Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 27

Magnetism - all with Video Answers

Educators

+ 5 more educators

Chapter Questions

Problem 1

(I) $(a)$ What is the force per meter of length on a straight
wire carrying a 9.40 -A current when perpendicular to a
0.90 -T uniform magnetic field? (b) What if the angle
between the wire and field is $35.0^{\circ} ?$

Bruce Edelman

Bruce Edelman

Numerade Educator

Problem 2

(I) Calculate the magnitude of the magnetic force on
a 240 -m length of wire stretched between two towers and
carrying a $150-$ A current. The Earth's magnetic field of
$5.0 \times 10^{-5} \mathrm{T}$ makes an angle of $68^{\circ}$ with the wire.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 3

(I) A 1.6 -m length of wire carrying 4.5 $\mathrm{A}$ of current toward
the south is oriented horizontally. At that point on the
Earth's surface, the dip angle of the Earth's magnetic field makes an angle of $41^{\circ}$ to the wire. Estimate the magnitude
of the magnetic force on the wire due to the Earth's
magnetic field of $5.5 \times 10^{-5} \mathrm{T}$ at this point.

Bruce Edelman

Numerade Educator

Problem 4

(II) The magnetic force per meter on a wire is measured
to be only 25 percent of its maximum possible value. Sketch
the relationship of the wire and the field if the force
had been a maximum, and sketch the relationship as it
actually is, calculating the angle between the wire and the
magnetic field.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 5

(II) The force on a wire is a maximum of $7.50 \times 10^{-2} \mathrm{N}$ when
placed between the pole faces of a magnet. The current flows
horizontally to the right and the magnetic field is vertical. The
wire is observed to "jump" toward the observer when the current is turned on. (a) What type of magnetic pole is the
top pole face? (b) If the pole faces have a diameter of
$10.0 \mathrm{cm},$ estimate the current in the wire if the field is 0.220 $\mathrm{T}$ .
(c) If the wire is tipped so that it makes an angle of $10.0^{\circ}$ with
the horizontal, what force will it now feel?

Vishal Gupta

Numerade Educator

Problem 6

(II) Suppose a straight 1.00 -mm-diameter copper wire could
just "float" horizontally in air because of the force due to
the Earth's magnetic field $\vec{\mathbf{B}},$ which is horizontal, perpendic-
ular to the wire, and of magnitude $5.0 \times 10^{-5} \mathrm{T}$ . What
current would the wire carry? Does the answer seem
feasible? Explain briefly.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 7

(II) A stiff wire 50.0 $\mathrm{cm}$ long is bent at a right angle in the
middle. One section lies along the $z$ axis and the other is
along the line $y=2 x$ in the $x y$ plane. A current of 20.0 $\mathrm{A}$ flows in the wire-down the $z$ axis and out the line in the
$x y$ plane. The wire passes through a uniform magnetic field
given by $\vec{\mathbf{B}}=(0.318 \hat{\mathbf{i}}) \mathrm{T}$ . Determine the magnitude and
direction of the total force on the wire.

Narayan Hari

Numerade Educator

Problem 8

(II) A long wire stretches along the $x$ axis and carries a $3.0-\mathrm{A}$
current to the right $(+x) .$ The wire is in a uniform
magnetic field $\vec{\mathbf{B}}=(0.20 \hat{\mathbf{i}}-0.36 \hat{\mathbf{j}}+0.25 \hat{\mathbf{k}}) \mathrm{T}$ . Determine
the components of the force on the wire per cm of length.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 9

(II) A current-carrying circular loop of wire (radius $r,$
current $I$ ) is partially immersed in a magnetic field of constant
magnitude $B_{0}$ directed out of the page as shown in Fig. $39 .$ Determine the net force
on the loop due to the
field in terms of $\theta_{0}$
(Note that $\theta_{0}$ points to
the dashed line, above
which $B=0 . )$

