (2& (+3*745&=
q+
q+
q+
a21
0
a21
1x 2xx
P
3
4
30
2
6a8 D..5< (2& ^
-
8/12/2019 Cap21 Prob
8/92
L2*.(&+ I!d
>?.+&)) (2& &5&'(+3' 03&5/ *(P/C&
(,
qWZ PP
P
P r
kq
r
kq;WW
;W
W;WI
;W
WW %%"
12&+&
'&'&% ccc\c\I!
I!
;W axaxP
*7/
IV!II
I!I
;W \\ axaxrP
OC$)(3(C(374 0,+ qW; $%;W
,*7/ rW;P
:*5C*(374 \$"
-
8/12/2019 Cap21 Prob
9/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) e
15 D -,5&'C5& 2*) * /3.,5& -,-&7( 43:&7
$5&'(+3' 03&5/ 537&) 7&:&+
37(&+)&'(=
6d8 D55 -,5&'C5&) 2*:& /3.,5& -,-&7() 37
(2& .+&)&7'& ,0 *7 &?(&+7*5
&5&'(+3'
03&5/=
6a8 Y*5)&= @2& /3+&'(3,7 ,0 (2& 03&5/ 3)
(,1*+/ * 7&4*(3:& '2*+4&=
6b8 Y*5)&= @2& /3+&'(3,7 ,0 (2& &5&'(+3'
0,+'& *'(374 ,7 * .,37( '2*+4& /&.&7/) ,7
(2& )347 ,0 (2& .,37( '2*+4&=
6c8 Y*5)&= >5&'(+3' 03&5/ 537&)
37(&+)&'( *7< .,37( 37 ).*'& ,''C.3&/ $< *
.,37(
'2*+4&=
6d8 @+C&= D7 &5&'(+3' 03&5/ .*+(3*55<
.,5*+3A&) (2& -,5&'C5&)9 +&)C5(374 37
(2&)&.*+*(3,7 ,0 (2&3+ '2*+4&) *7/ (2&
'+&*(3,7 ,0 &5&'(+3' /3.,5& -,-&7()=
17 [SSM] @1, -,5&'C5&) 2*:& /3.,5& -,-&7()
,0 &QC*5 -*473(C/&=@2& /3.,5& -,-&7() *+&
,+3&7(&/ 37 :*+3,C) ',7034C+*(3,7) *) )2,17 37 Y34C+&
I!T
W]= P&(&+-37& (2& &5&'(+3'T03&5/
/3+&'(3,7 *( &*'2 ,0 (2& 7C-$&+&/
5,'*(3,7)=
>?.5*37
-
8/12/2019 Cap21 Prob
10/92
L2*.(&+ I!!\
Estimation and Approximation
18 >)(3-*(& (2& 0,+'& +&QC3+&/ (, $37/
(2& (1, .+,(,7) 37 (2& B& 7C'5&C)(,4&(2&+=
BJR@Z f,/&5 (2& .+,(,7) *) .,37( '2*+4&)= You will need
to have an
estimate of the distance between them.
Picture the ProblemU&'*C)& (2& 7C'5&C) 3) 37
&QC353$+3C-; (2& $37/374 0,+'&
-C)( $& &QC*5 (, (2& &5&'(+,)(*(3' 0,+'&
,0 +&.C5)3,7 $&(1&&7 (2& .+,(,7)=
D..5< \*
(, * .+,(,7 (, ,$(*37Z \(3'&5&'(+,)(*$37/374 FF
X)374 L,C5,-$M) 5*1; )C$)(3(C(&
0,+ (3'&5&'(+,)(*F Z\(3'&5&'(+,)(*I
I
Fr
kq
I
I
$37/374r
kqF
D))C-374 (2& /3*-&(&+ ,0 (2& B& 7C'5&C)
(, $& *..+,?3-*(&5< !\!]
-; )C$)(3(C(&
7C-&+3'*5 :*5C&) *7/ &:*5C*(&
(3'&5&'(+,)(*F Z
NRI=\-!\
L!\_\I=!Lg-R!\edd=dI!]
I!eIIe
$37/374
F
19 D .,.C5*+ '5*))+,,- /&-,7)(+*(3,7 ',7)3)() ,0 +C$$374 *
.5*)(3' +,/13(2 0C+ (, 43:& (2& +,/ '2*+4&; *7/
(2&7 .5*'374 (2& +,/ 7&*+ *7 &-.(< ),/* '*7
(2*( 3) ,7 3() )3/& 6Y34C+& I!TW_8= >?.5*37 12<
(2& '*7 1355 +,55 (,1*+/ (2& +,/=
Determine the Concept U&'*C)& (2& '*7 3)
4+,C7/&/; (2& .+&)&7'& ,0 (2&
7&4*(3:&5< '2*+4&/ .5*)(3' +,/ 37/C'&) *
.,)3(3:& '2*+4& ,7 3(= @2& .,)3(3:&
'2*+4&) 37/C'&/ ,7 (2& '*7 *+& *((+*'(&/;
:3* (2& L,C5,-$ 37(&+*'(3,7; (, (2&
7&4*(3:& '2*+4&) ,7 (2& .5*)(3' +,/= X753N&
'2*+4&) *((+*'(; ), (2& '*7 1355 +,55(,1*+/ (2&
+,/=
20 O.*+N) 37 *3+ ,''C+ 12&7 3,7) 37 (2& *3+ *+&
*''&5&+*(&/ (, * )C'2 *2342 ).&&/ $< *7
&5&'(+3' 03&5/ (2*( 12&7 (2&< 3-.*'( ,7
7&C(+*5 4*) -,5&'C5&) (2&
7&C(+*5 -,5&'C5&) $&',-& 3,7)= J0 (2&
&5&'(+3' 03&5/ )(+&74(2 3) 5*+4& &7,C42;
(2&
3,73A&/ ',553)3,7 .+,/C'() *+& (2&-)&5:&)
*''&5&+*(&/ *7/ .+,/C'& -,+& 3,7) ,7
3-.*'(; *7/ ), 0,+(2= @23) *:*5*7'2& ,0 3,7) 3) 12*( 1&
'*55 * ).*+N= 6a8 D))C-&(2*( *7 3,7 -,:&); ,7
*:&+*4&; &?*'(5< ,7& -&*7 0+&& .*(2
(2+,C42 (2& *3+ $&0,+&
23((374 * -,5&'C5&= J0 (2& 3,7 7&&/) (,
*'QC3+& *..+,?3-*(&5< !=\ &h ,0 N37&(3'
&7&+4< 37 ,+/&+ (, 3,73A& * -,5&'C5&;
&)(3-*(& (2& -373-C- )(+&74(2 ,0
(2&&5&'(+3' 03&5/ +&QC3+&/ *( )(*7/*+/ +,,-
.+&))C+& *7/ (&-.&+*(C+&= D))C-& (2*(
(2&
'+,))T)&'(3,7*5 *+&* ,0 *7 *3+ -,5&'C5& 3) *$,C(
\=!\ 7-I= 6b8 B,1 /,&) (2&
)(+&74(2 ,0 (2& &5&'(+3' 03&5/ 37 6a8
/&.&7/ ,7 (&-.&+*(C+&F 6c8 B,1 /,&)
(2&)(+&74(2 ,0 (2& &5&'(+3' 03&5/ 37 6a8
/&.&7/ ,7 .+&))C+&F
-
8/12/2019 Cap21 Prob
11/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) !!
Picture the ProblemE& '*7 C)& (2& /&0373(3,7 ,0
&5&'(+3' 03&5/ (, &?.+&)) E 37
(&+-) ,0 (2& 1,+N /,7& ,7 (2& 3,73A374
&5&'(+,7) *7/ (2& /3)(*7'& (2&<
(+*:&5
$&(1&&7 ',553)3,7)= E& '*7 C)& (2&
3/&*5T4*) 5*1 (, +&5*(& (2& 7C-$&+
/&7)3(< ,0
-,5&'C5&) 37 (2& 4*) *7/ (2& )'*((&+374
'+,))T)&'(3,7 (, (2& -&*7 0+&& .*(2
*7/; 2&7'&; (, (2& &5&'(+3' 03&5/=
6a8 D..5< ',7)&+:*(3,7 ,0 &7&+4<
(,+&5*(& (2& 1,+N /,7& ,7 (2&
&5&'(+,7)
$< (2& &5&'(+3' 03&5/ (, (2& '2*74&
37
(2&3+ N37&(3' &7&+4
-
8/12/2019 Cap21 Prob
12/92
L2*.(&+ I!!I
Charge
21 D .5*)(3' +,/ 3) +C$$&/ *4*37)( * 1,,5 )23+(;
(2&+&$< *'QC3+374 * '2*+4&
,0 H\=d\ L= B,1 -*7< &5&'(+,7) *+&
(+*7)0&++&/ 0+,- (2& 1,,5 )23+( (, (2& .5*)(3'
+,/F
Picture the Problem @2& '2*+4& *'QC3+&/ $<
(2& .5*)(3' +,/ 3) *7 37(&4+*5
7C-$&+ ,0 &5&'(+,73' '2*+4&); (2*( 3); q[
n&6e8=
b&5*(& (2& '2*+4& *'QC3+&/ $< (2&
.5*)(3' +,/ (, (2& 7C-$&+ ,0
&5&'(+,7) (+*7)0&++&/ 0+,- (2& 1,,5
)23+(Z
enq & e
qn
&
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(& n&Z
&5&'(+,7)!\\=]
&5&'(+,7L!\!=_\I
Ld\=\
!I
!e
&
n
22 D '2*+4& &QC*5 (, (2& '2*+4& ,0 D:,4*/+,M)
7C-$&+ ,0 .+,(,7)
6ND[ _=\I !\IW
8 3) '*55&/ *faraday= L*5'C5*(& (2& 7C-$&+ ,0
',C5,-$) 37 *
0*+*/*
'*+$,7F
Picture the Problem E& '*7 037/ (2& 7C-$&+ ,0
',C5,-$) ,0 .,)3(3:& '2*+4&
(2&+& *+& 37 !=\\ N4 ,0 '*+$,7 0+,- enQ L_ ;
