Chapter: Chapter 22 Learning Objectives LO 22.1.0 Solve problems related to the electric field. LO 22.1.1 Identify that at every point in the space srronding a charged particle! the particle sets p an electric field
E
! "hich is a vector #antity and ths has both magnitde and
direction. LO 22.1.2 $%plain ho" an electric field e%ert an electric force
F
E
can be sed to e%plain ho" a charged particle can
on a second charged particle even thogh there is no contact bet"een
the particles. LO 22.1.& $%plain ho" a small positive test charge is sed 'in principle( to measre the electric field at any given point. LO 22.1.) $%plain electric field lines! inclding "h ere they originate and terminate and "hat "h at their spacing represents. LO 22.2.0 Solve problems related to the electric field de to a charged particle. LO 22.2.1 In a s*etch! dra" a charged particle! indicate its sign! pic* a nearby point! and then dra" the electric field vector
E
at that point! "ith its tail anchored on the point.
LO 22.2.2 +or a given point in the electric field of a charged cha rged particle! identify the direction of the field vector
E
"hen the particle is positively charged and "hen it is negatively charged.
LO 22.2.& +or a given point in the electric field of a charged cha rged particle! apply the relationship bet"een the field magnitde E magnitde E ! the charge magnitde q! and the distance r bet"een the point and the particle. LO 22.2.) Identify that the e#ation given here for the magnitde of an electric field applies only to a particle! not an e%tended object. LO 22.2., If more than one electric field is set p at a point! dra" each electric field vector and then find the net electric field by b y adding the individal electric fields as vectors 'not as scalars(. LO 22.&.0 Solve problems related to the electric field de to an electric dipole. LO 22.&.1 -ra" an electric dipole! identifying the charges 'sies and signs(! dipole a%is! and direction of the electric dipole moment. LO 22.&.2 Identify the direction of the electric field at any given point along the dipole a%is. LO 22.&.& Otline ho" the e#ation e #ation for the electric field de to an a n electric dipole is derived from the e#ations for the electric field de to the individal charged particles. LO 22.&.) +or a single charged particle and an electric dipole! compare the rate at "hich the electric field magnitde decreases "ith increase in distance. LO 22.&., /pply the relationship bet"een the magnitde p magnitde p of of dipole moment! the charge separation d ! and the magnitde q of either of the charges. LO 22.&. +or any distant point along a dipole a%is! apply the relationship bet"een the electric field magnitde E magnitde E ! the distance z distance z from from the center of the dipole! and either the dipole moment magnitde p magnitde p or or the prodct of charge magnitde q and charge separation d . LO 22.).0 Solve problems related to the electric field de to a line of charge. LO 22.).1 +or a niform n iform distribtion of charge! find the linear charge density λ density λ for for charge along a line! the srface charge density σ for for charge on a srface! and the volme charge density ρ density ρ for for charge in a volme. LO 22.).2 +or charge that is distribted niformly along along a line! find the net electric field at a given point near the line by splitting the distribtion p into charge elements dq and dq and then
smming 'by integration( the electric field vectors
E
set p at the point by each element.