Bruce Edelman

Numerade Educator

Problem 10

(II) A 2.0 -m-long wire carries a current of 8.2 $\mathrm{A}$ and is
immersed within a uniform magnetic field $\vec{\mathbf{B}}$ . When this wire lies along the $+x$ axis, a magnetic force $\vec{\mathbf{F}}=(-2.5 \hat{\mathbf{j}}) \mathrm{N}$ acts
on the wire, and when it lies on the $+y$ axis, the force is
$\vec{\mathbf{F}}=(2.5 \hat{\mathbf{i}}-5.0 \hat{\mathbf{k}}) \mathrm{N} .$ Find $\vec{\mathbf{B}}$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 11

(III) A curved wire, connecting two points a and b, lies in a
plane perpendicular to a uniform magnetic field $\vec{\mathbf{B}}$ and
carries a current $I .$ Show that the resultant magnetic force on the wire, no matter what its
shape, is the same as that on a
straight wire connecting the two
points carrying the same current I.
See Fig. $40 .$

Bruce Edelman

Numerade Educator

Problem 12

(III) A circular loop of wire, of radius $r,$ carries current $I$ It is placed in a magnetic
field whose straight lines
seem to diverge from a point
a distance $d$ below the loop
on its axis. CThat is, the field makes an angle $\theta$ with the
loop at all points, Fig. 41
where $\tan \theta=r / d .$ ) Deter-
mine the force on the loop.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 13

(1) Determine the magnitude and direction of the force on
an electron traveling $8.75 \times 10^{5} \mathrm{m} / \mathrm{s}$ horizontally to the east
in a vertically upward magnetic field of strength 0.45 $\mathrm{T}$ .

Vishal Gupta

Numerade Educator

Problem 14

(I) An electron is projected vertically upward with a speed
of $1.70 \times 10^{6} \mathrm{m} / \mathrm{s}$ into a uniform magnetic field of 0.480 $\mathrm{T}$
that is directed horizontally away from the observer.
Describe the electron's path in this field.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 15

(I) Alpha particles of charge $q=+2 e$ and mass
$m=6.6 \times 10^{-27} \mathrm{kg}$ are emitted from a radioactive source
at a speed of $1.6 \times 10^{7} \mathrm{m} / \mathrm{s}$ . What magnetic field strength
would be required to bend them into a circular path of
radius $r=0.18 \mathrm{m} ?$

Bruce Edelman

Numerade Educator

Problem 16

(I) Find the direction of the force on a negative charge for
each diagram shown in Fig, $42,$ where $\overline{\mathbf{v}}$ (green) is the velocity of the charge and $\vec{\mathbf{B}}$ (blue) is the direction of the
magnetic field. $(\otimes)$ means the vector points inward. O means
it points outward, toward you.)

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 17

(I) Determine the direction of $\vec{\mathbf{B}}$ for each case in Fig. $43,$
where $\vec{\mathbf{F}}$ represents the maximum magnetic force on a positively charged particle
moving with velocity $\vec{\mathbf{v}}$

Bruce Edelman

Numerade Educator

Problem 18

(1I) What is the velocity of a beam of electrons that goes
undeflected when passing through perpendicular electric
and magnetic fields of magnitude $8.8 \times 10^{3} \mathrm{V} / \mathrm{m}$ and
$7.5 \times 10^{-3} \mathrm{T}$ , respectively? What is the radius of the elec-
tron orbit if the electric field is turned off?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 19

(II) A doubly charged helium atom whose mass is
$6.6 \times 10^{-27} \mathrm{kg}$ is accelerated by a voltage of 2700 $\mathrm{V}$ .
(a) What will be its radius of curvature if it moves in a plane
perpendicular to a uniform $0.340-\mathrm{T}$ field? (b) What is its
period of revolution?