12&+& nL3) (2& 7C-$&+ ,0 *(,-) 37
!=\\ N4 ,0 '*+$,7 *7/ (2& 0*'(,+ ,0 _ 3) .+&)&7( (,
*'',C7( 0,+ (2& .+&)&7'& ,0 _
.+,(,7) 37 &*'2 *(,-= E& '*7 037/ (2& 7C-$&+ ,0
*(,-) 37 !=\\ N4 ,0 '*+$,7 $5&'(+3' Y3&5/ !Z
P3)'+&(& L2*+4& P3)(+3$C(3,7) !W
>?.+&)) (2& .,)3(3:& '2*+4& 37 (&+-)
,0 (2& &5&'(+,73' '2*+4&; (2& 7C-$&+
,0 .+,(,7) .&+ *(,-; *7/ (2& 7C-$&+
,0 *(,-) 37 !=\\ N4 ,0 '*+$,7Z
enQ L_
X)374 * .+,.,+(3,7; +&5*(& (2&7C-$&+ ,0 *(,-) 37
!=\\ N4 ,0
'*+$,7 nL; (, D:,4*/+,M) 7C-$&+
*7/ (2& -,5&'C5*+ -*))M ,0 '*+$,7Z
Mm
Nn L
D
L M
mNn LDL
OC$)(3(C(& 0,+ Ln (, ,$(*37Z
M
emNQ LD
_
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
QZ
L!\dI=V
-,5
N4\!I\!=\
L!\!=_\IN4!=\\-,5
*(,-)!\_=\II_a
!eIW
Q
24 OC..,)& * 'C$& ,0 *5C-37C- 123'2 3) !=\\ '- ,7 *
)3/& *''C-C5*(&) *7&( '2*+4& ,0 GI=]\ .L= 6a8E2*(
.&+'&7(*4& ,0 (2& &5&'(+,7) ,+3437*55<
37 (2&
'C$& 1*) +&-,:&/F 6b8 U< 12*(
.&+'&7(*4& 2*) (2& -*)) ,0 (2& 'C$&
/&'+&*)&/
$&'*C)& ,0 (23) +&-,:*5F
Picture the Problem 6a8 @2& .&+'&7(*4& ,0
(2& &5&'(+,7) ,+3437*55< 37 (2& 'C$&(2*( 1*)
+&-,:&/ '*7 $& 0,C7/ 0+,- (2& +*(3, ,0 (2&
7C-$&+ ,0 &5&'(+,7) +&-,:&/
(, (2& 7C-$&+ ,0 &5&'(+,7) ,+3437*55< 37
(2& 'C$&= 6b8 @2& .&+'&7(*4&
/&'+&*)& 37
(2& -*)) ,0 (2& 'C$& '*7 $& 0,C7/ 0+,- (2&
+*(3, ,0 (2& -*)) ,0 (2& &5&'(+,7)
+&-,:&/ (, (2& -*)) ,0 (2& 'C$&=
6a8 >?.+&)) (2& +*(3, ,0 (2&
&5&'(+,7)
+&-,:&/ (, (2& 7C-$&+ ,0
&5&'(+,7),+3437*55< 37 (2& 'C$&Z
*(,-)*(,-.&+
&5&'(+,7)
/*''C-C5*(&
373
+&-
NN
e
Q
N
N 6!8
@2& 7C-$&+ ,0 *(,-) 37 (2& 'C$& 3)
(2& +*(3, ,0 (2& -*)) ,0 (2& 'C$& (,(2& -*))
,0 *7 *5C-37C- *(,-Z
*(,-D5
'C$&D5
*(,-D5
'C$&
*(,-)m
V
m
mN
@2& -*)) ,0 *7 *5C-37C- *(,- 3) 3()-,5*+ -*)) /3:3/&/
$< D:,4*/+,M)
7C-$&+ZD
D5*(,-D5
N
Mm
-
8/12/2019 Cap21 Prob
14/92
L2*.(&+ I!!V
OC$)(3(C(374 *7/ )3-.530?.+&)) (2& +*(3, ,0 (2& -*))
,0(2& &5&'(+,7) +&-,:&/ (, (2& -*)) ,0
(2& 'C$&Z'C$&D5
&5&'(+,7+&-
'C$&
+&-
V
mN
m
m
Y+,- 6a8; (2& 7C-$&+ ,0 &5&'(+,7)
+&-,:&/ 3) 43:&7 $
-
8/12/2019 Cap21 Prob
15/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) !]
25 PC+374 * .+,'&)) /&)'+3$&/ $<
(2&photoelectric effect; C5(+*T:3,5&(
5342( '*7 $& C)&/ (, '2*+4& * .3&'& ,0
-&(*5= 6a8 J0 )C'2 5342( 3) 37'3/&7( ,7 * )5*$
,0 ',7/C'(374 -*(&+3*5 *7/ &5&'(+,7) *+&
&%&'(&/ 13(2 &7,C42 &7&+4< (2*(
(2&?.+&)) (2& 0,+'& (2*( qW&?&+() ,7q!Z
&* cI
!;W
W!!;W
rqqk
OC$)(3(C(& *7/ )3-.530< (, ,$(*37Z
&
&&*
c
cc
I
!;W
W
I
!;I
I
!
I
!;W
W!
I
!;I
I!
!
r
q
r
qqk
r
qqk
r
qqk
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
!*
Z
&&* cR!\]=!c-\=_
L\=_
-\=W
L\=VL\=_gL-R!\edd=d
I
II
IIe
!
28 D I=\ L .,37( '2*+4& *7/ * V=\ L .,37( '2*+4& *+&
* /3)(*7'&L
*.*+(= E2&+& )2,C5/ * (23+/ .,37( '2*+4& $&
.5*'&/ ), (2*( (2& &5&'(+3' 0,+'& ,7(2*( (23+/
'2*+4& 3) A&+,F
Picture the Problem@2& (23+/ .,37( '2*+4& )2,C5/ $&
.5*'&/ *( (2& 5,'*(3,7 *(
123'2 (2& 0,+'&) ,7 (2& (23+/ .,37( '2*+4&
/C& (, &*'2 ,0 (2& ,(2&+ (1, .,37(
'2*+4&) '*7'&5= @2&+& '*7 $& 7, )C'2
.5*'& &?'&.( ,7 (2& 537& $&(1&&7
(2& (1,
.,37( '2*+4&)= P&7,(& (2& I=\ L *7/ V=\ L .,37(
'2*+4&) $< (2& 7C-&+*5) I *7/
V; +&).&'(3:&5
-
8/12/2019 Cap21 Prob
18/92
L2*.(&+ I!!d
D..5< (2& ',7/3(3,7 0,+ (+*7)5*(3,7*5
&QC353$+3C- (, (2& (23+/ .,37( '2*+4&Z\W;IW;V **
,+
W;IW;V FF 6!8
l&((374 (2& /3)(*7'& 0+,- (2& (23+/
.,37( '2*+4& (, (2& V=\ L .,37('2*+4&
$&x,&?.+&)) (2& 0,+'& (2*(
(2& V=\ L .,37( '2*+4& &?&+() ,7 (2&(23+/
.,37( '2*+4&Z
IVW
W;V xL
qkqF
@2& 0,+'& (2*( (2& I=\ L .,37(
'2*+4& &?&+() ,7 (2& (23+/ '2*+4&
3)43:&7 $
-
8/12/2019 Cap21 Prob
19/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) !e
.,37( '2*+4&); 2,1&:&+; 1& '*7 037/ * .5*'&
12&+& (2&)& 0,+'&) '*7'&5
$&'*C)&
(2&< *+& &QC*5 37 -*473(C/& *7/
,..,)3(&5< /3+&'(&/= P&7,(& (2& I=\ L
*7/ V=\L .,37( '2*+4&) $< (2& 7C-&+*5) I *7/ V;
+&).&'(3:&5
-
8/12/2019 Cap21 Prob
20/92
L2*.(&+ I!I\
Picture the Problem @2& ',7034C+*(3,7 ,0 (2& .,37(
'2*+4&) *7/ (2& 0,+'&) ,7 (2&
0,C+(2 .,37( '2*+4& *+& )2,17 37 (2& 034C+& m *)
3) * ',,+/37*(& )?.+&)) (2& 0,+'& (2*( .,37(
'2*+4&
q!&?&+() ,7 .,37( '2*+4& qVZ'* c
I
V;!
V!V;!
r
qkq
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
V;!*
Z
''* cR!\IW_=Wc-\]\\=\
7L\\=W7L\\=W
L
-R!\edd=d ]
II
Ie
V;!
>?.+&)) (2& 0,+'& (2*( .,37( '2*+4&
qI&?&+() ,7 .,37( '2*+4& qVZ&* c
I
V;I
VIV;I
r
qkq
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
V;I*
Z
&&* cR!\IW_=Wc-\]\\=\
7L\\=W7L\\=W
L
-R!\edd=d ]
II
Ie
V;I
-
8/12/2019 Cap21 Prob
21/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) I!
>?.+&)) (2& 0,+'& (2*( .,37( '2*+4& qW
&?&+() ,7 .,37( '2*+4& qVZV;WI
V;W
VWV;W
c%*r
qkq
; 12&+& V;Wc% 3) * C73(
:&'(,+ .,37(374 0+,- qW(, qV=
>?.+&)) V;W%
37 (&+-) ,0 !;W%
*7/ V;!%
Z
'&
%%%
c-\]\\=\c-\]\\=\
V;!!;WV;W
L,7:&+( V;W%
(, V;Wc% Z
'&
'&
%
%% ca\a=\ca\a=\
-\]\\=\-\]\\=\
c-\]\\=\c-\]\\=\c
IIV;W
V;W
V;W
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
V;W*
Z
'&
'&*
cR!\!V=!cR!\!V=!