LO 22.).& $%plain ho" symmetry can be sed to simplify the calclation of the electric field at a point near a line of niformly distribted charge. LO 22.,.0 Solve problems related to the electric field de to a charged dis*. LO 22.,.1 S*etch a dis* "ith niform charge and indicate the direction of the electric field at a point on the central a%is if the charge is positive and if it is negative. LO 22.,.2 $%plain ho" the e#ation for the electric field on the central a%is of a niformly charged ring can be sed to find the e#ation for the electric field on the central a%is of a niformly charged dis*. LO 22.,.& +or a point on the central a%is of a niformly charged dis*! apply the relationship bet"een the srface charge density σ ! the dis* radis R! and the distance z to that point. LO 22..0 Solve problems related to a point charge in an electric field. LO 22..1 +or a charged particle placed in an e%ternal electric field 'a field de to other charged objects(! apply the relationship bet"een the electric field q! and the electrostatic force
F
E
at that point! the particles charge
that acts on the particle! and identify the relative directions of
the force and the field "hen the particle is positively charged and negatively charged. LO 22..2 $%plain illi*ans procedre of measring the elementary charge. LO 22..& $%plain the general mechanism of in*3jet printing. LO 22.4.0 Solve problems related to a dipole in an electric field. LO 22.4.1 On a s*etch of an electric dipole in an e%ternal electric field! indicate the direction of the field! the direction of the dipole moment! the direction of the electrostatic forces on the t"o ends of the dipole! and the direction in "hich those forces tend to rotate the dipole! and identify the vale of the net force on the dipole. LO 22.4.2 Calclate the tor#e on an electric dipole in an e%ternal electric field by evalating a cross prodct of the dipole moment vector and the electric field vector! in magnitde3angle notation and nit3vector notation. LO 22.4.& +or an electric dipole in an e%ternal electric field! relate the potential energy of the dipole to the "or* done by a tor#e as the dipole rotates in the electric field LO 22.4.) +or an electric dipole in an e%ternal electric field! calclate the potential energy by ta*ing a dot prodct of the dipole moment vector and the electric field vector! in magnitde3 angle notation and nit3vector notation. LO 22.4., +or an electric dipole in an e%ternal electric field! identify the an gles for the minimm and ma%imm potential energies and the angles for the minimm and ma%imm tor#e magnitdes. ltiple Choice 1. 5he nits of the electric field are: /( 6⋅C2 7( C86 C( 6 -( 68C $( C8m2
/ns: -ifficlty: $ Section: 2231 Learning Objective 22.1.0 2. 5he nits of the electric field are: /( 98'C⋅m( 7( 98C C( 9⋅C -( 98m $( none of these /ns: / -ifficlty: $ Section: 2231 Learning Objective 22.1.0 &. /n electric field is most directly related to: /( the momentm of a test charge 7( the *inetic energy of a test charge C( the potential energy of a test charge -( the force acting on a test charge $( the charge carried by a test charge /ns: -ifficlty: $ Section: 2231 Learning Objective 22.1.& ). /s sed in the definition of electric field! a test charge: /( has ero charge 7( mst be a proton C( has charge of magnitde 1. × 10 ;1< C -( mst be an electron $( none of the above /ns: $ -ifficlty: $ Section: 2231 Learning Objective 22.1.& ,. $%perimenter / ses a test charge q0 and e%perimenter 7 ses a test charge 2q0 to measre an electric field prodced by stationary charges. / finds a field that is: /( the same as the field fond by 7 7( greater than the field fond by 7 C( less than the field fond by 7