Bruce Edelman

Numerade Educator

Problem 20

(II) A proton (mass $m_{\mathrm{p}} ),$ a deuteron $\left(m=2 m_{\mathrm{p}}, Q=e\right)$
and an alpha particle $\left(m=4 m_{\mathrm{p}}, Q=2 e\right)$ are accelerated
by the same potential difference $V$ and then enter a uniform
magnetic field $\vec{\mathbf{B}},$ where they move in circular paths for the deuteron and alpha particle in terms of that for
the proton.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 21

(II) For a particle of mass $m$ and charge $q$ moving in a
circular path in a magnetic field $B,(a)$ show that its
kinetic energy is proportional to $r^{2},$ the square of the radius of curvature of its path, and (b) show that its
angular momentum is $L=q B r^{2},$ about the center of the
circle.

Bruce Edelman

Numerade Educator

Problem 22

(II) An electron moves with velocity $\vec{\mathbf{v}}=(7.0 \hat{\mathbf{i}}-6.0 \hat{\mathbf{j}}) \times 10^{4} \mathrm{m} / \mathrm{s}$
in a magnetic field $\vec{\mathbf{B}}=(-0.80 \hat{\mathrm{i}}+0.60 \hat{\mathrm{j}}) \mathrm{T}$ . Determine the
magnitude and direction of the force on the electron.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 23

(II) A 6.0 -MeV (kinetic energy) proton enters a $0.20-\mathrm{T}$
field, in a plane perpendicular to the field. What is the
radius of its path?

Bruce Edelman

Numerade Educator

Problem 24

(II) An electron experiences the greatest force as it travels
$2.8 \times 10^{6} \mathrm{m} / \mathrm{s}$ in a magnetic field when it is moving north-
ward. The force is vertically upward and of magnitude
$8.2 \times 10^{-13} \mathrm{N} .$ What is the magnitude and direction of the
magnetic field?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 25

(II) A proton moves through a region of space where there
is a magnetic field $\vec{\mathbf{B}}=(0.45 \hat{\mathbf{i}}+0.38 \hat{\mathbf{j}}) \mathrm{T}$ and an electric
field $\quad \overline{\mathbf{E}}=(3.0 \hat{\mathbf{i}}-4.2 \hat{\mathbf{j}}) \times 10^{3} \mathrm{V} / \mathrm{m} .$ At a given instant, the proton's velocity is $\vec{\mathbf{v}}=(6.0 \hat{\mathbf{i}}+3.0 \hat{\mathbf{j}}-5.0 \hat{\mathbf{k}}) \times 10^{3} \mathrm{m} / \mathrm{s}$ .
Determine the components of the total force on the
proton.

Bruce Edelman

Numerade Educator

Problem 26

(II) An electron experiences a force $\vec{\mathbf{F}}=(3.8 \hat{\mathbf{i}}-2.7 \hat{\mathbf{j}}) \times 10^{-13} \mathrm{N}$
when passing through a magnetic field $\quad \vec{\mathbf{B}}=(0.85 \mathrm{T}) \hat{\mathbf{k}}$ .
Determine the components of the electron's velocity.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 27

(II) A particle of charge $q$ moves in a circular path of
radius $r$ in a uniform magnetic field $\vec{\mathbf{B}}$ . If the magnitude
of the magnetic field is doubled, and the kinetic energy of
the particle remains constant, what happens to the angular
momentum of the particle?

Bruce Edelman

Numerade Educator

Problem 28

(II) An electron enters a uniform magnetic field $B=0.28 \mathrm{T}$
at a $45^{\circ}$ angle to $\vec{\mathbf{B}}$ . Determine the radius $r$ and pitch $p$ (distance between loops) of the electron's helical
path
Fig. 44 .

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 29

(II) A particle with charge $q$ and momentum $p,$ initially
moving along the $x$ axis, enters a region where a uniform
magnetic field $\vec{\mathbf{B}}=B_{0} \hat{\mathbf{k}}$ extends over a width $x=\ell$ as shown in Fig. $45 .$ The
particle is deflected a
distance $d$ in the $+y$
direction as it traverses
the field. Determine
(a) whether $q$ is posi-
tive or negative, and
(b) the magnitude of
its momentum p.