ca\a=\ca\a=\-I\]\\=\
7L\\=W7L\\=W
L
-R!\edd=d
]]
II
Ie
V;W
OC$)(3(C(& 7C-&+3'*5 :*5C&) 37 &QC*(3,7 6!8 *7/
)3-.530< (, 037/ V*
Z
'&
'&&'*
cR!\!\=IcR!\!\=I
cR!\!V=!cR!\!V=!cR!\IV=WcR!\IV=W
]]
]]]]
V
@23) +&)C5( (&55) C) (2*( (2& 7&( 0,+'& 3)
I=ea !\]R *5,74 (2& /3*4,7*5 37 (2&
/3+&'(3,7 *1*< 0+,- (2& W=\ 7L '2*+4&=
31 D .,37( '2*+4& ,0 ]=\\ L 3) ,7 (2&y*?3) *(y[ W=\\ '-;
*7/ * )&',7/
.,37( '2*+4& ,0 H]=\\ L 3) ,7 (2&y*?3) *(y[ HW=\\ '-=
Y37/ (2& &5&'(+3' 0,+'&
,7 * .,37( '2*+4& ,0 I=\\ L ,7 (2&x*?3) *(x[ d=\\
'-=
Picture the Problem @2& ',7034C+*(3,7 ,0 (2& .,37(
'2*+4&) *7/ (2& 0,+'&) ,7
.,37( '2*+4& qW *+& )2,17 37 (2& 034C+& m *) 3)
* ',,+/37*(& )
-
8/12/2019 Cap21 Prob
22/92
L2*.(&+ I!II
cm,x
cm,y
C00.51 =q
C00.52
=q
C00.23
=q
00.80
3,1Fr
3,2Fr
00.3
00.3
@2& 7&( 0,+'& *'(374 ,7 .,37( '2*+4&
qW3) 43:&7 $
-
8/12/2019 Cap21 Prob
23/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) IW
32 D .,37( .*+(3'5& (2*( 2*) * '2*+4& ,0 HI=] L 3)
5,'*(&/ *( (2& ,+3437= D
)&',7/ .,37( .*+(3'5& (2*( 2*) * '2*+4& ,0 _=\ L 3)
*(x[ !=\ -;y[ \=]\ -= D
(23+/ .,37( .*+(3'5&; *7/ &5&'(+,7; 3) *( * .,37(
13(2 ',,+/37*(&) 6x;y8= Y37/ (2&:*5C&) ,0x*7/y)C'2 (2*(
(2& &5&'(+,7 3) 37 &QC353$+3C-=
Picture the Problem @2& .,)3(3,7) ,0 (2& .,37(
.*+(3'5&) *+& )2,17 37 (2&
/3*4+*-= J( 3) *..*+&7( (2*( (2& &5&'(+,7 -C)(
$& 5,'*(&/ *5,74 (2& 537& %,37374 (2&
(1, .,37( .*+(3'5&)= f,+&,:&+; $&'*C)& 3( 3)
7&4*(3:&5< '2*+4&/; 3( -C)( $& '5,)&+
(, (2& .*+(3'5& 13(2 * '2*+4& ,0 I=] L (2*7 (,
(2& .*+(3'5& 13(2 * '2*+4& ,0
_=\ L; *) 3) 37/3'*(&/ 37 (2& 034C+&= E& '*7
037/ (2& x*7/ y',,+/37*(&) ,0 (2&
&5&'(+,7M) .,)3(3,7 $< &QC*(374 (2& (1,
&5&'(+,)(*(3' 0,+'&) *'(374 ,7 3( *7/ ),5:374
0,+ 3() /3)(*7'& 0+,- (2& ,+3437= E& '*7 C)&
)3-35*+ (+3*745&) (, &?.+&)) (23) +*/3*5
/3)(*7'& 37 (&+-) ,0 (2&x*7/y',,+/37*(&) ,0
(2& &5&'(+,7=
m,x
m,y
50.0
0.1
C5.22
=q
ex
ey
e
eF,1r
eF ,2r
C0.61
=q
>?.+&)) (2& ',7/3(3,7 (2*( -C)( $&)*(3)03&/
30 (2& &5&'(+,7 3) (, $& 37
&QC353$+3C-Z
ee FF
;I;!
l&((374 r +&.+&)&7( (2& /3)(*7'&
0+,-
(2& ,+3437 (, (2& &5&'(+,7; &?.+&))
(2&
-*473(C/& ,0 (2& 0,+'& (2*( (2&
.*+(3'5& 12,)& '2*+4& 3) q!&?&+() ,7
(2& &5&'(+,7Z
I!
;!
-I]=!
r
ekqFe
>?.+&)) (2& -*473(C/& ,0 (2& 0,+'&(2*(
(2& .*+(3'5& 12,)& '2*+4& 3)qI
&?&+() ,7 (2& &5&'(+,7Z
II
;Ir
eqkF e
OC$)(3(C(& *7/ )3-.530< (, ,$(*37Z
II
I
!
-I]=! r
q
r
q
-
8/12/2019 Cap21 Prob
24/92
L2*.(&+ I!IV
OC$)(3(C(& 0,+ q!*7/ qI*7/ )3-.5305&'(+3' Y3&5/ !Z
P3)'+&(& L2*+4& P3)(+3$C(3,7) I]
F3,2
F1,2
F2,1
F3,1
F1,3
F2,3
m,x
m,y
0.10
0.20
C1.01 =q C4.03 =q
C2.02 =q
>?.+&)) (2& 7&( 0,+'& *'(374 ,7 (2&
.,37( .*+(3'5& 12,)& '2*+4& 3) q!Z!;W!;I! ***
>?.+&)) (2& 0,+'& (2*( (2& .,37(
.*+(3'5& 12,)& '2*+4& 3) qI&?&+() ,7
(2& .,37( .*+(3'5& 12,)& '2*+4& 3) q!Z
!;IW
!;I
!I
!;I
!;I
I
!;I
!I!;II
!;I
!I!;I c %%
%*
r
qkq
rr
qkq
r
qkq
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
!;I*
Z
''* cRd\=!c-!\=\-!\=\
L\=!L\=I
L
-R!\edd=d
WI
Ie
!;I
>?.+&)) (2& 0,+'& (2*( (2& .,37(
.*+(3'5& 12,)& '2*+4& 3) qW&?&+() ,7
(2& .,37( .*+(3'5& 12,)& '2*+4& 3) q!Z
!;WW
!;W
!W!;W %*
r
qkq
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
!;W*
Z
&&* cRdee=\c-I\=\-I\=\
L\=!L\=V
L
-R!\edd=d
WI
Ie
!;W
OC$)(3(C(& (, 037/!
*
Z
'&* c
Rd=!c
Re\=\!
-
8/12/2019 Cap21 Prob
26/92
L2*.(&+ I!I_
>?.+&)) (2& 7&( 0,+'& *'(374 ,7 (2&
.,37( .*+(3'5& 12,)& '2*+4& 3) qIZ
'*
**
***
cRd\=!I;W
!;II;W
I;!I;WI
$&'*C)& I;!*
*7/ !;I*
*+& *'(3,7T*7/T+&*'(3,7 0,+'&)=
>?.+&)) (2& 0,+'& (2*( (2& .,37(
.*+(3'5& 12,)& '2*+4& 3) qW&?&+() ,7
(2& .,37( .*+(3'5& 12,)& '2*+4& 3) qIZ
'&
%*
c-!\=\c-I\=\W
I;W
IW
I;WW
I;W
IWI;W
r
qkq
r
qkq
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
I;W*
Z
'&
'&*
cR_V\=\cRId=!
c-!\=\c-I\=\-IIV=\
L\=IL\=V
L
-R!\edd=d
WI
Ie
I;W
Y37/ (2& 7&( 0,+'& *'(374 ,7 (2& .,37(
.*+(3'5& 12,)& '2*+4& 3) qIZ
'&
''&'**
cRI=!
cRW=!
cRd\=!cR_V\=\cRId=!cRd\=!I;WI
R,(374 (2*( W;!*
*7/ !;W*
*+& *7 *'(3,7T*7/T+&*'(3,7 .*3+; *) *+& W;I*
*7/ I;W*
;
&?.+&)) (2& 7&( 0,+'& *'(374 ,7 (2&
.,37( .*+(3'5& 12,)& '2*+4& 3) qWZ
'&
'&&*****
cR_V=\cRV=\
cR_V\=\cRId=!cRdee=\I;W!;WW;IW;!W
34 D .,37( .*+(3'5& (2*( 2*) * '2*+4& ,0 ]=\\ L 3)
5,'*(&/ *(x[ \;y[ \
*7/ * .,37( .*+(3'5& (2*( 2*) * '2*+4& q3) 5,'*(&/
*(x[ V=\\ '-;y[ \= @2&
&5&'(+3' 0,+'& ,7 * .,37( .*+(3'5& (2*( 2*) *
'2*+4& ,0 I=\\ L*(x[ d=\\ '-;y[ \
3) H6!e=a R8c&= E2&7 (23) I=\\TL '2*+4& 3)
+&.,)3(3,7&/ *(x[ !a=d '-;y[ \; (2&
&5&'(+3' 0,+'& ,7 3( 3) A&+,=
P&(&+-37& (2& '2*+4& q=
-
8/12/2019 Cap21 Prob
27/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) Ia
Picture the Problem l&( q! +&.+&)&7( (2&
'2*+4& ,0 (2& .,37( .*+(3'5& *( (2&
,+3437; qI[ q,*7/ qW(2& '2*+4& ,0 (2& .,37(
.*+(3'5& 373(3*55< *( 6d=\\ '-; \8= @2&
/3*4+*- )2,1) (2& 0,+'&) *'(374 ,7 (2& .,37(
.*+(3'5& 12,)& '2*+4& 3) qW12&7 3(
5,'*(&/ *( 6d=\\ '-; \8= E& '*7 *..5< L,C5,-$M) 5*1
*7/ (2& .+37'3.5& ,0
)C.&+.,)3(3,7 ,0 0,+'&) (, 037/ (2& '2*+4&
q=
L\\=]!q
\ V=\\ d=\\x; '-
y qq
I L\\=IW q
W;!
*
W;I*
>?.+&)) (2& 7&( 0,+'& ,7 (2& .,37(
.*+(3'5& 12,)& '2*+4& 3) qW12&7 3( 3)
*( 6d=\\ '-; \8Z
W;IWW;I
IW;!
WW;!
!W
W;IW
W;I
WIW;!W
W;!
W!
W;IW;!
%%
%%
***
r
q
r
qkq
r
qkq
r
qkq
OC$)(3(C(& 7C-&+3'*5 :*5C&) (, ,$(*37Z
&&
&
c-\V\\=\-\V\\=\
c-\d\\=\-\=\d\\
L\\=]
L\\=IL
-R!\edd=dcRa=!e
WW
I
Ie
q
O,5:374 0,+ q
-
8/12/2019 Cap21 Prob
28/92
L2*.(&+ I!Id
>?.+&)) (2& 7&( 0,+'& *'(374 ,7 (2&
.,37( '2*+4& qZqQqxQq ;V]*(*?3);,7 I ***
>?.+&)) (2& 0,+'& ,7 .,37( '2*+4& q
/C& (, (2& .,37( '2*+4& Q,7 (2&x
*?3)Z
&* cI*?3);,7 R
kqQqxQ
>?.+&)) (2& 7&( 0,+'& ,7 .,37( '2*+4&
q/C& (, (2& .,37( '2*+4&) *( V]Z
&
&*
c
I
I
cV]',)II
I
I;V]*(
R
kqQ
R
kqQqQ
OC$)(3(C(& 0,+ qxQ *?3);,7*
*7/
qQ ;V]*(I *
(, ,$(*37Z
&
&&*
cI!
cI
Ic
I
II
R
kqQ
R
kqQ
R
kqQq
36 @2& )(+C'(C+& ,0 (2& RBW-,5&'C5& 3)
*..+,?3-*(&5< (2*( ,0 *7&QC35*(&+*5
(&(+*2&/+,7; 13(2 (2+&& B
G3,7) 0,+-374 (2& $*)& *7/ *7 R
WH3,7 *( (2&
*.&? ,0 (2& (&(+*2&/+,7= @2& 5&74(2 ,0
&*'2 )3/& 3) !=_V !\H!\-= L*5'C5*(&
(2&&5&'(+3' 0,+'& (2*( *'() ,7 &*'2 3,7=
Picture the Probleml&( (2& BG3,7) $& 37
(2&x-y.5*7& 13(2 B!*( 6\; \; \8; BI*(
6a; \; \8; *7/ BW*(
\;
I
W;
I
aa= @2& R
W3,7; 13(2 '2*+4& qV37 ,C+ 7,(*(3,7; 3)
(2&7 *(
W
I;
WI;
Ia
aa12&+& a [!=_V !\!\-= @, )3-.530< ,C+
'*5'C5*(3,7)
1&M55 )&( R!\]_=d eII Cake = E& '*7 *..5<
L,C5,-$M) 5*1 *7/ (2&
.+37'3.5& ,0 )C.&+.,)3(3,7 ,0 0,+'&) (, 037/ (2&
7&( 0,+'& *'(374 ,7 &*'2 3,7=
-
8/12/2019 Cap21 Prob
29/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) Ie
z
a
y
a
a
(I!
