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Physics For Scientists And Engineers 6E - part 225

SECTION 29.1 • Magnetic Fields and Forces

897

• When the particle’s velocity vector makes any angle ! ! 0 with the magnetic field,

the magnetic force acts in a direction perpendicular to both

v and B; that is, F

perpendicular to the plane formed by

v and B (Fig. 29.3a).

• The magnetic force exerted on a positive charge is in the direction opposite the

direction of the magnetic force exerted on a negative charge moving in the same
direction (Fig. 29.3b).

• The magnitude of the magnetic force exerted on the moving particle is propor-

tional to sin !, where ! is the angle the particle’s velocity vector makes with the
direction of

We can summarize these observations by writing the magnetic force in the form

(29.1)

v ! B

(a)

(b)

(c)

Figure 29.2 (a) Magnetic field pattern surrounding a bar magnet as displayed with

iron filings. (b) Magnetic field pattern between opposite poles (N–S) of two bar magnets.

(a)

+ q

(b)

–

Figure 29.3 The direction of the magnetic force F

acting on a charged particle

moving with a velocity v in the presence of a magnetic field B. (a) The magnetic force is

perpendicular to both v and B. (b) Oppositely directed magnetic forces F

are exerted

on two oppositely charged particles moving at the same velocity in a magnetic field. The

dashed lines show the paths of the particles, which we will investigate in Section 29.4.

Henry Leap and Jim Lehman

Vector expression for the

magnetic force on a charged

particle moving in a magnetic

field

which by definition of the cross product (see Section 11.1) is perpendicular to both

and

B. We can regard this equation as an operational definition of the magnetic field

at some point in space. That is, the magnetic field is defined in terms of the force
acting on a moving charged particle.

Figure 29.4 reviews two right-hand rules for determining the direction of the cross

product

v ! B and determining the direction of F

. The rule in Figure 29.4a

depends on our right-hand rule for the cross product in Figure 11.2. Point the four
fingers of your right hand along the direction of

v with the palm facing B and curl

them toward

B. The extended thumb, which is at a right angle to the fingers, points

in the direction of

v ! B. Because F

v ! B, F

is in the direction of your thumb

if q is positive and opposite the direction of your thumb if q is negative. (If you need
more help understanding the cross product, you should review pages 337 to 339, in-
cluding Fig. 11.2.)

An alternative rule is shown in Figure 29.4b. Here the thumb points in the direc-

tion of

v and the extended fingers in the direction of B. Now, the force F

on a posi-

tive charge extends outward from your palm. The advantage of this rule is that the
force on the charge is in the direction that you would push on something with your
hand—outward from your palm. The force on a negative charge is in the opposite
direction. Feel free to use either of these two right-hand rules.

The magnitude of the magnetic force on a charged particle is

(29.2)

where ! is the smaller angle between

v and B. From this expression, we see that F

zero when

v is parallel or antiparallel to B (! " 0 or 180°) and maximum when v is

perpendicular to

B (! " 90°).

There are several important differences between electric and magnetic forces:

• The electric force acts along the direction of the electric field, whereas the mag-

netic force acts perpendicular to the magnetic field.

• The electric force acts on a charged particle regardless of whether the particle is

moving, whereas the magnetic force acts on a charged particle only when the parti-
cle is in motion.

!vB

sin !

898

C H A P T E R 2 9 • Magnetic Fields

(a)

(b)

Figure 29.4 Two right-hand rules for determining the direction of the magnetic force

q v ! B acting on a particle with charge q moving with a velocity v in a magnetic

field B. (a) In this rule, the fingers point in the direction of v, with B coming out of

your palm, so that you can curl your fingers in the direction of B. The direction of

v ! B, and the force on a positive charge, is the direction in which the thumb points.

(b) In this rule, the vector v is in the direction of your thumb and B in the direction of

your fingers. The force F

on a positive charge is in the direction of your palm, as if you

are pushing the particle with your hand.

Magnitude of the magnetic

force on a charged particle

moving in a magnetic field

SECTION 29.1 • Magnetic Fields and Forces

899

• The electric force does work in displacing a charged particle, whereas the magnetic

force associated with a steady magnetic field does no work when a particle is dis-
placed because the force is perpendicular to the displacement.

From the last statement and on the basis of the work–kinetic energy theorem, we

conclude that the kinetic energy of a charged particle moving through a magnetic
field cannot be altered by the magnetic field alone. In other words, when a charged
particle moves with a velocity

v through a magnetic field, the field can alter the

direction of the velocity vector but cannot change the speed or kinetic energy of the
particle.