Bruce Edelman

Numerade Educator

Problem 30

(II) The path of protons emerging from an accelerator must
be bent by $90^{\circ}$ by a "bending magnet" so as not to strike a
barrier in their path a distance $d$ from their exit hole in the
accelerator. Show that the field $\vec{\mathbf{B}}$ in the bending magnet, which we assume is uniform and can extend over an area
$d \times d,$ must have magnitude $B \geqslant\left(2 m K / e^{2} d^{2}\right)^{\frac{1}{2}}$ , where $m$ is
the mass of a proton and $K$ is its kinetic energy.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 31

(III) Suppose the Earth's magnetic field at the equator has
magnitude $0.50 \times 10^{-4} \mathrm{T}$ and a northerly direction at all
points. Estimate the speed a singly ionized uranium ion
$(m=238 \mathrm{u}, q=e)$ would need to circle the Earth 5.0 $\mathrm{km}$
above the equator. Can you ignore gravity? [lgnore relativity.]

Bruce Edelman

Numerade Educator

Problem 32

(III) $\mathrm{A} 3.40$ -g bullet moves with a speed of 155 $\mathrm{m} / \mathrm{s}$ perpen-
dicular to the Earth's magnetic field of $5.00 \times 10^{-9} \mathrm{T}$ . If the
bullet possesses a net charge of $18.5 \times 10^{-9} \mathrm{C},$ by what distance will it be deflected from its path due to the Earth's
magnetic field after it has traveled 1.00 $\mathrm{km} ?$

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 33

(III) A proton moving with speed $v=1.3 \times 10^{5} \mathrm{m} / \mathrm{s}$ in a
field-free region abruptly enters an essentially uniform magnetic field
$B=0.850 \mathrm{T}(\vec{\mathbf{B}} \perp \vec{\mathbf{v}}) .$ If the proton
enters the magnetic field region at a
$45^{\circ}$ angle as shown in Fig. $46,(a)$ at
what angle does it leave, and $(b)$ at
what distance $x$ does it exit the field?

Bruce Edelman

Numerade Educator

Problem 34

(III) A particle with charge $+q$ and mass $m$ travels in a
uniform magnetic field $\vec{\mathbf{B}}=B_{0} \hat{\mathbf{k}} .$ At time $t=0,$ the
particle's speed is $v_{0}$ and its velocity vector lies in the $x y$ plane
directed at an angle of $30^{\circ}$ with respect to the $y$ axis as shown in Fig, $47 .$ At a
later time $t=t_{\alpha},$ the
particle will cross the $x$ axis at $x=\alpha, \quad$ In
terms of $q, m, x_{0},$ and
$B_{0}$ , determine $(a) \quad \alpha$
and $(b) l_{\alpha}$ .

Keshav Singh

Numerade Educator

Problem 35

(1) How much work is required to rotate the current
loop (Fig. 22$)$ in a uniform magnetic field $\vec{\mathbf{B}}$ from
$(a) \theta=0^{\circ}(\vec{\boldsymbol{\mu}} | \mathbf{B})$ to $\theta=180^{\circ},(b) \theta=90^{\circ}$ to $\theta=-90^{\circ} ?$

Bruce Edelman

Numerade Educator

Problem 36

(1) A 13.0 -cm-diameter circular loop of wire is placed with
the plane of the loop parallel to the uniform magnetic field
between the pole pieces of a large magnet. When 4.20 $\mathrm{A}$
flows in the coil, the torque on it is 0.185 $\mathrm{m} \cdot \mathrm{N}$ . What is the
magnetic field strength?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 37

(II) A circular coil 18.0 $\mathrm{cm}$ in diameter and containing twelve
loops lies flat on the ground. The Earth's magnetic field at
this location has magnitude $5.50 \times 10^{-5} \mathrm{T}$ points into
the Earth at an angle of $66.0^{\circ}$ below a line pointing due north. If a $7.10-$ A clockwise current passes through the coil,
determine $(a)$ the torque on the coil, and $(b)$ which edge of
the coil rises up, north, east, south, or west.