\;\;(
\;
I
W;
I
((
W
I;WI;I (
((
q
I
q
q
q
W
!
V
>?.+&)) (2& 7&( 0,+'& *'(374 ,7 .,37(
'2*+4& q!Z!;V!;W!;I! ****
Y37/ !;I*
Z &&%* ccc !;II!;I
I!!;I CC
r
qkq
Y37/ !;W*
Z
'&
'&
%*
cI
WcI
!
cI
W\cI
\
c!;WI
!;W
!W!;W
C
a
aa
C
r
qkq
-
8/12/2019 Cap21 Prob
30/92
L2*.(&+ I!W\
R,(374 (2*( (2& -*473(C/& ,0 .,37( '2*+4& qV3)
(2+&& (3-&) (2*( ,0 (2& ,(2&+ .,37(
'2*+4&) *7/ (2*( 3( 3) 7&4*(3:&; &?.+&))
!;V*
Z
.'&
.'&
.'&
%*
cW
IcWI
!cI
!W
cW
IcWI
cI
W
W
I
WII
c
W
I\c
WI\c
I\
WcWIII
!;V!;V
Ca
aaa
C
aaa
aaa
CC
OC$)(3(C(& 37 (2& &?.+&))3,7 0,+ !*
(,
,$(*37Z
.
.'&
'&&*
c_
cW
IcWI
!cI
!W
cI
WcI
!c!
C
C
CC
Y+,- )
-
8/12/2019 Cap21 Prob
31/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) W!
6a8 >?.+&)) (2& &5&'(+3' 03&5/ *( *
.,37( ^ 5,'*(&/ * /3)(*7'&x0+,- *
.,37( '2*+4& qZ
^;\I c%" xkq
x
>:*5C*(& (23) &?.+&))3,7 0,+x[ _=\ -Z
&&" cNRgL\=!c
-\=_
L\=VL
-R!\edd=d
-\=_I
I
Ie
6b8 >:*5C*(& "
*(x [ !\ -Z
&&" cNRgLW_=\c-!\
L\=VL
-R!\edd=d
-!\I
I
Ie
6c8 @2& 0,55,1374 4+*.2 1*) .5,((&/ C)374 *
).+&*/)2&&( .+,4+*-Z
-500
-250
0
250
500
-2 -1 0 1 2
x 6-8
Ex6
NRgL8
38 @1, .,37( '2*+4&); &*'2 GV=\ L; *+& ,7
(2&x*?3)9 ,7& .,37( '2*+4& 3)
*( (2& ,+3437 *7/ (2& ,(2&+ 3) *(x[ d=\ -= Y37/
(2& &5&'(+3' 03&5/ ,7 (2&x*?3) *(6a8x[ HI=\ -;
6b8x[ I=\ -; 6c8x[ _=\ -; *7/ 6d8x[ !\ -= 6e8 D( 12*( .,37( ,7
(2&x*?3) 3) (2& &5&'(+3' 03&5/ A&+,F 6f8
ON&('2Ex:&+)C)x0,+ HW=\ - `x ` !! -=
Picture the Problem l&( q +&.+&)&7( (2&
.,37( '2*+4&) ,0 GV=\ L *7/ C)&
L,C5,-$M) 5*1 0,+ "
/C& (, * .,37( '2*+4& *7/ (2& .+37'3.5& ,0
)C.&+.,)3(3,7 0,+
03&5/) (, 037/ (2& &5&'(+3' 03&5/ *( (2&
5,'*(3,7) ).&'303&/=
-
8/12/2019 Cap21 Prob
32/92
L2*.(&+ I!WI
R,(374 (2*( q![ qI; C)& L,C5,-$M) 5*1 *7/ (2&
.+37'3.5& ,0 )C.&+.,)3(3,7 (,
&?.+&)) (2& &5&'(+3' 03&5/ /C& (,
(2& 43:&7 '2*+4&) *( .,37( ^ * /3)(*7'&x0+,-
(2&
,+3437Z
^;I^;I
I
^;I^;I!
^;II
^;I!
I!
I!
I!I!
c-\=d
!c
!gL-NRW_
c-\=d
!c
!
c-\=d
c
qq
qq
qqqq
xx
xxkq
xkq
xkqxxx
%%
%%
%%"""
6a8 D..5< (23) &QC*(3,7 (, (2& .,37( *(x [ I=\ -Z
&&&" cNRgLV=ec-!\
!c-\=I
!gL-NRW_-\=I
II
I
6b8 >:*5C*(&"
*(x [ I=\ -Z
&&&" cNRgL\=dc-\=_
!c-\=I
!gL-NRW_-\=I
II
I
6c8 >:*5C*(&"
*(x [ _=\ -Z
&&&" cNRgL\=dc-\=I
!c-\=_
!gL-NRW_-\=_
II
I
6d8 >:*5C*(&"
*(x [ !\ -Z
&&&" cNRgLV=ec-\=I
!c-!\
!gL-NRW_-!\
II
I
6e8 Y+,- )
-
8/12/2019 Cap21 Prob
33/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) WW
6f8 @2& 0,55,1374 4+*.2 1*) .5,((&/ C)374 *
).+&*/)2&&( .+,4+*-Z
-100
-75
-50
-25
0
25
50
75
100
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
x 6-8
Ex6
NR-
IgL8
39 E2&7 * I=\T7L .,37( '2*+4& 3) .5*'&/ *( (2&
,+3437; 3( &?.&+3&7'&) *7
&5&'(+3' 0,+'& ,0 d=\ !\HVR37 (2&
Gy/3+&'(3,7= 6a8 E2*( 3) (2& &5&'(+3' 03&5/
*((2& ,+3437F 6b8 E2*( 1,C5/ $& (2& &5&'(+3'
0,+'& ,7 * HV=\T7L .,37( '2*+4& .5*'&/*( (2&
,+3437F 6c8 J0 (23) 0,+'& 3) /C& (, (2&
&5&'(+3' 03&5/ ,0 * .,37( '2*+4& ,7 (2&y
*?3) *(y[ W=\ '-; 12*( 3) (2& :*5C& ,0 (2*(
'2*+4&F
Picture the Problem E& '*7 037/ (2& &5&'(+3'
03&5/ *( (2& ,+3437 0+,- 3()
/&0373(3,7 *7/ (2& &5&'(+3' 0,+'& ,7 * .,37(
'2*+4& .5*'&/ (2&+& C)374 "*
q = E&
'*7 *..5< L,C5,-$M) 5*1 (, 037/ (2& :*5C& ,0 (2&
.,37( '2*+4& .5*'&/ *(y[ W '-=
6a8 D..5< (2& /&0373(3,7 ,0 &5&'(+3'
03&5/ (, ,$(*37Z
'
'*"
cRgL!\\=V
7L\=I
cR!\\=d\;\\;\
]
V
\
q
6b8 @2& 0,+'& ,7 * .,37( '2*+4& 37 *7
&5&'(+3' 03&5/ 3) 43:&7 $
-
8/12/2019 Cap21 Prob
34/92
L2*.(&+ I!WV
O,5:374 0,+ q*+(2M) 4+*:3(*(3,7*5 0,+'& ,7 3(=
6a8 >?.+&)) (2& -*473(C/& ,0 (2&
&5&'(+3' 0,+'& *'(374 ,7 (2&
&5&'(+,7Z
eEFe
>?.+&)) (2& -*473(C/& ,0 (2&
4+*:3(*(3,7*5 0,+'& *'(374 ,7 (2&
&5&'(+,7Z
gmF eg
@2& +*(3, ,0 (2&)& 0,+'&) 3)Z
mg
eE
F
F
g
e
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(&FegFgZ
!I
IW!