-( either greater or less than the field fond by 7! depending on the masses of the test charges $( either greater or less than the field fond by 7! depending on the accelerations of the test charges /ns: / -ifficlty: $ Section: 2231 Learning Objective 22.1.& . $lectric field lines: /( are trajectories of a test charge 7( are vectors in the direction of the electric field C( form closed loops -( cross each other in the region bet"een t"o point charges $( are none of the above /ns: $ -ifficlty: $ Section: 2231 Learning Objective 22.1.) 4. 5"o thin spherical shells! one "ith radis R and the other "ith radis 2 R! srrond an isolated charge point particle. 5he ratio of the nmber of field lines throgh the larger sphere to the nmber throgh the smaller is: /( 1 7( 2 C( ) -( 182 $( 18) /ns: / -ifficlty: $ Section: 2231 Learning Objective 22.1.) =. / certain physics te%tboo* sho"s a region of space in "hich t"o electric field lines cross each other. >e conclde that: /( at least t"o point charges are present 7( an electrical condctor is present C( an inslator is present -( the field points in t"o directions at the same place $( the athor made a mista*e /ns: $ -ifficlty: $ Section: 2231
Learning Objective 22.1.) <. Choose the correct statement concerning electric field lines: /( field lines may cross 7( field lines are close together "here the field is large C( field lines point a"ay from negatively charged particle -( a point charge particle released from rest moves along a field line $( none of these are correct /ns: 7 -ifficlty: $ Section: 2231 Learning Objective 22.1.) 10. 5he diagram sho"s the electric field lines de to t"o charged parallel metal plates. >e conclde that:
/( 7( C( -( $(
the pper plate is positive and the lo"er plate is negative a proton at ? "old e%perience the same force if it "ere placed at @ a proton at ? e%periences a greater force than if it "ere placed at A a proton at ? e%periences less force than if it "ere placed at A an electron at ? cold have its "eight balanced by the electrical force
/ns: 7 -ifficlty: $ Section: 2231 Learning Objective 22.1.) 11. 5he diagram sho"s the electric field lines in a region of space containing t"o small charged spheres '@ and A(. 5hen:
/( 7( C( -( $(
@ is negative and A is positive the magnitde of the electric filed is the same every"here the electric field is strongest mid"ay bet"een @ and A @ is positive and A is negative @ and A mst have the same sign
/ns: -
-ifficlty: $ Section: 2231 Learning Objective 22.1.) 12. 5"o protons 'p1 and p2( are on the x a%is! as sho"n belo". 5he directions of the electric field at points 1! 2! and & respectively! are:
/( 7( C( -( $(
→! ←! → ←! →! ← ←! →! → ←! ←! ← ←! ←! →
/ns: $ -ifficlty: Section: 2232 Learning Objective 22.2.2 1&. Let k denote 18)πε0. 5he magnitde of the electric field at a distance r from an isolated point charge q is: /( kq8r 7( kr 8q C( kq8r & -( kq8r 2 $( kq28r 2 /ns: -ifficlty: $ Section: 2232 Learning Objective 22.2.& 1). 5he electric field at a distance of 10 cm from an isolated point particle "ith a charge of 2 × 10 ;< C is: /( 1.= 68C 7( 1= 68C C( 1=0 68C -( 1=00 68C $( none of these /ns: -ifficlty: Section: 2232 Learning Objective 22.2.&
1,. /n isolated point charged point particle prodces an electric field "ith magnitde E at a point 2 m a"ay from the charge. / point at "hich the field magnitde is E 8) is: /( 0., m a"ay from the charge 7( 1 m a"ay from the charge C( 2 m a"ay from the charge -( ) m a"ay from the charge $( = m a"ay from the charge /ns: -ifficlty: Section: 2232 Learning Objective 22.2.& 1. /n isolated charged point particle prodces an electric field "ith magnitde E at a point 2 m a"ay. /t a point 1 m from the particle the magnitde of the field is: /( E 7( 2 E C( ) E -( E 82 $( E 8) /ns: C -ifficlty: Section: 2232 Learning Objective 22.2.& 14. 5he e#ation E B kq8r 2 applies to: /( only pointli*e particles 7( any symmetric objects C( all objects -( any spherical object $( it applies to all objects as long as they are not moving /ns: / -ifficlty: $ Section: 2232 Learning Objective 22.2.) 1=. 5"o point particles! "ith charges of q1 and q2! are placed a distance r apart. 5he electric field is ero at a point bet"een the particles on the line segment connecting them. >e conclde that: /( q1 and q2 mst have the same magnitde and sign 7( mst be mid"ay bet"een the particles C( q1 and q2 mst have the same sign bt may have different magnitdes -( q1 and q2 mst have e#al magnitdes and opposite signs $( q1 and q2 mst have opposite signs and may have different magnitdes
/ns: C -ifficlty: Section: 2232 Learning Objective 22.2., 1<. 5he diagrams belo" depict for different charge distribtions. 5he charged particles are all the same distance from the origin. 5he electric field at the origin:
/( 7( C( -( $(