Bruce Edelman

Numerade Educator

Problem 38

(II) Show that the magnetic dipole moment $\mu$ of an electron
orbiting the proton nucleus of a hydrogen atom is related to
the orbital momentum $L$ of the electron by
$$
\mu=\frac{e}{2 m} L
$$

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 39

$$
\begin{array}{l}{\text { (II) A } 15 \text { -loop circular coil } 22 \mathrm{cm} \text { in diameter lies in the } x y} \\ {\text { plane. The current in each loop of the coil is } 7.6 \mathrm{A} \text { clockwise, }} \\ {\text { and an external magnetic field } \vec{\mathbf{B}}=(0.5 \hat{\mathbf{i}}+0.60 \hat{\mathbf{j}}-0.65 \hat{\mathbf{j}}) \mathrm{T}} \\ {\text { passes through the coil. Determine }(a) \text { the magnetic }}\end{array}
$$ $$
\begin{array}{l}{\text { moment of the coil, } \vec{\boldsymbol{\mu}} ;(b) \text { the torque on the coil due to the }} \\ {\text { external magnetic field; }(c) \text { the potential energy } U \text { of the }} \\ {\text { coil in the field (take the same zero for } U \text { as we did in our }} \\ {\text { discussion of Fig. } 22}\end{array}
$$

Bruce Edelman

Numerade Educator

Problem 40

(III) Suppose a nonconducting rod of length $d$ carries a
uniformly distributed charge $Q$ . It is rotated with angular
velocity $\omega$ about an axis perpendicular to the rod at one
end, Fig. $48 .$ Show that the magnetic dipole moment of this rod is 6$Q \omega d^{2} .$ [Hint:
Consider the motion
of each infinitesimal
length of the rod.]

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 41

(1) If the current to a motor drops by $12 \%,$ by what factor
does the output torque change?

Bruce Edelman

Numerade Educator

Problem 42

(I) A galvanometer needle deflects full scale for a $63.0-\mu \mathrm{A}$
current. What current will give full-scale deflection if the
magnetic field weakens to 0.800 of its original value?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 43

(I) If the restoring spring of a galvanometer weakens by
15$\%$ over the years, what current will give full-scale deflection
if it originally required 46$\mu \mathrm{A}$ ?

Bruce Edelman

Numerade Educator

Problem 44

(I) What is the value of $q / m$ for a particle that moves in a
circle of radius 8.0 $\mathrm{mm}$ in a 0.46 -T magnetic field if a crossed
$260-\mathrm{V} / \mathrm{m}$ electric field will make the path straight?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 45

(1I) An oil drop whose mass is determined to be
$3.3 \times 10^{-15} \mathrm{kg}$ is held at rest between two large plates separated by 1.0 $\mathrm{cm}$ as in
Fig. $31 .$ If the potential
difference between the
plates is $340 \mathrm{V},$ how many
excess electrons does this
drop have?

Bruce Edelman

Numerade Educator

Problem 46

(II) A Hall probe, consisting of a rectangular slab of
current-carrying material, is calibrated by placing it in a
known magnetic field of magnitude 0.10 T. When the field is
oriented normal to the slab's rectangular face, a Hall emf of
12 $\mathrm{mV}$ is measured across the slab's width. The probe is then placed in a magnetic field of unknown magnitude $B$ , and a
Hall emf of 63 $\mathrm{mV}$ is measured. Determine $B$ assuming that
the angle $\theta$ between the unknown field and the plane of the
slab's rectangular face is $(a) \theta=90^{\circ},$ and $(b) \theta=60^{\circ} .$