!e
!\_e=I
-g)e=d!N4!\e=!\e
RgL!]\L!\!=_\I
g
e
F
F
,+
ge FF !I!\_e=I @2C); (2& &5&'(+3' 0,+'& 3)
4+&*(&+ $< *
0*'(,+ ,0 I=_e!\!I=
6b8 l&((374 (2& C.1*+/ /3+&'(3,7 $&
.,)3(3:&; *..5< (2& ',7/3(3,7 0,+
)(*(3' &QC353$+3C- (, (2& .374 .,74
$*55 (, ,$(*37Z
\ ge FF
,+
\ mgqE E
mgq
-
8/12/2019 Cap21 Prob
35/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) W]
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(& qZ
-L!aa=\
RgL!]\
-g)e=d!N4!\a\=I IW
q
41 [SSM] @1, .,37( '2*+4&) q!*7/ q
I$,(2 2*:& * '2*+4& &QC*5 (,
G_=\ 7L *7/ *+& ,7 (2&y *?3) *(y![ GW=\ '- *7/yI[ HW=\
'- +&).&'(3:&5?.+&)) x"
0,+ $,(2 '2*+4&)Z &" c',)II
r
kqx
-
8/12/2019 Cap21 Prob
36/92
L2*.(&+ I!W_
OC$)(3(C(& 0,+ ',)*7/ r; )C$)(3(C(& 7C-&+3'*5
:*5C&); *7/ &:*5C*(& (, ,$(*37Z
&
&
&&"
cNRgL]WcNRgL]=VW
c-\=\V\-\=\W\
-\=\V\7L_=\gL-R!\d=eddI
c-\V\=\Ic-\V\=\I'-\=V
IWII
IIe
WI
i
r
kq
rr
kqx
@2& -*473(C/& *7/ /3+&'(3,7 ,0 (2&
&5&'(+3' 03&5/ *(x [ V=\ '- 3)Z\nNRgLW]
6b8 D..5< "*
q (, 037/ (2& 0,+'&
,7 * .,37( '2*+4& q\.5*'&/ ,7 (2&
x*?3) *(x [ V=\ '-Z
&
&*
cR_e
cNRgLV=]W7L\=I
42 D .,37( '2*+4& ,0 G]=\ L 3) 5,'*(&/ ,7 (2&x*?3)
*(x[ HW=\ '-; *7/
* )&',7/ .,37( '2*+4& ,0 Hd=\ L 3) 5,'*(&/ ,7
(2&x*?3) *(x[ GV=\ '-= E2&+&
)2,C5/ * (23+/ '2*+4& ,0 G_=\ L $& .5*'&/ ), (2*(
(2& &5&'(+3' 03&5/ *( (2& ,+3437 3)
A&+,F
Picture the Problem J0 (2& &5&'(+3' 03&5/ *( x [
\ 3) A&+,; $,(2 3() x *7/ y
',-.,7&7() -C)( $& A&+,= @2& ,75< 1*< (23)
',7/3(3,7 '*7 $& )*(3)03&/ 13(2 .,37(
'2*+4&) ,0 G]=\L *7/ d=\ L ,7 (2& x *?3) 3) 30 (2&
.,37( '2*+4& G_=\ L 3)
*5), ,7 (2& x*?3)= l&( (2& )C$)'+3.() ]; d; *7/ _
3/&7(30< (2& .,37( '2*+4&) *7/(2&3+ 03&5/)=
E& '*7 C)& L,C5,-$M) 5*1 0,+"
/C& (, * .,37( '2*+4& *7/ (2&
.+37'3.5& ,0 )C.&+.,)3(3,7 0,+ 03&5/) (,
/&(&+-37& 12&+& (2& G_=\ L .,37(
'2*+4&
)2,C5/ $& 5,'*(&/ ), (2*( (2& &5&'(+3'
03&5/ *(x [ \ 3) A&+,=
>?.+&)) (2& &5&'(+3' 03&5/ *(x[ \ 37
(&+-) ,0 (2& 03&5/) /C& (, (2& .,37(
'2*+4&) ,0 G]=\L; d=\ L; *7/
G_=\ LZ
\
\ L_LdL]
""""
OC$)(3(C(& 0,+ &*'2 ,0 (2& 03&5/) (,,$(*37Z
\ccc dId
d_I
_
_]I
]
]
%%%r
kqrkq
rkq
,+
\cccI
d
d
I
_
_
I
]
]
&&&r
kq
r
kq
r
kq
-
8/12/2019 Cap21 Prob
37/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) Wa
P3:3/& ,C( (2& C73( :&'(,+ &c (,
,$(*37Z\
I
d
d
I
_
_
I
]
]
r
q
r
q
r
q
OC$)(3(C(& 7C-&+3'*5 :*5C&) (,
,$(*37Z \
'-\=V
d_
'-\=W
]II
_
I
%
O,5:374 0,+ r_
-
8/12/2019 Cap21 Prob
38/92
L2*.(&+ I!Wd
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
!"
Z
'&
'&
'&%"
cNRgL]a]=\cNRgLVV=!
cWa!=\ceId=\RgL!\]]=!
-I=\-]=\
c-\=Ic-\=]
-I=\-]=\
L\=]gL-R!\edd=dc
W
IIII
IIe
^!;I
^!;
!!
r
kq
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
I"
Z
'&
'&
'&%"
cNRgL]V=ecNRgL]V=e
ca\a=\ca\a=\RgL!\]=!W
-I=\-I=\
c-I=\c-\=I
-I=\-I=\
L!IgL-R!\edd=dc
W
IIII
IIe
^I;I
^I;
II
r
kq
OC$)(3(C(& 0,+ !"
*7/ I"
*7/ )3-.530< (, 037/ ^"
Z
'&
'&'&"
cNRgL!=!\cNRgL!\=d
cNRgL]V=ecNRgL]V=ecNRgL]a]=\cNRgLVV=!^
@2& -*473(C/& ,0^
"
3)Z
NRgL!W
NRgL!\=!NRgLd=!\
II
^
E
@2& /3+&'(3,7 ,0 ^"
3)Z
IW\
NRgLd=!\
NRgL!\=!(*7 !E
R,(& (2*( (2& *745& +&(C+7&/ $<
-
8/12/2019 Cap21 Prob
39/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) We
Y37/ (2& -*473(C/& ,0 *
Z
R!\!=I
R!\_I=!R!\W\=!
!]
I!]I!]
F
Y37/ (2& /3+&'(3,7 ,0 *
Z
]!
R!\!=W
R!\!=_I(*7
!]
!]!
F
44 @1, &QC*5 .,)3(3:& '2*+4&) q *+& ,7
(2&y*?3)9 ,7& .,37( '2*+4& 3) *(
y[ Ga*7/ (2& ,(2&+ 3) *(y[ Ha= 6a8 O2,1 (2*( ,7 (2&x
*?3) (2&x',-.,7&7( ,0(2& &5&'(+3' 03&5/ 3)
43:&7 $?.5*37 12< * .&+),7 -342( &?.&'( (23)
+&)C5( &:&7
13(2,C( /&+3:374 3( $< (*N374 (2& *..+,.+3*(&
53-3(=
Picture the Problem @2& /3*4+*- )2,1) (2& 5,'*(3,7) ,0
.,37( '2*+4&) q!*7/ qI
*7/ (2& .,37( ,7 (2& x *?3) *( 123'2 1& *+& (,
037/"
= Y+,- )?.+&)) (2&xT',-.,7&7( ,0 (2&
&5&'(+3' 03&5/ /C& (, (2& .,37(
'2*+4&)
*(y [ a *7/y [ a *) * 0C7'(3,7 ,0
(2& /3)(*7'& r 0+,- &3(2&+ '2*+4& (,
.,37(PZ
&" c',)I I r
kqx
-
8/12/2019 Cap21 Prob
40/92
L2*.(&+ I!V\
OC$)(3(C(& 0,+ ',)*7/ r(, ,$(*37Z
&
&&&"
cI
cIcIcI
IWII
IWIIWI
ax
kqx
ax
kqx
r
kqx
r
x
r
kqx
@2& -*473(C/& ,0 #"
3)Z
IWIII
ax
kqxEx
6b8 Y,+ a,x xIG aIaI; ),Z
WIWIII
a
kqx
a
kqxEx
6c8 Y,+xoo a,(2& '2*+4&) )&.*+*(&/ $< a1,C5/
*..&*+ (, $& * )3745& '2*+4& ,0
-*473(C/& Iq.J() 03&5/ 1,C5/ $& 43:&7
$:*5C*(& !"
Z
'&'&
'&%"
cNRgL]VV=\cNRgLe\a=\c]!V=\cd]a=\NRgL\_=!
-W=\-]=\
c-\=Wc-\=]
-W=\-]=\
L\=VgL-R!\edd=dc
IIII
IIe
^!;I
!=^
!!
r
kq
>:*5C*(& I"
Z
'&'&
'&%"
cNRgL\!=!cNRgL\!=IcVVa=\cdeV=\NRgLI]=I
-I=\-V=\
c-I=\c-\=V
-I=\-V=\
L\=]gL-R!\edd=dc
IIII
IIe
^I;I
^I;
II
r
kq
OC$)(3(C(& *7/ )3-.530< (, 037/ ^"
Z
'&
'&'&"
cNRgL]]=!cNRgL!\=!
cNRgL\!=!cNRgL\!=IcNRgL]VV=\cNRgLe\d=\^
@2& -*473(C/& ,0 ^"
3)Z
NRgLe=!
NRgL]]=!NRgL!\=! II
^
E
-
8/12/2019 Cap21 Prob
42/92
L2*.(&+ I!VI
@2& /3+&'(3,7 ,0 ^"
3)Z
IW]
NRgL!\=!
NRgL]]=!(*7 !E
R,(& (2*( (2& *745& +&(C+7&/ $<
-
8/12/2019 Cap21 Prob
43/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) VW
6a8 >:*5C*(&x
Ex
(, ,$(*37Z
IWIII]III
IWIII]II
IWIIIWII
IWIIIWII
WI
II
WI
I
II
axaxxkq
axxaxxkq
axaxdxdxkq
axxdx
dkqaxkqx
dx
d
x
Ex
O&( (23) /&+3:*(3:& &QC*5 (, A&+, 0,+
&?(+&-& :*5C&)Z \W IWIII]III axaxx
O,5:374 0,+x
-
8/12/2019 Cap21 Prob
44/92
L2*.(&+ I!VV
Picture the Problem @2& &5&'(+3' 03&5/ ,0
(2& V(2
'2*+4&/ .,37( .*+(3'5& -C)(
'*7'&5 (2& )C- ,0 (2& &5&'(+3' 03&5/)
/C& (, (2& ,(2&+ (2+&& '2*+4&/ .,37(
.*+(3'5&)=
U< )
-
8/12/2019 Cap21 Prob
45/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) V]
@2& .,)3(3:& ),5C(3,7 ',++&).,7/) (, (2& V(2
.,37( .*+(3'5& $&374 * /3)(*7'&
WL *$,:& (2& $*)& ,0 (2& (+3*745&;
12&+& 3( .+,/C'&) (2& )*-& )(+&74(2 *7/
)*-& /3+&'(3,7 &5&'(+3' 03&5/ '*C)&/
$< (2& (2+&& '2*+4&) *( (2& ',+7&+) ,0
(2&
(+3*745&= O, (2& '2*+4&/ .,37( .*+(3'5& -C)(
$& .5*'&/ * /3)(*7'& WL $&5,1 (2&
-3/.,37( ,0 (2& (+3*745&=
48 @1, .,37( .*+(3'5&); &*'2 2*:374 * '2*+4&
&QC*5 (, q,)3( ,7 (2& $*)& ,0
*7 &QC35*(&+*5 (+3*745& (2*( 2*) )3/&) ,0
5&74(2L *) )2,17 37 Y34C+& I!TWd= D (23+/
.,37( .*+(3'5& (2*( 2*) * '2*+4& &QC*5 (, Iq)3() *(
(2& *.&? ,0 (2& (+3*745&= D 0,C+(2
.,37( .*+(3'5& (2*( 2*) '2*+4& q3) .5*'&/ *( (2&
-3/.,37( ,0 (2& $*)&537& -*N374(2& &5&'(+3'
03&5/ *( (2& '&7(&+ ,0 (2& (+3*745&
&QC*5 (, A&+,= E2*( 3) (2& :*5C& ,0
qF 6@2& '&7(&+ 3) 37 (2& .5*7& ,0 (2&
(+3*745& *7/ &QC3/3)(*7( 0+,- *55
(2+&&:&+(3'&)=8
Picture the Problem@2& &5&'(+3' 03&5/ ,0 V(2
'2*+4& -C)( '*7'&5 (2& )C- ,0 (2&
&5&'(+3' 03&5/) /C& (, (2& ,(2&+
(2+&& '2*+4&)= X)374 (+34,7,-&(+
-
8/12/2019 Cap21 Prob
46/92
L2*.(&+ I!V_
OC$)(3(C(374 0,+ !"