is least for sitation 1 is greatest for sitation & is ero for sitation ) is do"n"ard for sitation 1 is do"n"ard for sitation &
/ns: C -ifficlty: Section: 2232 Learning Objective 22.2., 20. 5he diagram sho"s a particle "ith positive charge Q and a particle "ith negative charge −Q. 5he electric field at point on the perpendiclar bisector of the line joining them is:
/( 7( C( -( $(
↑ ↓ → ← ero
/ns: / -ifficlty: $ Section: 2232 Learning Objective 22.2., 21. 5he diagram sho"s t"o identical particles! each "ith positive charge Q. 5he electric field at point on the perpendiclar bisector of the line joining them is:
/( 7( C( -( $(
↑ ↓ → ← ero
/ns: C -ifficlty: $ Section: 2232 Learning Objective 22.2., 22. 5"o point particles! one "ith charge D= × 10 ;< C and the other "ith charge;2 × 10 ;< C! are separated by ) m. 5he electric field mid"ay be t"een them is: /( < × 10< 68C 7( 1&!,00 68C C( 1&,!000 68C -( & × 10 ;< 68C $( 22., 68C /ns: $ -ifficlty: Section: 2232 Learning Objective 22.2., 2&. 5"o charged point particles are located at t"o vertices of an e#ilateral triangle and the electric field is ero at the third verte%. >e conclde: /( the t"o particles have charges "ith opposite signs and the same magnitde 7( the t"o particles have charges "ith opposite signs and different magnitdes C( the t"o particles have identical charges -( the t"o particles have charges "ith the same sign bt different magnitdes $( at least one other charged particle is present /ns: $ -ifficlty: $ Section: 2232 Learning Objective 22.2., 2). 5"o point particles! "ith the same charge! are located at t"o vertices of an e#ilateral triangle. / third charged particle is placed so the electric field at the third verte% is ero. 5he third particle mst: /( be on the perpendiclar bisector of the line joining the first t"o charges 7( be on the line joining the first t"o charges
C( be identical to the first t"o charges -( have the same magnitde as the first t"o charges bt may have a different sign $( be at the center of the triangle /ns: / -ifficlty: $ Section: 2232 Learning Objective 22.2., 2,. 5"o point charges are arranged as sho"n. In "hich region cold a third charge D1 C be placed so that the net electrostatic force on it is eroE
/( 7( C( -( $(
I only I and II only III only I and III only II only
/ns: / -ifficlty: $ Section: 2232 Learning Objective 22.2., 2. /n electric dipole consists of a particle "ith a charge of D × 10 ; C at the origin and a particle "ith a charge of ; × 10 ; C on the x a%is at x B & × 10 ;& m. 5he direction of the electric field de to the dipole at points on the x a%is is: /( in the positive x direction 7( in the negative x direction C( in the positive y direction -( in the negative y direction $( in the positive x direction bet"een the charges and in the negative x direction else"here /ns: $ -ifficlty: $ Section: 223& Learning Objective 22.&.2 24. Comparing the field of a single point charge "ith the field of an electric dipole! /( the field of the point charge decreases more rapidly "ith distance 7( the field of the point charge decreases less rapidly "ith distance C( the field of the point charge decreases more rapidly "ith distance bt only along the dipole a%is -( the field of the point charge decreases less rapidly "ith distance bt only perpendiclar to the dipole a%is
$( the fields decrease e#ally rapidly "ith distance /ns: 7 -ifficlty: $ Section: 223& Learning Objective 22.&.) 2=. /n electric dipole consists of a particle "ith a charge of D × 10 ; C at the origin and a particle "ith a charge of ; × 10 ; C on the x a%is at x B & × 10 ;& m. Its dipole moment is: /( 1.= × 10 ;= C⋅m! in the positive x direction 7( 1.= × 10 ;= C⋅m! in the negative x direction C( 0 C⋅m! becase the net charge is 0 -( 1.= × 10 ;= C⋅m! in the positive y direction $( 1.= × 10 ;= C⋅m! in the negative y direction /ns: 7 -ifficlty: $ Section: 223& Learning Objective 22.&., 2<. /n electric dipole consists of t"o charges! F2., GC! separated by 1.0 % 103) m and centered on the origin. If the dipole is oriented along the x a%is! "hat is the electric field at x B 1, cmE /( 0 68C 7( 100 68C C( 200 68C -( 40 68C $( 1&00 68C /ns: $ -ifficlty: $ Section: 223& Learning Objective 22.&. &0. / total charge of .& × 10 ;= C is distribted niformly throghot a 2.43cm radis sphere. 5he volme charge density is: /( &.4 × 10 ;4 C8m& 7( .< × 10 ; C8m& C( .< × 10 ; C8m2 -( 2., × 10 ;) C8m& $( 4. × 10 ;) C8m& /ns: $ -ifficlty: $ Section: 223) Learning Objective 22.).1