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 47

(II) A Hall probe used to measure magnetic field strengths
consists of a rectangular slab of material (free-electron
density $n )$ with width $d$ and thickness $t,$ carrying a current $I$
along its length $\ell$ . The slab is immersed in a magnetic field of magnitude $B$ oriented perpendicular to its rectangular
face (of area $\ell d ),$ so that a Hall emf $\mathscr{E}_{\mathrm{H}}$ is produced across
its width $d .$ The probe's magnetic sensitivity, defined as
$K_{\mathrm{H}}=8_{\mathrm{H}} / I B,$ indicates the magnitude of the Hall emf achieved for a given applied magnetic field and current.
A slab with a large $K_{H}$ is a good candidate for
use as a Hall probe. (a) Show that $K_{H}=1 /$ ent. Thus, a
good Hall probe has small values for both $n$ and $t$ . $(b)$ As possible candidates for the material used in a Hall probe,
consider $\left($ i) a typical metal $\left(n \approx 1 \times 10^{29} / \mathrm{m}^{3}\right)$ and \right.
(ii) a (doped) semiconductor $\left(n \approx 3 \times 10^{22} / \mathrm{m}^{3}\right) .$ Given that
a semiconductor slab can be manufactured with a thickness of $0.15 \mathrm{mm},$ how thin $(\mathrm{nm})$ should a metal slab be to
yield a $K_{\mathrm{H}}$ value equal to that of the semiconductor slab?
Compare this metal slab thickness with the 0.3 -nm size of a typical metal atom. (c) For the typical semiconductor slab
described in part $(b),$ what is the expected value for $\mathscr{E}_{\mathrm{H}}$
when $I=100 \mathrm{mA}$ and $B=0.1 \mathrm{T}$ ?

Bruce Edelman

Numerade Educator

Problem 48

(II) A rectangular sample of a metal is 3.0 $\mathrm{cm}$ wide and
680$\mu \mathrm{m}$ thick. When it carries a $42-\mathrm{A}$ current and is placed
in a $0.80-\mathrm{T}$ magnetic field it produces a $6.5-\mu \mathrm{V}$ Hall emf.
Determine: (a) the Hall field in the conductor; (b) the drift
speed of the conduction electrons; (c) the density of free
electrons in the metal.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 49

(1I) In a probe that uses the Hall effect to measure magnetic
fields, a 12.0 -A current passes through a 1.50 -cm-wide 1.30 -mm-thick strip of sodium metal. If the Hall emf is 1.86$\mu V$ ,
what is the magnitude of the magnetic field (take it perpendic-
ular to the flat face of the strip)? Assume one free electron per
atom of Na, and take its specific gravity to be $0.971 .$

Bruce Edelman

Numerade Educator

Problem 50

(II) The Hall effect can be used to measure blood flow rate
because the blood contains ions that constitute an electric
current. $(a)$ Does the sign of the ions influence the emf? (b) Determine the flow velocity in an artery 3.3 $\mathrm{mm}$ in
diameter if the measured emf is 0.13 $\mathrm{mV}$ and $B$ is 0.070 $\mathrm{T}$ .
(In actual practice, an alternating magnetic field is used.)

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 51

(I) In a mass spectrometer, germanium atoms have radii of
curvature equal to $21.0,21.6,21.9,22.2,$ and 22.8 $\mathrm{cm} .$ The
largest radius corresponds to an atomic mass of 76 $\mathrm{u}$ . What
are the atomic masses of the other isotopes?

Bruce Edelman

Numerade Educator

Problem 52

(II) One form of mass spectrometer accelerates ions by a
voltage $V$ before they enter a magnetic field $B$ . The ions are
assumed to start from rest. Show that the mass of an ion is
$m=q B^{2} R^{2} / 2 V,$ where $R$ is the radius of the ions' path in
the magnetic field and $q$ is their charge.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 53

(II) Suppose the electric field between the electric plates in
the mass spectrometer of Fig. 33 is $2.48 \times 10^{4} \mathrm{V} / \mathrm{m}$ and the
magnetic fields are $B=B^{\prime}=0.58 \mathrm{T}$ . The source contains
carbon isotopes of mass numbers $12,13,$ and 14 from a long dead piece of a tree. (To estimate atomic masses, multiply by
$1.66 \times 10^{-27} \mathrm{kg} .$ ) How far apart are the lines formed by the
singly charged ions of each type on the photographic film?
What if the ions were doubly charged?

