N S

Figure 7.1

A small compass can be used totrace the magnetic field lines of abar magnet.

Magnetic Forces and Mag -netic Fields

Chapter 7

FIGURE 7.2 (a) Magnetic field patterns surrounding a bar magnet as displayed with iron filings. (b) Magnetic fieldpatterns between dissimilar poles of two bar magnets. (c) Magnetic field pattern between similar poles of two bar magnets.

(Co u

rtes y

ofHe

nry

Leap

a nd

J im

Lehm

an)

(a) (b) (c)

(a)

FB

+ q

v

θ

(b)

FB

BF

B

B

v

v

+

–

FIGURE 7.3 The direction of themagnetic force on a charged particle movingwith a velocity in the presence of amagnetic field . (a) When is at an angle to , the magnetic force is perpendicular toboth and . (b) Oppositely directedmagnetic forces are exerted on twooppositely charged particles moving with thesame velocity in a magnetic field. Thebroken lines suggest the paths followed bythe particles after the instant shown in the figure.

B:

v:B:

v:B:

v:

BFB

v

(a)

v

(b)

FB

B

FIGURE 7.4 Two right-hand rules for determiningthe direction of the magnetic force actingon a particle with charge q moving with a velocity in amagnetic field . (a) In this rule, the fingers point in thedirection of , with coming out of your palm, so thatyou can curl your fingers in the direction of . Thedirection of , and the force on a positive charge, isthe direction in which the thumb points. (b) In this rule,the vector is in the direction of your thumb and is inthe direction of your fingers. The force on a positivecharge is in the direction of your palm, as if you arepushing the particle with your hand.

F:

B

B:

v:

v: B:

B:

B:

v:B:

v:F:

B q v: B:

(a)

(b)

out of page:

into page:

××××××

××××××

××××××

××××××

××××××

××××××

××××××

B

B

F IGURE 7.5 (a)Magnetic field lines coming outof the paper are indicated by dots, representing thetips of arrows coming outward. (b) Magnetic fieldlines going into the paper are indicated by crosses,representing the feathers of arrows going inward.

z

y

x

B

60

e

B

F

v

FIGURE 7.6 (Example 7.1) Themagnetic force on the electron isin the negative z direction when and lie in the xy plane.B

:v:

F:

B

r

v

v

v

q

q

q

B in

FB

FB

FB

+

+

+

Figure 7.7

When the velocity of a charged particleis perpendicular to a uniform magneticfield, the particle moves in a circularpath in a plane perpendicular to . Themagnetic force acting on the chargeis always directed toward the center ofthe circle

F:

B

B:

Helicalpath

B

x

+q

z

y

+

Figure 7.8

A charged particle having a velocityvector with a component parallel to auniform magnetic field moves in ahelical path.

F

v

Magnetic fieldregion (out of page)

Particlemotion

B

FIGURE 7.9 (ThinkingPhysics 7.2) A positivelycharged particle enters aregion of magnetic fielddirected out of the page.

FIGURE 7.10 (Example 7.3) Thebending of an electron beam in amagnetic field.

(Cou

rt esy

of H

enry

Lea

p a n

d Ji

m L

ehm

an)

Bin

+

E

Source

Slit

(a)

++++++

–––––

v

(b)

+ q

qv × B

qE

×

×

Figure 7.11

(a) A velocity selector. When a positively charged particle is inthe presence of a magnetic field directed into the page and anelectric field directed downward, it experiences a downwardelectric force and an upward magnetic force .(b) When these forces balance, the particle moves in a straightline through the fields.

q v: B:

q E:

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

r

P

Bin

Velocity selector

E

0, in

q

Detectorarray

B

v

Fgure 7.12

A mass spectrometer. Positively charged particles are sentfirst through a velocity selector and then into a regionwhere the magnetic field causes the particles to move ina semicircular path and strike a detector array at P.

B:

0

FIGURE 7.13 (a) Thomson’s apparatus for measuring e/me . Electrons are accelerated from the cathode, pass through two slits,and are deflected by both an electric field and a magnetic field (directed perpendicular to the electric field). The electrons thenstrike a fluorescent screen. (b) J. J. Thomson (left) in the Cavendish Laboratory, University of Cambridge. It is interesting to notethat the man on the right, Frank Baldwin Jewett, is a distant relative of John W. Jewett Jr., coauthor of this text.

