28. sources of magnetic field.doc

8/20/2019 28. Sources of Magnetic Field.doc

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[ Problem View ]

Magnetic Field near a Moving Charge

Description: Use Biot-Savart law to find the magnetic field at various points due to a

charge moving along the z axis

! particle with positive charge is moving with speed along the z axis toward positive

!t the time of this problem it is located at the origin" #our tas$ is to find themagnetic field at various locations in the three-dimensional space around the moving

charge

Part A

%hich of the following expressions gives the magnetic field at the point due to the

moving charge&

!

B

'

(

ANSWER: ! onl) B onl) ' onl) ( onl)

both ! and B both ' and ( both ! and ' both B and (

*he main point here is that the r -dependence is reall) *he results from using

http://session.masteringphysics.com/myct/problemView?assignmentProblemID=1206664http://session.masteringphysics.com/myct/problemView?assignmentProblemID=1206664


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in the numerator rather than the unit vector

! second point is that the order of the cross product must be such that the right-hand

rule wor$s+ ,f )our right thumb is along the direction of the current" " )our fingersmust curl along the direction of the magnetic field

Part B

ind the magnetic field at the point

Part B.1

Part not displayed

.xpress )our answer in terms of " " " and " and use " " and for the three unit

vectors

ANSWER: / 0mu123045pi665075v3x189:65)1unit

Part C


.xpress )our answer in terms of " " " " and " and use " " and for the threeunit vectors

ANSWER:

/

-0mu123045pi665075v 3)189:65x1unit

Part D


Part D.1 Evalate the cross prodct

*o find the magnetic field for this part" it is convenient to use expression B from Part

!+

;nowing and " find

.xpress )our answer in terms of given 7uantities and " " and3or

ANSWER: / v5x85)1unit


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Part D.! Find the distance "ro# the charge

Part not displayed

.xpress )our answer in terms of " " " " " and " and use " " and for the threeunit vectors

ANSWER: / 0mu123045pi665075v5x1830x189: < z189:690=3:665)1unit

Part E

*he field found in this problem for a moving charge is the same as the field from a

current element of length carr)ing current provided that the 7uantit) is replaced b)

which 7uantit)&

$int E.1

Hint not displayed

ANSWER: i5dl

[ Print ]

[ Problem View ]

Magnetic Field "ro# Crrent Seg#ents



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Description: >agnetic field calculations for z-axis as a function of currents in x) plane

%earning &oal: *o appl) the Biot-Savart law to find the magnetic field produced on the z

axis from current elements in the xy plane

,n this problem )ou are to find the magnetic field component along the z axis that results

from various current elements in the xy plane 0ie" at 6

*he field at a point due to a current-carr)ing wire is given b) the Biot-Savart law"

"

where and " and the integral is done over the current-carr)ing wire

.valuating the vector integral will t)picall) involve the following steps+

•

'hoose a convenient coordinate s)stem--t)picall) rectangular" sa) with coordinateaxes " " and

• %rite in terms of the coordinate variables and directions 0 " " etc6 *o

do this" )ou must find and !gain" finding the cross product can be doneeither

o geometricall) 0b) finding the direction of the cross product vector first"

then chec$ing for cancellations from an) other portion of the wire" and then

finding the magnitude or relevant component6 or

o algebraicall) 0b) using " etc6

• .valuate the integral for the component0s6 of interest

,n this problem" )ou will focus on the second of these steps and find the integrand forseveral different current elements #ou ma) use either of the two methods suggested for

doing this

Part A


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*he field at the point shown in thefigure due to a single current element is given b)

"

where and ,n this expression" what is the variable &

$int A.1 Ma'ing sense o" s(scripts

• points from the origin to the point where )ou want to find the magnetic field

• points from the origin to the current element in 7uestion

• points from the current element to the point where )ou want to find the

magnetic field

ANSWER:

Part B

ind " the z component of the magnetic field at the point from the

current flowing over a short distance located at the point


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$int B.1 Cross prodct

*he $e) here is the cross product in the Biot-Savart law %hat is the cross product

when and are parallel&

.xpress )our answer in terms of " " " " " and the unit vectors " " and3or

ANSWER: /2

Part C




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Part C.1 Deter#ine the displace#ent "ro# the crrent ele#ent