&1. Charge is placed on the srface of a 2.43cm radis isolated condcting sphere. 5he srface charge density is niform and has the vale .< × 10 ; C8m2. 5he total charge on the sphere is: /( ,.4 × 10 ;10 C 7( 1. × 10 ;= C C( &.2 × 10 ;= C -( .& × 10 ;= C $( 4., × 10 ;) C /ns: -ifficlty: $ Section: 223) Learning Objective 22.).1 &2. / spherical shell has an inner radis of &.4 cm and an oter radis of )., cm. If charge is distribted niformly throghot the shell "ith a volme density of .1 × 10 ;) C8m& the total charge is: /( 1.0 × 10 ;4 C 7( 1.& × 10 ;4 C C( 2.0 × 10 ;4 C -( 1.4 × 10 ; C $( ,.0 × 10 ; C /ns: / -ifficlty: $ Section: 223) Learning Objective 22.).1 &&. / cylinder has a radis of 2.1 cm and a length of =.= cm. 5otal charge .1 × 10 ;4 C is distribted niformly throghot. 5he volme charge density is: /( 4.) × 10 ;11 C8m& 7( ,.& × 10 ;, C8m& C( ,.& × 10 ;, C8m2 -( =., × 10 ;) C8m& $( ,.0 × 10 ;& C8m& /ns: $ -ifficlty: $ Section: 223) Learning Objective 22.).1 &). ositive charge Q is niformly distribted on a semicirclar rod. >hat is the direction of the electric field at point ! the center of the semicircleE
/( 7( C( -( $(
↑ ↓ ← →
/ns: -ifficlty: $ Section: 223) Learning Objective 22.).& &,. ositive charge DQ is niformly distribted on the pper half of a semicirclar rod and negative charge ; Q is niformly distribted on the lo"er half. >hat is the direction of the electric field at point ! the center of the semicircleE
/( 7( C( -( $(
↑ ↓ ← →
/ns: 7 -ifficlty: $ Section: 223) Learning Objective 22.).& &. ositive charge DQ is niformly distribted on the pper half of a rod and a negative charge –Q is niformly distribted on the lo"er half. >hat is the direction of the electric field at point ! on the perpendiclar bisector of the rodE
/( 7( C( -(
↑ ↓ ← →
$( /ns: 7 -ifficlty: $ Section: 223) Learning Objective 22.).& &4. / dis* "ith a niform positive srface charge density lies in the x-y plane! centered on the origin. /long the positive z a%is! the direction of the electric field is: /( in the D z direction 7( in the ; z direction C( in the D x direction -( in the D y direction $( there is no field along the positive z a%is /ns: / -ifficlty: $ Section: 223, Learning Objective 22.,.1 &=. / dis* "ith a niform positive srface charge density lies in the x-y plane! centered on the origin. 5he dis* contains 2., % 103 C8m2 of charge! and is 4., cm in radis. >hat is the electric field at z B 1, cmE /( &0 68C 7( &00 68C C( &000 68C -( &.0 % 10) 68C $( &.0 % 104 68C /ns: -ifficlty: Section: 223, Learning Objective 22.,.& &<. 5he electric field de to a niform distribtion of charge on a spherical shell is ero: /( every"here 7( no"here C( only at the center of the shell -( only inside the shell $( only otside the shell /ns: -ifficlty: $ Section: 223 Learning Objective 22..1
)0. / charged particle is placed in an electric field that varies "ith location. 6o force is e%erted on this charge: /( at locations "here the electric field is ero 7( at locations "here the electric field strength is 18'1. × 10 ;1<( 68C C( if the particle is moving along a field line -( if the particle is moving perpendiclar to a field line $( if the field is cased by an e#al amont of positive and negative charge /ns: / -ifficlty: $ Section: 223 Learning Objective 22..1 )1. 5he magnitde of the force of a )00368C electric field on a 0.023C point charge is: /( = 6 7( , × 10 ;, 6 C( = × 10 ;& 6 -( 0.0= 6 $( 2 × 10& 6 /ns: / -ifficlty: $ Section: 223 Learning Objective 22..1 )2. / 200368C electric field is in the positive x direction. 5he force on an electron in this field is: /( 200 6 in the positive x direction 7( 200 6 in the negative x direction C( &.2 × 10 ;14 6! in the positive x direction -( &.2 × 10 ;14 6! in the negative x direction $( 0 6 /ns: -ifficlty: $ Section: 223 Learning Objective 22..1 )&. /n electron traveling north enters a region "here the electric field is niform and points north. 5he electron: /( speeds p 7( slo"s do"n C( veers east -( veers "est $( contines "ith the same speed in the same direction /ns: 7