Fluorescentcoating

–

SlitsCathode

–

+

+

+

Deflectionplates

Magnetic field coil

Deflected electron beam

Undeflectedelectronbeam

(a)

(Bel

l Tel

epho

ne L

abs/

Cou r

tes y

of E

mi li

o Se

grè

Visu

al A

rchi

ves)

(b)

B

P

D1

D2

North pole of magnet

Particle exits here

Alternating ∆V

)b()a(

FIGURE 7.14 (a) Acyclotron consists of an ionsource at P, two hollow sectionscalled dees, D1 and D2, acrosswhich an alternating potentialdifference is applied, and auniform magnetic field. (Thesouth pole of the magnet is notshown.) The red dashedcurved lines represent the pathof the particles. (b) The firstcyclotron, invented by E. O.Lawrence and M. S. Livingstonin 1934.

( Cou

r tes y

of L

awre

nce

Ber k

ele y

Lab

ora t

ory ,

Uni v

ersi

ty o

f Cal

if orn

ia)

××××××

××××××

××××

××××××

×××××

×

(b)

Bin

I = 0

Bin

××××××

××××××

××××

××××××

×××××

× ××××××

××××××

××××

××××××

×××××

×

I

Bin

I

)d()c((a)

FIGURE 7.15 (a) A wiresuspended vertically betweenthe poles of a magnet. (b) Thesetup shown in (a) as seenlooking at the south pole ofthe magnet so that themagnetic field (green crosses)is directed into the page.When no current is flowing inthe wire, it remains vertical.(c) When the current isupward, the wire deflects tothe left. (d) When the currentis downward, the wire deflectsto the right.

qvd

ABin

FB

+

FIGURE 7.16 A section of a wirecontaining moving charges in amagnetic field . The magneticforce on each charge is , andthe net force on a segment of length

is .I:

B:

q v:d B:

B:

Bd

I

s

FIGURE 7.17 A wire segment ofarbitrary shape carrying a current Iin a magnetic field experiences amagnetic force. The force on anylength element is andis directed out of the page.

Id s: d B:

d s:

B:

R

I

θdθ

d

θ

B

θ

s

F IGURE 7.18 (Example 7.4) Thenet force on a closed current loop in auniform magnetic field is zero. For theloop shown here, the force on thestraight portion is 2IRB and out of thepage, whereas the force on the curvedportion is 2IRB and into the page.

(a)

b

a

I

B

(b)

B

F2

O

F4

b2

I

I

I

FIGURE 7.19 (a) Overhead viewof a rectangular current loop in auniform magnetic field. No magneticforces are exerted on sides and because these sides are parallel to .Forces are exerted on sides and ,however. (b) Edge view of the loopsighting down and shows thatthe forces and exerted on thesesides create a torque that tends torotate the loop clockwise. The purpledot in the left circle representscurrent in wire coming towardyou; the purple in the right circlerepresents current in wire movingaway from you.

F:

4F:

2

B:

(a)

b

a

I

B

(b)

B

F2

O

F4

b2

I

I

I

FIGURE 7.19 (a) Overhead viewof a rectangular current loop in auniform magnetic field. No magneticforces are exerted on sides and because these sides are parallel to .Forces are exerted on sides and ,however. (b) Edge view of the loopsighting down and shows thatthe forces and exerted on thesesides create a torque that tends torotate the loop clockwise. The purpledot in the left circle representscurrent in wire coming towardyou; the purple in the right circlerepresents current in wire movingaway from you.

F:

4F:

2

B:

F2

F4

OB

A

b2– sin θ

b2

θ θ

θ

×

Figure 7.20

An end view of the loop in Figure 7.19brotated through an angle with respect tothe magnetic field. If is at an angle uwith respect to vector , which isperpendicular to the plane of the loop,the torque is IAB sin u.

A:

B:

A

I

µ

FIGURE 7.21 Right-hand rule fordetermining the direction of the vector . The direction of the magnetic moment

is the same as the direction of .A::

A:

Pd Bout

r

θ d

PdBin

Ir

r

s

F IGURE 7.22 The magnetic fieldat a point P due to a current I

through a length element is givenby the Biot–Savart law. The field isout of the page at P and into the pageat P . (Both P and P are in the planeof the page.)

d s:d B

:

r

I

B

FIGURE 7.23 The right-handrule for determining the direction ofthe magnetic field surrounding along, straight wire carrying a current.Note that the magnetic field linesform circles around the wire. Themag-nitude of the magnetic field at adistance r from the wire is given byEquation 7.21.

:

O

R

θ

d

y

z

I

r

r

xθ

PxdBx

dBydB

s

FIGURE 7.24 (Example 7.6) The geometry forcalculating the magnetic field at a point P lying on theaxis of a current loop. By symmetry, the total field isalong this axis.

B:

(© R

icha

rd M

egna

, Fun

dam

enta

l Pho

togr

aphs

)

(a) (b) (c)

S

N

IS

N

FIGURE 7.25 (Example 7.6) (a) Magnetic field lines surrounding a current loop. (b) Magnetic field lines surrounding acurrent loop displayed with iron filings. (c) Magnetic field lines surrounding a bar magnet. Note the similarity between thisline pattern and that of a current loop.