; I"
; W"
; *7/ V"
-
8/12/2019 Cap21 Prob
47/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) Va
6c8 >?.+&)) (2& 7&( 0,+'& *'(374 ,7
(2&
'2*+4& *(y[ GaZ \
I I
I
I
\*(
a
kq
a
kqqF ayq
O,5:& 0,+ q\(, ,$(*37Z qqV!
\
Remarks: In Part (0), we could just as well have expressed the
net force
acting on the charge at1= (. Due to the symmetric distribution
of the
charges at1= (and1= +(2summing the forces acting on -0at the
origin
does not lead to a relationship between -0and -+
50 @1, .,)3(3:& .,37( '2*+4&) Gq*+& ,7 (2&y*?3)
*(y[ Ga*7/y[ Ha= D$&*/ ,0 -*)) m *7/ '2*+4& Hq)53/&)
13(2,C( 0+3'(3,7 *5,74 * (*C( (2+&*/ (2*( +C7)
*5,74 (2&x*?3)= l&(x$& (2& .,)3(3,7 ,0 (2&
$&*/= 6a8 O2,1 (2*( 0,+x`` a; (2&
$&*/ &?.&+3&7'&) * 537&*+ +&)(,+374
0,+'& 6* 0,+'& (2*( 3) .+,.,+(3,7*5 (,x*7/
/3+&'(&/ (,1*+/ (2& &QC353$+3C- .,)3(3,7 *(x[ \8
*7/ (2&+&0,+& C7/&+4,&) )3-.5&
2*+-,73' -,(3,7= 6b8 Y37/ (2& .&+3,/ ,0 (2&
-,(3,7=
Picture the Problem J7 ^+,$5&- VV 3( 3) )2,17 (2*( (2&
&5&'(+3' 03&5/ ,7 (2& x
*?3); /C& (, &QC*5 .,)3(3:& .,37( '2*+4&)
5,'*(&/ *( 6\; a8 *7/ 6\;a8; 3) 43:&7 $5&'(+3'
Y3&5/ !Z P3)'+&(& L2*+4& P3)(+3$C(3,7) Ve
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(& tZ
)I=\
)!a\V=\-g)!\!=a]e
-g)!\eed=I\=\!
I!W
d
t
6d8 X)& * ',7)(*7(T*''&5&+*(3,7&QC*(3,7 (,
&?.+&)) (2& /3)(*7'&
(2& &5&'(+,7 (+*:&5) 37 * 43:&7
(3-& 37(&+:*5Z
I
I
!
3 tatvx ,+; $&'*C)& v3[\;
II! tax
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(& xZ
-W=\
)!a\V=\-g)!\a]e=! II!W
I!
x
52 @2& *''&5&+*(3,7 ,0 * .*+(3'5& 37 *7
&5&'(+3' 03&5/ /&.&7/) ,7 (2&
'2*+4&T(,T-*)) +*(3, ,0 (2& .*+(3'5&= 6a8
L,-.C(& qgm0,+ * .+,(,7; *7/ 037/ 3()
*''&5&+*(3,7 37 * C730,+- &5&'(+3' 03&5/
(2*( 2*) * -*473(C/& ,0 !\\ RgL= 6b8 Y37/(2& (3-& 3(
(*N&) 0,+ * .+,(,7 373(3*55< *( +&)( 37 )C'2 * 03&5/
(, +&*'2 * ).&&/ ,0
\=\!c612&+& c 3) (2& ).&&/ ,0 5342(8=
6E2&7 (2& ).&&/ ,0 *7 &5&'(+,7
*..+,*'2&)
(2& ).&&/ ,0 5342( c; +&5*(3:3)(3'
N37&-*(3') -C)( $& C)&/ (, '*5'C5*(& 3() -,(3,7;$C(
*( ).&&/) ,0 \=\!c,+ 5&)); 7,7T+&5*(3:3)(3'
N37&-*(3') 3) )C003'3&7(5< *''C+*(&
0,+ -,)( .C+.,)&)=8
Picture the Problem E& '*7 C)& R&1(,7M) )&',7/
5*1 ,0 -,(3,7 (, 037/ (2&
*''&5&+*(3,7 ,0 (2& .+,(,7 37 (2& C730,+-
&5&'(+3' 03&5/ *7/ ',7)(*7(T*''&5&+*(3,7
&QC*(3,7) (, 037/ (2& (3-& +&QC3+&/ 0,+ 3(
(, +&*'2 * ).&&/ ,0 \=\!c *7/ (2&
/3)(*7'& 3( (+*:&5) 1235& *'QC3+374 (23)
).&&/=
6a8 X)& /*(* 0,C7/ *( (2& $*'N ,0
-
8/12/2019 Cap21 Prob
50/92
L2*.(&+ I!]\
@2& /3+&'(3,7 ,0 (2& *''&5&+*(3,7 ,0 *
.+,(,7 3) 37 (2& /3+&'(3,7 ,0 (2& &5&'(+3'
03&5/=
6b8 X)374 (2& /&0373(3,7 ,0
*''&5&+*(3,7; +&5*(& (2& (3-&
+&QC3+&/
0,+ *7 &5&'(+,7 (, +&*'2 \=\!c(,
3()*''&5&+*(3,7Z
a
c
a
vt
\!=\
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(& tZ
-)W=\-g)!\]a_=e
-g)!\eed=I\=\!
Ie
d
t
53 D7 &5&'(+,7 2*) *7 373(3*5 :&5,'3(< ,0 I=\\
!\_-g) 37 (2& Gx/3+&'(3,7=
J( &7(&+) * +&43,7 (2*( 2*) * C730,+-
&5&'(+3' 03&5/ '" cRgLW\\
= 6a8 Y37/ (2&
*''&5&+*(3,7 ,0 (2& &5&'(+,7= 6b8 B,1 5,74
/,&) 3( (*N& 0,+ (2& &5&'(+,7 (, (+*:&5
!\=\ '- 37 (2&x/3+&'(3,7 37 (2& +&43,7 (2*( 2*)
(2& 03&5/F 6c8 @2+,C42 12*( *745&;
*7/ 37 12*( /3+&'(3,7; 3) (2& &5&'(+,7
/&05&'(&/ 1235& (+*:&5374 (2& !\=\ '- 37
(2& x/3+&'(3,7F
Picture the Problem @2& &5&'(+3' 0,+'& *'(374 ,7
(2& &5&'(+,7 3) ,..,)3(& (2&
/3+&'(3,7 ,0 (2& &5&'(+3' 03&5/= E& '*7
*..5< R&1(,7M) )&',7/ 5*1 (, 037/ (2&
&5&'(+,7M) *''&5&+*(3,7 *7/ C)& ',7)(*7(
*''&5&+*(3,7 &QC*(3,7) (, 037/ 2,1 5,74 3(
(*N&) (2& &5&'(+,7 (, (+*:&5 * 43:&7
/3)(*7'& *7/ 3() /&05&'(3,7 /C+374 (23)
37(&+:*5
,0 (3-&= Y37*55
-
8/12/2019 Cap21 Prob
51/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) ]!
6a8 X)& R&1(,7M) )&',7/ 5*1 (,
+&5*(& (2& *''&5&+*(3,7 ,0 (2&
&5&'(+,7
03+)( (, (2& 7&( 0,+'& *'(374 ,7 3( *7/
(2&7 (2& &5&'(+3' 03&5/ 37 123'2 3(
037/) 3()&50Z
ee m
e
m
"*(
7&(
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(& a
Z
'
'
'(
c-g)!\Id=]
c-g)!\Ia_=]
cRgLW\\N4!\e=!\e
L!\!=_\I
I!W
I!W
W!
!e
6b8 b&5*(& (2& (3-& (, (+*:&5 *
43:&7
/3)(*7'& 37 (2&x/3+&'(3,7 (, (2&
&5&'(+,7M) ).&&/ 37 (2&x/3+&'(3,7Z
7)\=]\-g)!\I=\\
-\=!\\
_
xv
xt
6c8 @2& *745& (2+,C42 123'2 (2&
&5&'(+,7 3) /&05&'(&/ *) 3( (+*:&5)
*
2,+3A,7(*5 /3)(*7'& x 3) 43:&7 $
-
8/12/2019 Cap21 Prob
52/92
L2*.(&+ I!]I
Picture the ProblemU&'*C)& (2& &5&'(+3'
03&5/ 3) 37 (2& y/3+&'(3,7; (2& 0,+'& 3(
&?&+() ,7 (2& &5&'(+,7 3) 37 (2&
Gy/3+&'(3,7= D..5
-
8/12/2019 Cap21 Prob
53/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) ]W
X)374 (2& 1,+NTN37&(3' &7&+4:*5C*(& U 0,+ [
\Z
j!\I=W\',)RgL!\\=V7-]\=\\ IVV eU
>:*5C*(& U 0,+ [ W\Z
j!\d=IW\',)RgL!\\=V7-]\=\W\ IVV eU
General Problems
61 [SSM] O2,1 (2*( 3( 3) ,75< .,))3$5& (, .5*'&
,7& 3),5*(&/ .+,(,7 37*7 ,+/37*+< &-.(<
',00&& 'C. $< ',7)3/&+374 (2& 0,55,1374
)3(C*(3,7= D))C-& (2&
03+)( .+,(,7 3) 03?&/ *( (2& $,((,- ,0 (2& 'C.=
P&(&+-37& (2& /3)(*7'& /3+&'(5?.+&))
(2& &5&'(+3' 03&5/ *(P*)
(2& )C- ,0 (2& 03&5/) /C& (, (2& .,37(
'2*+4&) 5,'*(&/ *(x [ !=\\ '- *7/
x [ !=\\ '-Z
&
&&
"""
c
cc
I
'-\\=!
I
I
'-\\=!
!
I
'-\\=!
I
I
'-\\=!
!
I!
PP
PP
qqP
x
q
x
qk
x
kq
x
kq
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
$"
Z
&&" cRgL!\dW=!c
'-e=\\
L]=\\
'-!!=\\
L\\=]
L
-R!\edd=d _
III
Ie
P
*7/
RgL!\dW=! _PE
6b8 @2& &5&'(+3' 03&5/ )(+&74(2 /C& (,
*/3.,5& 3) 43:&7 $< >QC*(3,7 I!T!\Z W
I
x
kpE
U&'*C)& qap I ; 12&+& Ia3) (2&
)&.*+*(3,7 ,0 (2& '2*+4&) (2*(
',7)(3(C(& (2& /3.,5&;E3) *5), 43:&7$
-
8/12/2019 Cap21 Prob
64/92
L2*.(&+ I!_V
X)& L,C5,-$M) 5*1 (, &?.+&)) (2&
7&( 0,+'& ,7 (2& '2*+4& GqZqq FFT VI!
OC$)(3(C(& *7/ )3-.530< (, ,$(*37Z I
I
II!