-ifficlty: $ Section: 223 Learning Objective 22..1 )). /n electron traveling north enters a region "here the electric field is niform and points "est. 5he electron: /( speeds p 7( slo"s do"n C( veers east -( veers "est $( contines "ith the same speed in the same direction /ns: C -ifficlty: $ Section: 223 Learning Objective 22..1 ),. / charged oil drop "ith a mass of 2 × 10 ;) *g is held sspended b y a do"n"ard electric field of &00 68C. 5he charge on the drop is: /( D1., × 10 C 7( ;1., × 10 C C( D., × 10 ; C -( ;., × 10 ; C $( 0 C /ns: -ifficlty: $ Section: 223 Learning Objective 22..1 ). 5he prpose of illi*enHs oil drop e%periment "as to determine: /( the mass of an electron 7( the charge of an electron C( the ratio of charge to mass for an electron -( the sign of the charge on an electron $( viscosity /ns: 7 -ifficlty: $ Section: 223 Learning Objective 22..2 )4. 5he force e%erted by a niform electric field on a dipole is: /( parallel to the dipole moment 7( perpendiclar to the dipole moment C( parallel to the electric field
-( perpendiclar to the electric field $( none of the above /ns: $ -ifficlty: $ Section: 2234 Learning Objective 22.4.1 )=. /n electric field e%erts a tor#e on a dipole only if: /( the field is parallel to the dipole moment 7( the field is not parallel to the dipole moment C( the field is perpendiclar to the dipole moment -( the field is not perpendiclar to the dipole moment $( the field is niform /ns: 7 -ifficlty: $ Section: 2234 Learning Objective 22.4.1 )<. 5he tor#e e%erted by an electric field on a dipole is: /( parallel to the field and perpendiclar to the dipole moment 7( parallel to both the field and dipole moment C( perpendiclar to both the field and dipole moment -( parallel to the dipole moment and perpendiclar to the field $( not related to the directions of the field and dipole moment /ns: C -ifficlty: $ Section: 2234 Learning Objective 22.4.2 ,0. / niform electric field of &00 68C ma*es an angle of 2,° "ith the dipole moment of an electric dipole. If the tor#e e%erted by the field has a magnitde of 2., × 10 ;4 6⋅m! the dipole moment mst be: /( =.& × 10 ;10 C⋅m 7( <.2 × 10 ;10 C⋅m C( 2.0 × 10 ;< C⋅m -( =.& × 10 ;, C⋅m $( 1.= × 10 ;) C⋅m /ns: C -ifficlty: Section: 2234 Learning Objective 22.4.2
,1. >hen the dipole moment of a dipole in a niform electric field rotates to become more nearly aligned "ith the field: /( the field does positive "or* and the potential energy increases 7( the field does positive "or* and the potential energy decreases C( the field does negative "or* and the potential energy increases -( the field does negative "or* and the potential energy decreases $( the field does no "or* /ns: 7 -ifficlty: $ Section: 2234 Learning Objective 22.4.& ,2. 5he dipole moment of a dipole in a &00368C electric field is initially perpendiclar to the field! bt it rotates so it is in the same direction as the field. If the moment has a magnitde of 2 × 10 ;< C⋅m the "or* done by the field is: /( ;12 × 10 ;4 9 7( ; × 10 ;4 9 C( 0 9 -( × 10 ;4 9 $( 12 × 10 ;4 9 /ns: -ifficlty: Section: 2234 Learning Objective 22.4.& ,&. /n electric dipole is oriented parallel to a niform electric field! as sho"n.
It is rotated to one of the five orientations sho"n belo". an* the final orientations according to the change in the potential energy of the dipole3field system! most negative to most positive.
/( 7( C( -( $(
1! 2! &! ) )! &! 2! 1 1! 2! )! & &! then 2 and ) tie! then 1 1! then 2 and ) tie! then &
/ns: / -ifficlty: $ Section: 2234
Learning Objective 22.4.& ,). /n electric dipole of dipole moment )., % 103< CJm is in a 2,0 68C electric field. If the electric field points east! and the dipole moment vector points &4K "est of soth! "hat is the potential energyE /( ;.= % 103= 9 7( ;<.0 % 1034 9 C( 0 9 -( .= % 103= 9 $( <.0 % 1034 9 /ns: -ifficlty: $ Section: 2234 Learning Objective 22.4.) ,,. 5he diagrams sho" for possible orientations of an electric dipole in a niform electric field E
an* them according to the magnitde of the tor#e e%erted on the dipole by the field!
least to greatest.
/( 7( C( -( $(
1! 2! &! ) )! &! 2! 1 1! 2! )! & &! then 2 and ) tie! then 1 1! then 2 and ) tie! then &
/ns: $ -ifficlty: $ Section: 2234 Learning Objective 22.4.,