%hat is " the distance 0magnitude6 from the current element to the point in

7uestion&

ANSWER: / x18

Part C.! Find the direction "ro# the cross prodct

%hat is the unit vector that describes the direction of the magnetic field at the origin

&

.xpress )our answer in terms of " " or

ANSWER: z1unit


ANSWER: /0mu1263045pi6 5,5dl 3x189:0mu1263045pi6 5,5dl 3x189:5z1unit

Part D

ind " the z component of the magnetic field at the point P located at

from the current flowing over a short distance located at the


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point

Part D.1 Deter#ine the displace#ent "ro# the crrent ele#ent

Part not displayed

Part D.! )se the cross prodct to get the direction

Part not displayed

.xpress )our answer in terms of " " " " " " and the unit vectors " " and3or

ANSWER: / 2

Part E




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Part E.1 Deter#ine the displace#ent "ro# the crrent ele#ent

Part not displayed

Part E.! Find the direction o" the #agnetic "ield vector

Part not displayed


ANSWER: /-mu12 5 , 5 dl 3 04 5 pi 5 )189:6

-mu12 5 , 5 dl 3 04 5 pi 5 )189:65 z1unit

Part F

ind " the z component of the magnetic field at the point P located at

from the current flowing over a short distance located at the


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point

Part F.1 Deter#ine the displace#ent "ro# the crrent ele#ent

%hat is " the displacement 0magnitude6 from the current element to the point in

7uestion& *he figure shows another perspective of the same situation to ma$e thiscalculation easier


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ANSWER: / s7rt0x189: < z189:6

Part F.! Deter#ine *hich nit vector to se

!nother wa) to write the Biot-Savart law is

"

where )ou replace with *his eliminates the problem of finding and can ma$ecomputation easier

#ou are as$ed for the z component of the magnetic field points in the direction

%hich component 0 " " or 6 must )ou cross with to get a vector in the z

direction& ?ecall that

.xpress )our answer in terms of " " or 0ignoring the sign6

ANSWER: x1unit

Part F.+ Evalate the cross prodct

So" the z component of the magnetic field results from the cross product of and the

x component of *hese two vectors are orthogonal" so finding the cross product is

relativel) straightforward %hat is the value of &

@ive )our answer in terms of " " " and

ANSWER: / dl 5 x18 5 z1unit

Substitute this expression into the formula for the magnetic field given in the last hint

Abserve that it has in the denominator since in the original e7uation was replaced

with in the numerator

.xpress )our answer in terms of " " " " " " and the unit vectors " " and3or

ANSWER:

/

mu12 5 dl 3 04 5 pi6 5 , 5 x18 3 0x189: < z189:690=3:6

mu12 5 dl 3 04 5 pi6 5 , 5 x18 3 0x189: < z189:690=3:65z1unit


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[ Print ]

orce

/0,185,1:5mu125a9:630:5pi50d9:-a9:3466

/0m5,1:5mu12630:5pi50d9:-a9:3466

[ Problem View ]

Magnetic Field inside a ,er- %ong Solenoid

Description: eads through steps of using !mpereCs law to find field inside solenoid a

long solenoid

%earning &oal: *o appl) !mpDreCs law to find the magnetic field inside an infinite

solenoid

,n this problem we will appl) !mpDreCs law" written

"

to calculate the magnetic field inside a ver) long solenoid *he solenoid has length "

diameter " and turns per unit length with each carr)ing current

,t is usual to assume that the component

of the current along the z axis is negligible 0*his ma) be assured b) winding two la)ers of

closel) spaced wires that spiral in opposite directions6

rom s)mmetr) considerations it is possible to show that far from the ends of the

solenoid" the magnetic field is axial

Part A

%hich figure shows the loop that the must be used as the !mpDrean loop for finding



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for inside the solenoid&

Part A.1 Choice o" path "or loop integral

%hich of the following choices are a re7uirement of the !mpDrean loop that would

allow )ou to use !mpDreCs law to find &

a *he path must pass through the point

b *he path must have enough s)mmetr) so that is constant along large

parts of it

c *he path must be a circle

ANSWER: a onl) a and b a and c b and c

,f possible" choose the loop so that the desired field component runs parallel to theloop and other sides of the loop have zero field component along them