W
I
VI
d
kq
d
qkq
d
qkqT
X)& L,C5,-$M) 5*1 (, &?.+&)) (2&
7&( 0,+'& ,7 (2& '2*+4& GVqZqq FFT II
OC$)(3(C(& *7/ )3-.530< (, ,$(*37Z I
I
III
e
I
VVI
d
kq
d
qkq
d
qqkT
65 [SSM] D .,)3(3:& '2*+4& Q 3) (, $& /3:3/&/
37(, (1, .,)3(3:& .,37('2*+4&) q!*7/ qI= O2,1 (2*(; 0,+ *
43:&7 )&.*+*(3,7D; (2& 0,+'& &?&+(&/
$< ,7&
'2*+4& ,7 (2& ,(2&+ 3) 4+&*(&)( 30 q![
qI[
!
IQ=
Picture the Problem E& '*7 C)& L,C5,-$M) 5*1 (,
&?.+&)) (2& 0,+'& &?&+(&/ ,7
,7& '2*+4& $< (2& ,(2&+ *7/ (2&7 )&(
(2& /&+3:*(3:& ,0 (23) &?.+&))3,7 &QC*5 (,
A&+,
(, 037/ (2& /3)(+3$C(3,7 ,0 (2& '2*+4& (2*(
-*?3-3A&) (23) 0,+'&=
X)374 L,C5,-$M) 5*1; &?.+&)) (2&
0,+'& (2*( &3(2&+ '2*+4& &?&+() ,7
(2&
,(2&+Z
I
I!
D
qkqF
>?.+&)) qI37 (&+-) ,0 Q*7/ q!Z !I qQq
OC$)(3(C(& 0,+ qI(, ,$(*37Z I
!!
D
qQkqF
P300&+&7(3*(&F13(2 +&).&'( (, q!*7/
)&( (23) /&+3:*(3:& &QC*5 (, A&+, 0,+
&?(+&-& :*5C&)Z
&?(+&-*0,+\
! !!I
!!
!
I
!
qQqD
k
qQqdq
d
D
k
dq
dF
O,5:& 0,+ q!(, ,$(*37Z Qq I!
! QqQq I!
!I
-
8/12/2019 Cap21 Prob
65/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) _]
@, /&(&+-37& 12&(2&+ * -*?3-C- ,+
* -373-C- &?3)() *( QqI!
! ;
/300&+&7(3*(&F* )&',7/ (3-& *7/
&:*5C*(& (23) /&+3:*(3:& *( QqI!
! Z
=,0(5
-
8/12/2019 Cap21 Prob
66/92
L2*.(&+ I!__
6a8 X)& L,C5,-$M) 5*1 (, &?.+&))
(2& +&.C5)3:& &5&'(+3' 0,+'&
&*'2
.*+(3'5& &?&+() ,7 (2& ,(2&+Z
I
I;!
I!
r
qkqF
>?.+&)) Iq 37 (&+-) ,0 (2& (,(*5'2*+4& *7/
q!Z
!I qQq
OC$)(3(C(& 0,+ Iq (, ,$(*37Z I
I;!
I
!!
I
I;!
!!
r
qQqk
r
qQkqF
E+3(374 (23) &QC*(3,7 &?.53'3(5< *) *
QC*/+*(3' &QC*(3,7 43:&)Z\
I
I;!
!
I
! k
FrQqq
OC$)(3(C(& 7C-&+3'*5 :*5C&) (, ,$(*37Z
\gL-R!\d=edd
-_\=\Rd\LI\\
IIe
I
!
I
!
qq
,+
\L!\I\=WL!\\\=I Ie!VI! qq
X)& (2& QC*/+*(3' 0,+-C5* ,+
-
8/12/2019 Cap21 Prob
67/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) _a
/,171*+/= @2& .*+(3'5& 23() (2& 4+,C7/ *( *
).&&/ vI gh; 12&+& h3) (2& 373(3*5
2&342( ,0 (2& .*+(3'5&= Y37/ m 37 (&+-) ,0E; q;
*7/g=
Picture the Problem L2,,)& (2&
',,+/37*(& )*+(2= @2&7 (2& 1,+N /,7&
,7 (2& .,37( .*+(3'5& $< (2& &5&'(+3'
03&5/ 1355 '2*74& (2& &7&+4< ,0
(2&
)
-
8/12/2019 Cap21 Prob
68/92
L2*.(&+ I!_d
Picture the ProblemE& '*7 C)& L,C5,-$M) 5*1; (2&
/&0373(3,7 ,0 (,+QC&; *7/
(2& ',7/3(3,7 0,+ +,(*(3,7*5 &QC353$+3C- (, 037/ (2&
&5&'(+,)(*(3' 0,+'& $&(1&&7 (2&(1,
'2*+4&/ $,/3&); (2& (,+QC& (23) 0,+'&
.+,/C'&) *$,C( *7 *?3) (2+,C42 (2&
'&7(&+ ,0 (2& +,/; *7/ (2& -*)) +&QC3+&/
(, -*37(*37 &QC353$+3C- 12&7 3( 3) 5,'*(&/
&3(2&+ I]=\ '- (, (2& +342( ,+ (, (2& 5&0(
,0 (2& -3/T.,37( ,0 (2& +,/=
6a8 X)374 L,C5,-$M) 5*1; &?.+&))(2& &5&'(+3'
0,+'& $&(1&&7 (2& (1,
'2*+4&)Z
I
I!
dqkqF
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(&FZ
RII]=\RIIVa=\-!\\=\
L!\\\=]Lg-R!\edd=dI
IaIIe
F ; /,171*+/
6b8 @2& (,+QC& 6-&*)C+&/ *$,C( (2&
+,(*(3,7 *?3)8 /C& (, (2& 0,+'&F 3)Z
F
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(& Z
;-R!!I=\-R!!IV=\
-]\\=\RIIVa=\
',C7(&+'5,'N13)&=
6c8 D..5< \+,/(2&
,0'&7(&+ (, (2& +,/Z \ 'mg 'g
m
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/
&:*5C*(& mZ 4d=V]N4\V]dW=\
-I]\=\-g)d!=e
-R!!IV=\I
m
6d8 D..5< \+,/(2&
,0'&7(&+ (, (2& +,/Z \ 'mg
OC$)(3(C(& 0,+ (, ,$(*37Z \ 'mgF
OC$)(3(C(374 0,+F43:&)Z\
I
I! 'mgd
'qkq
!
I
Ikq
'mgd'q
12&+& q3) (2& +&QC3+&/ '2*+4&=
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
qIZ
L!\\\=]-]\\=\L!\\\=]Lg-R!\d=edd
-I]\=\-g)d!=eN4\V]dI=\-!\\=\ aaIIe
II
I
'q
-
8/12/2019 Cap21 Prob
69/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) _e
70 @1, W=\TL .,37( '2*+4&) *+& 5,'*(&/ *(x[ \;y[ I=\
- *7/ *(x[ \;y[ HI=\ -= @1, ,(2&+ .,37( '2*+4&); &*'2
13(2 '2*+4& &QC*5 (, Q;*+& 5,'*(&/ *(
x[ V=\ -;y[ I=\ - *7/ *(x[ V=\ -;y[ HI=\ - 6Y34C+& I!TVW8=
@2& &5&'(+3'
03&5/ *(x[ \;y[ \ /C& (, (2& .+&)&7'& ,0
(2& 0,C+ '2*+4&) 3) 6V=\
!\WRgL8c&=P&(&+-37& Q=
Picture the Problem @2& 03&5/ *( 6\; \8 .+,/C'&/
$< (2& (1, W=\ L '2*+4&) 3)
A&+,= @2C) (2& &7(3+& 03&5/ *( 6\; \8 3)
/C& (, (2& (1, .,37( '2*+4&) 12,)& '2*+4&)
*+& Q. l&( (2& 7C-&+*5 ! +&0&+ (,
(2& .,37( '2*+4& 37 (2& !)( QC*/+*7( *7/ (2&
7C-&+*5 I (, (2& .,37( '2*+4& 37 (2& V(2
QC*/+*7(= E& '*7 C)& L,C5,-$M) 5*1 0,+
(2& &5&'(+3' 03&5/ /C& (, * .,37( '2*+4&
*7/ (2& )C.&+.,)3(3,7 ,0 0,+'&) (, &?.+&))
(2& 03&5/ *( (2& ,+3437 *7/ C)& (23)
&QC*(3,7 (, ),5:& 0,+ Q.
>?.+&)) (2& &5&'(+3' 03&5/ *( (2&
,+3437 /C& (, (2& .,37( '2*+4&) QZ
&
&'&'&
%%"""
c
c-\=dc-\=Ic-\=Vc-\=Ic-\=V
cc\;\
WWW
\;II
\;I
\;!I
\;!
I!
xE
r
kQ
r
kQ
r
kQ
rkQ
rkQ
12&+& r3) (2& /3)(*7'& 0+,- &*'2 '2*+4&
(, (2& ,+3437 *7/
W
-\=d
r
kQEx =
>?.+&)) r37 (&+-) ,0 (2& ',,+/37*(&)
6x;y8 ,0 (2& .,37( '2*+4&)Z
II yxr
OC$)(3(C(& 0,+ r (, ,$(*37Z
IWII-\=d
yx
kQEx
O,5:374 0,+ Q
-
8/12/2019 Cap21 Prob
70/92
L2*.(&+ I!a\
Picture the Probleml&( (2& 7C-&+*5 ! /&7,(&
,7& ,0 (2& .,37( '2*+4&) *7/ (2&
7C-&+*5 I (2& ,(2&+= i7,1374 (2& (,(*5
'2*+4& ,7 (2& (1, ).2&+&); 1& '*7 C)&
L,C5,-$M) 5*1 (, 037/ (2& '2*+4& ,7 &*'2 ,0
(2&-= D )&',7/ *..53'*(3,7 ,0
L,C5,-$M) 5*1 12&7 (2& ).2&+&) '*++< (2&
)*-& '2*+4& *7/ *+& \=_\\ - *.*+(
1355
-
8/12/2019 Cap21 Prob
71/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) a!