2

1

B2

a

I1

I2

F1

a

Figure 7.26

Two parallel wires that each carry asteady current exert a force on eachother. The field due to the currentin wire 2 exerts a force of magnitudeF1 5 I1,B2 on wire 1. The force isattractive if the currents are parallel(as shown) and repulsive if thecurrents are antiparallel.

B:

2

)b()a(

I = 0

I

d

B

s

Figure 7.27

(a) When no current is present in the vertical wire, all compass needles point in the same direction (toward the Earth’sNorth Pole). (b) When the wire carries a strong current, the compass needles deflect in a direction tangent to the circle,which is the direction of the magnetic field created by the current. (c) Circular magnetic field lines surrounding a current-carrying conductor, displayed with iron filings.

( © R

icha

rd M

egna

, Fun

dam

e nta

l Ph o

t ogr

aph s

)

(c)

2R

r

1 I

d s

FIGURE 7.28 (Example 7.7)A long, straight wire of radius Rcarrying a steady current Iuniformly distributed across thewire. The magnetic field at anypoint can be calculated fromAmpère’s law using a circular pathof radius r, concentric with the wire.

Rr

B 1/r

B r

B

FIGURE 7.29 (Example 7.7)Magnitude of the magnetic fieldversus r for the wire described inFigure 7.31. The field isproportional to r inside the wireand varies as 1/r outside the wire.

B

ca

d

I

I

r

b

s

F IGURE 7.30 (Example 7.8)A toroid consisting of many turnsof wire wrapped around adoughnut-shaped structure (calleda torus). If the coils are closelyspaced, the field in the interior ofthe toroid is tangent to the dashedcircle and varies as 1/r. Thedimension a is the cross-sectionalradius of the torus. The fieldoutside the toroid is very small andcan be described by using theamperian loop at the right side,perpendicular to the page.

(a)

S

N

(Hen

ry L

eap

a nd

Jim

Le h

man

)

F IGURE 7.31 (a) Magnetic field linesfor a tightly wound solenoid of finite lengthcarrying a steady current. The field in thespace enclosed by the solenoid is nearlyuniform and strong. Note that the field linesresemble those of a bar magnet and that thesolenoid effectively has north and southpoles. (b) The magnetic field pattern of a barmagnet, displayed with iron filings.

(b)

B

×

×××

×

3

2

4

1

w

×

FIGURE 7.32 Cross-sectional view of an ideal solenoid, where the interiormagnetic field is uniform and the exterior field is close to zero. Ampère’s lawapplied to the circular path near the bottom whose plane is perpendicular tothe page can be used to show that there is a weak field outside the solenoid.Ampère’s law applied to the rectangular dashed path in the plane of the pagecan be used to calculate the magnitude of the interior field.

r

µ

L

I

FIGURE 7.33 Anelectron moving in a circularorbit of radius r has an angularmomentum in onedirection and a magneticmoment in the oppositedirection. The motion of theelectron in the direction of thegray arrow results in a currentin the direction shown.

:

L:

(c)

(b)

(a)

B

B

F IGURE 7.34 (a) Random orientation of atomicmagnetic dipoles in the domains of an unmagne-tized substance. (b) When an external field isapplied, the domains with components of magneticmoment in the same direction as grow larger. (c) Asthe field is made even stronger, the domains withmagnetic moment vectors not aligned with theexternal field become very small.

B:

B:

Figure Q7.3 Bending of a beam ofelectrons in a magnetic field.

(Co u

rt esy

of C

ENCO

)

(Cou

r tes y

of C

ENCO

)Figure Q7.17 Magnetic levitation

using two ceramic magnets.

y

vd

x

z

a

I

t

d

c

I

B

B

F

+

Figure P7.13

y

x

I

a

B

b

c

z

d

Figure P7.16

I

Figure P7.23

x

I2 I1

2a–2a 0

Figure P7.25

P30.0

I

I

Figure P7.27

I

I

aa

a

a

aB

AC

I

Figure P7.28

Figure P7.29

I1

c a

I2

Figure P7.31

0.200 m

0.200 m

A

B

C

P

D

×

×

Figure P7.33

FIGURE 7.1 (Quick Quiz 7.4) Where is themagnetic field the greatest?

Ad

CB

Is

1 A5 A

b

a

d

c

2 A

FIGURE 7.2 (Quick Quiz 7.6) Four closed pathsaround three current-carrying wires.

FIGURE 7.3 (Quick Quiz 7.6) Four closed pathsnear a single current-carrying wire.

a

b

c

d