From Equation 29.2, we see that the SI unit of magnetic field is the newton per

coulomb-meter per second, which is called the

tesla (T):

Because a coulomb per second is defined to be an ampere, we see that

A non-SI magnetic-field unit in common use, called the gauss (G), is related to the

tesla through the conversion 1 T " 10

G. Table 29.1 shows some typical values of mag-

netic fields.

1 T " 1

A#m

1 T " 1

C#m/s

Source of Field

Field Magnitude (T)

Strong superconducting laboratory magnet

Strong conventional laboratory magnet

Medical MRI unit

1.5

Bar magnet

Surface of the Sun

Surface of the Earth

0.5 % 10

Inside human brain (due to nerve impulses)

Some Approximate Magnetic Field Magnitudes

Table 29.1

Quick Quiz 29.1

The north-pole end of a bar magnet is held near a posi-

tively charged piece of plastic. Is the plastic (a) attracted, (b) repelled, or (c) unaf-
fected by the magnet?

Quick Quiz 29.2

A charged particle moves with velocity

v in a magnetic

field

B. The magnetic force on the particle is a maximum when v is (a) parallel to B,

(b) perpendicular to

B, (c) zero.

Quick Quiz 29.3

An electron moves in the plane of this paper toward the

top of the page. A magnetic field is also in the plane of the page and directed
toward the right. The direction of the magnetic force on the electron is (a) toward
the top of the page, (b) toward the bottom of the page, (c) toward the left edge of
the page, (d) toward the right edge of the page, (e) upward out of the page,
(f) downward into the page.

The tesla

900

C H A P T E R 2 9 • Magnetic Fields

29.2 Magnetic Force Acting on a Current-Carrying

Conductor

If a magnetic force is exerted on a single charged particle when the particle moves
through a magnetic field, it should not surprise you that a current-carrying wire also
experiences a force when placed in a magnetic field. This follows from the fact that the
current is a collection of many charged particles in motion; hence, the resultant force
exerted by the field on the wire is the vector sum of the individual forces exerted on all
the charged particles making up the current. The force exerted on the particles is
transmitted to the wire when the particles collide with the atoms making up the wire.

Before we continue our discussion, some explanation of the notation used in

this book is in order. To indicate the direction of

B in illustrations, we sometimes

present perspective views, such as those in Figure 29.5. If

B lies in the plane of the

page or is present in a perspective drawing, we use blue vectors or blue field lines
with arrowheads. In non-perspective illustrations, we depict a magnetic field

Example 29.1 An Electron Moving in a Magnetic Field

An electron in a television picture tube moves toward the
front of the tube with a speed of 8.0 % 10

m/s along the x

axis (Fig. 29.5). Surrounding the neck of the tube are coils
of wire that create a magnetic field of magnitude 0.025 T,
directed at an angle of 60° to the x axis and lying in the xy
plane.

(A)

Calculate the magnetic force on the electron using

Equation 29.2.

Solution Using Equation 29.2, we find the magnitude of
the magnetic force:

2.8 % 10

" (1.6 % 10

C)(8.0 % 10

m/s)(0.025 T)(sin 60&)

!vB

sin !

Because

v ! B is in the positive z direction (from the right-

hand rule) and the charge is negative,

is in the negative z

direction.

(B)

Find a vector expression for the magnetic force on the

electron using Equation 29.1.

Solution We begin by writing a vector expression for the
velocity of the electron:

and one for the magnetic field:

The force on the electron, using Equation 29.1, is

where we have used Equations 11.7a and 11.7b to eva-
luate

and

. Carrying out the multiplication, we

find,

This expression agrees with the result in part (A). The mag-
nitude is the same as we found there, and the force vector is
in the negative z direction.

($2.8 % 10

N)ˆ

iˆ ! jˆ

iˆ ! iˆ

($1.6 % 10

C)(8.0 % 10

m/s)(0.022 T) ˆ

($e)(8.0 % 10

m/s)(0.022 T)(ˆ

i ! ˆj)

($e)(8.0 % 10

m/s)(0.013 T)(ˆ

i ! ˆi)

($e)[(8.0 % 10

ˆi) m/s] % [(0.022ˆj)T]

($e)[(8.0 % 10

ˆi) m/s] % [(0.013ˆi)T]

" ($e)[(8.0 % 10

ˆi) m/s] % [(0.013ˆi ' 0.022 jˆ)T]

v % B

" (0.013ˆ

i ' 0.022 ˆj)T

B " (0.025 cos 60&ˆi ' 0.025 sin 60&ˆj)T

v " (8.0 % 10

i) m/s

–e

Figure 29.5 (Example 29.1) The magnetic force F

acting on

the electron is in the negative z direction when v and B lie in

the xy plane.

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