72 @1, .,37( '2*+4&) 2*:& * (,(*5 '2*+4& ,0 I\\ L
*7/ *+& )&.*+*(&/ $*'2 .,37(
.*+(3'5& 3) 37 )(*(3' &QC353$+3C- C7/&+
(2& 3705C&7'& ,0 (2& (&7)3,7 8
; (2&
4+*:3(*(3,7*5 0,+'& 4*
; *7/ (2& &5&'(+3'
0,+'& >*
= E& '*7 C)& L,C5,-$M) 5*1 (,
+&5*(& (2& &5&'(+3' 0,+'& (, (2&
'2*+4& ,7
&*'2 .*+(3'5& *7/ (2&3+ )&.*+*(3,7 *7/
(2& ',7/3(3,7) 0,+ )(*(3' &QC353$+3C- (,
+&5*(& (2&)& 0,+'&) (, (2& '2*+4& ,7
&*'2
.*+(3'5&=
x
y
8
69*
4
&*
6a8 D..5< (2& ',7/3(3,7) 0,+ )(*(3'
&QC353$+3C- (, (2& .,37( .*+(3'5&
12,)& 0+&&T$,/< /3*4+*- 3) )2,17
*$,:&Z
\)37)37I
I
> Tr
kqTFFx
*7/
\',) mgTFy
>53-37*(& T$&(1&&7 (2&)&
&QC*(3,7) (, ,$(*37Z I
I
(*7mgr
kq
k
mgrq
(*7
b&0&++374 (, (2& 034C+&; +&5*(&
(2&
)&.*+*(3,7 ,0 (2& ).2&+&) r(, (2&
5&74(2 ,0 (2& .&7/C5C-LZ
)37ILr
OC$)(3(C(& 0,+ r (, ,$(*37Z
kmgLq (*7)37I
6b8 >:*5C*(& q0,+ m [ !\=\ 4;L [ ]\=\ '-; *7/ [ !\Z
LIV!=\gL-R!\edd=d
\=!\(*7-g)d!=eN4\!\\=\\=!\)37-]\\=\I
IIe
I
q
75 OC..,)& (2*( 37 ^+,$5&- aVL [ !=] - *7/ m [ \=\!\ N4=
6a8 E2*( 3)
(2& *745& (2*( &*'2 )(+374 -*N&) 13(2 (2&
:&+(3'*5 30 q [ \=a] LF 6b8 E2*( 3) (2&*745& (2*(
&*'2 )(+374 -*N&) 13(2 (2& :&+(3'*5 30 ,7&
.*+(3'5& 2*) * '2*+4& ,0
\=]\ L; (2& ,(2&+ 2*) * '2*+4& ,0 !=\ LF
-
8/12/2019 Cap21 Prob
75/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) a]
Picture the Problem >*'2 ).2&+& 3) 37
)(*(3' &QC353$+3C- C7/&+ (2& 3705C&7'&
,0 (2& (&7)3,78
; (2& 4+*:3(*(3,7*5
0,+'& 4*
; *7/ (2& &5&'(+3' 0,+'& &*
= E&
'*7 C)& L,C5,-$M) 5*1 (, +&5*(& (2&
&5&'(+3' 0,+'& (, (2& '2*+4& ,7 &*'2
).2&+& *7/ (2&3+ )&.*+*(3,7 *7/ (2&
',7/3(3,7) 0,+ )(*(3' &QC353$+3C- (,
+&5*(& (2&)& 0,+'&) (, (2& '2*+4& ,7
&*'2
).2&+&=
x
y
8
69*
4
&*
6a8D..5< (2& ',7/3(3,7) 0,+ )(*(3'
&QC353$+3C- (, (2& '2*+4&/
).2&+&Z\)37)37
I
I
Tr
kqTFF ex
*7/
\',)
mgTF
y
>53-37*(& T$&(1&&7 (2&)&
&QC*(3,7)
(, ,$(*37Z I
I
(*7mgr
kq
b&0&++374 (, (2& 034C+& 0,+ ^+,$5&-
aV; +&5*(& (2& )&.*+*(3,7 ,0 (2&
).2&+&) r(, (2& 5&74(2 ,0 (2&
.&7/C5C-LZ
)37ILr
OC$)(3(C(& 0,+ r (, ,$(*37Z
II
I
)37V(*7
mgL
kq
,+
I
II
V(*7)37
mgL
kq 6!8
OC$)(3(C(& 7C-&+3'*5 :*5C&) *7/ &:*5C*(&
(*7)37I Z
W
II
IIIeI !\aW=]
-!=]-g)e=d!N4\=\!\V
La]=\gL-R!\edd=d(*7)37
U&'*C)& !(*7)37I Z (*7)37
*7/WW !\aW=]
-
8/12/2019 Cap21 Prob
76/92
L2*.(&+ I!a_
O,5:374 0,+ 43:&)Z !\+*/!ae=\
6b8 >:*5C*(& &QC*(3,7 6!8 13(2 +&.5*'374 qI13(2
q!qIZ
WW
II
IIeI
!\\e=]-!=]-g)e=d!N4\=\!\V
L\=!L]\=\gL-R!\edd=d
(*7)37
O,5:& 0,+ (, ,$(*37Z e=e+*/!aI=\
76 Y,C+ .,37( '2*+4&) ,0 &QC*5 -*473(C/& *+&
*++*74&/ *( (2& ',+7&+) ,0 *
)QC*+& ,0 )3/&L *) )2,17 37 Y34C+& I!TV]= 6a8 Y37/
(2& -*473(C/& *7/ /3+&'(3,7,0 (2& 0,+'&
&?&+(&/ ,7 (2& '2*+4& 37 (2& 5,1&+
5&0( ',+7&+ $< (2& ,(2&+ (2+&&
'2*+4&)= 6b8 O2,1 (2*( (2& &5&'(+3' 03&5/ *(
(2& -3/.,37( ,0 ,7& ,0 (2& )3/&) ,0 (2&
)QC*+& 3) /3+&'(&/ *5,74 (2*( )3/& (,1*+/
(2& 7&4*(3:& '2*+4& *7/ 2*) * -*473(C/&
E 43:&7 $< EkdqLI
! !] ]
=
Picture the Problem l&( (2& ,+3437 $&
*( (2& 5,1&+ 5&0(T2*7/ ',+7&+ *7/
/&)347*(& (2& .,37( '2*+4&) *) )2,17 37
(2& /3*4+*-= E& '*7 *..5< L,C5,-$M)
5*1 0,+ .,37( '2*+4&) (, 037/ (2& 0,+'&)
&?&+(&/ ,7 q! $< qI; qW; *7/ qV *7/
)C.&+3-.,)& (2&)& 0,+'&) (, 037/ (2&
7&(
0,+'& &?&+(&/ ,7 q!= J7 ^*+( 6b8; 1&M55
C)& L,C5,-$M) 5*1 0,+ (2& &5&'(+3' 03&5/
/C& (, * .,37( '2*+4& *7/ (2&
)C.&+.,)3(3,7 ,0 03&5/) (, 037/ (2&
&5&'(+3' 03&5/ *( .,37(P6\;LgI8=
E2
E3
F2,1
F3,1F4,1
E4
E1
L
L
y
xqq +=1
qq +=3qq =2
qq =4
)2/(0,LP
2L
6a8 X)374 )C.&+.,)3(3,7 ,0 0,+'&);
&?.+&)) (2& 7&( 0,+'& &?&+(&/ ,7
q!Z!;V!;W!;I! ****
D..5< L,C5,-$M) 5*1 (, &?.+&))
!;I* Z
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%%*
cc
c
I
I
W
!;IW!;I
!I
!;II!;I
!I
!;I
L
kqL
L
qqk
r
qkq
r
qkq
-
8/12/2019 Cap21 Prob
77/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) aa
D..5< L,C5,-$M) 5*1 (, &?.+&))
!;V*
Z
&&
%%*
cc
c
I
I
W
!;VW
!;V
!V!;VI
!;V
!V!;V
L
kqL
L
qqk
r
qkq
r
qkq
D..5< L,C5,-$M) 5*1 (, &?.+&))
!;W*
Z
'&
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ccI
ccI
c
IIW
I
WIW
I
!;WW
!;W
!W!;WI
!;W
!W!;W
L
kq
LLL
kq
r
qkq
r
qkq
OC$)(3(C(& *7/ )3-.530< (, ,$(*37Z
'&
'&'&
&'&'*
ccII
!!
ccI
cc
cccI
c
I
I
IIW
I
I
I
I
I
IIW
I
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I
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L
kq
L
kq
L
kq
L
kq
L
kq
L
kq
6b8 X)374 )C.&+.,)3(3,7 ,0 03&5/);
&?.+&)) (2& +&)C5(*7( 03&5/ *( .,37(
PZ
VWI! """""
P 6!8
X)& L,C5,-$M) 5*1 (, &?.+&)) !"
Z
''
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cVcI
I
cI
c
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W
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L
kqL
L
kq
Lrkq
rkq
P
P
P
X)& L,C5,-$M) 5*1 (, &?.+&)) I"
Z
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cVcI
I
cI
c
IW
W
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;II
;I
II
L
kqL
L
kq
L
r
qk
r
kq
P
P
P
-
8/12/2019 Cap21 Prob
78/92
L2*.(&+ I!ad
X)& L,C5,-$M) 5*1 (, &?.+&)) W"
Z
'&
'&%"
cI
!c]
d
cI
cc
IIW
W
;W
;WI
;W
WW
L
kq
LL
r
kq
r
kq
P
P
P
X)& L,C5,-$M) 5*1 (, &?.+&)) V" Z
'&
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cI
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d
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cc
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W
;V
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;V
VV
L
kq
LL
r
qk
r
kq
P
P
P
OC$)(3(C(& 37 &QC*(3,7 6!8 *7/ )3-.530< (,
,$(*37Z
''&'&''" c]]
!!
dcI
!c]
dcI
!c]
dcVcVIIIWIIWII
L
qk
L
kq
L
kq
L
kq
L
kqP
77 [SSM] Y34C+& I!TV_ )2,1) * /C-$$&55 ',7)3)(374 ,0 (1,
3/&7(3'*5
)-*55 .*+(3'5&); &*'2 ,0 -*)) m; *((*'2&/ (, (2&
&7/) ,0 * (237 6-*))5&))8 +,/ ,0
5&74(2 a (2*( 3) .3:,(&/ *( 3() '&7(&+= @2&
.*+(3'5&) '*++< '2*+4&) ,0 Gq*7/ Hq; *7/
(2& /C-$$&55 3) 5,'*(&/ 37 * C730,+-
&5&'(+3' 03&5/"
= O2,1 (2*( 0,+ )-*55 :*5C&)
,0 (2& *745& $&(1&&7 (2& /3+&'(3,7
,0 (2& /3.,5& *7/ (2& /3+&'(3,7 ,0 (2&
&5&'(+3'
03&5/; (2& )?.+&)) (2& -,-&7( ,0 37&+(3*
,0 (2&
/3.,5&Z
I
I! maI
-
8/12/2019 Cap21 Prob
79/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) ae
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8/12/2019 Cap21 Prob
84/92
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8/12/2019 Cap21 Prob
85/92
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43:&7 $
-
8/12/2019 Cap21 Prob
86/92
L2*.(&+ I!d_
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-
8/12/2019 Cap21 Prob
87/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
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D..5< (2& 1,+NTN37&(3' &7&+4:*5C*(374 (2&
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-
8/12/2019 Cap21 Prob
89/92
@2& >5&'(+3' Y3&5/ !Z P3)'+&(& L2*+4&
P3)(+3$C(3,7) de
Picture the Problem @2& 0+&& $,/