ANSWER: ! B ' (

Part B

!ssume that loop B 0in the Part B figure6 has length along %hat is the loop

integral in !mpDreCs law& !ssume that the top end of the loop is ver) far from the

solenoid 0even though it ma) not loo$ li$e it in the figure6" so that the field there is

assumed to be small and can be ignored

.xpress )our answer in terms of " " and other 7uantities given in the

introduction

ANSWER: /

B1z0r6 5 E1


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Part C

%hat ph)sical propert) does the s)mbol represent&

ANSWER: *he current along the path in the same direction as the magnetic

field

*he current in the path in the opposite direction from the

magnetic field

*he total current passing through the !mpDrean loop in either

direction

*he net current through the !mpDrean loop

*he positive direction of the line integral and the positive direction for the current arerelated b) the right-hand rule+

%rap )our right-hand fingers around the closed path" then the direction

of )our fingers is the positive direction for and the direction of )our

thumb is the positive direction for the net current

Fote also that the angle the current-carr)ing wire ma$es with the surface enclosed b)

the loop doesnCt matter 0,f the wire is at an angle" the normal component of the currentis decreased" but the area of intersection of the wire and the surface is correspondingl)

increased6

Part D

%hat is " the current passing through the chosen loop&

.xpress )our answer in terms of 0the length of the !mpDrean loop along the axis of

the solenoid6 and other variables given in the introduction

ANSWER: / , 5 n 5 E1

Part E

ind " the z component of the magnetic field inside the solenoid where !mpDreCs

law applies

.xpress )our answer in terms of " " " " and ph)sical constants such as

ANSWER: / mu125n5,

Part F


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%hat is " the z component of the magnetic field outside the solenoid&

Part F.1

Part not displayed

ANSWER:

2

Part &

*he magnetic field inside a solenoid can be found exactly using !mpDreCs law onl) if the

solenoid is infinitel) long Atherwise" the Biot-Savart law must be used to find an exactanswer ,n practice" the field can be determined with ver) little error b) using !mpDreCs

law" as long as certain conditions hold that ma$e the field similar to that in an infinitel)long solenoid

%hich of the following conditions must hold to allow )ou to use !mpDreCs law to find a

good approximation&

a 'onsider onl) locations where the distance from the ends is man) times

b 'onsider an) location inside the solenoid" as long as is much larger than forthe solenoid

c 'onsider onl) locations along the axis of the solenoid

$int &.1 #plications o" s-##etr-

,magine that the the solenoid is made of two e7ual pieces" one extending from to

and the other from to ,f both were present the field would have

its normal value" but if either is removed the field at drops to one-half of its

previous value *his shows that the field drops off significantl) near the ends of the

solenoid 0relative to its value in the middle6 Gowever" in doing this calculation" )ouassumed that the field is constant along the length of the !mpDrean loop So where

would this assumption brea$ down&

$int &.! /""0ais "ield dependence

#ou also used s)mmetr) considerations to sa) that the magnetic field is purel) axial

%here would this s)mmetr) argument not hold&


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Fote that far from the ends there cannot be a radial field" because it would impl) a

nonzero magnetic charge along the axis of the c)linder and no magnetic charges are

$nown to exist 0@aussCs aw for magnetic fields and charges6 ,n conHunction with!mpDreCs law" this allows us to conclude that the z component of the field cannot

depend on inside the solenoid

ANSWER: a onl) b onl) c onl) a and b a and c b and c

[ Print ]

orce

Fo tor7ue at all

%hichever pole of the magnet is slightl) closer to the iron than the other will be attracted

toward the iron

'onceptual

'loc$wise

no current

countercloc$wise

,ntroduction

if decreases with time*he surface can be an) surface whose edge is the loop

no matter how the field is generated

in either of the above two casesonl) if the field is generated b) the coulomb field of charges that are static or moving in astraight line

/0-!5B18630:5pi5?6

28. sources of magnetic field.doc

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