Bruce Edelman

Numerade Educator

Problem 54

(II) A mass spectrometer is being used to monitor air
pollutants. It is difficult, however, to separate molecules
with nearly equal mass such as $\mathrm{CO}(28.0106 \mathrm{u})$ and $\mathrm{N}_{2}(28.0134 \mathrm{u}) .$ How large a radius of curvature must a spec-
trometer have if these two molecules are to be separated at
the film or detectors by 0.65 $\mathrm{mm}$ ?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 55

(1I) An unknown particle moves in a straight line through
crossed electric and magnetic fields with $E=1.5 \mathrm{kV} / \mathrm{m}$
and $B=0.034 \mathrm{T}$ . If the electric field is turned off, the
particle moves in a circular path of radius $r=2.7 \mathrm{cm} .$
What might the particle be?

Bruce Edelman

Numerade Educator

Problem 56

Protons move in a circle of radius 5.10 $\mathrm{cm}$ in a 0.625 $\mathrm{T}$
magnetic field. What value of electric field could make their
paths straight? In what direction must the electric field point?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 57

Protons with momentum $3.8 \times 10^{-16} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ are magneti-
cally steered clockwise in a circular path 2.0 $\mathrm{km}$ in diameter
at Fermi National Accelerator Laboratory in Illinois. Deter-
mine the magnitude and direction of the field in the
magnets surrounding the beam pipe.

Bruce Edelman

Numerade Educator

Problem 58

A proton and an electron have the same kinetic energy
upon entering a region of constant magnetic field. What is
the ratio of the radii of their circular paths?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 59

Two stiff parallel wires a distance $d$ apart in a horizontal
plane act as rails to support a light metal rod of mass $m$
(perpendicular to each rail), Fig. $49 .$ A magnetic field $\vec{\mathbf{B}}$ directed vertically upward (outward in diagram), acts
throughout. At $t=0,$ a constant current $I$ begins to flow
through the system. Determine the speed of the rod, which starts from rest at $t=0,$ as a function of time $(a)$ assuming
no friction between the rod and the rails, and $(b)$ if the coef-
ficient of friction is $\mu_{k}$ . (c) In which direction does the rod
move, east or west, if the current through it heads north?

Bruce Edelman

Numerade Educator

Problem 60

Suppose the rod in Fig. 49 (Problem 59 ) has mass
$m=0.40 \mathrm{kg}$ and length 22 $\mathrm{cm}$ and the current through
it is $I=36 \mathrm{A}$ . If the coefficient of static friction is $\mu_{\mathrm{s}}=0.50,$ determine the minimum magnetic field $\vec{\mathbf{B}}$
(not necessarily vertical) that will just cause the rod to
slide. Give the magnitude of $\vec{\mathbf{B}}$ and its direction relative
to the vertical (outwards towards us)

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 61

Near the equator, the Earth's magnetic field points
almost horizontally to the north and has magnitude
$B=0.50 \times 10^{-4} \mathrm{T} .$ What should be the magnitude and
direction for the velocity of an electron if its weight is to be
exactly balanced by the magnetic force?

Prabhu Ramji

Numerade Educator

Problem 62

Calculate the magnetic force on an airplane which has acquired
a net charge of 1850$\mu C$ and moves with a speed of 120 $\mathrm{m} / \mathrm{s}$
perpendicular to the Earth's magnetic field of $5.0 \times 10^{-5} \mathrm{T}$ .

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 63

A motor run by a $9.0-\mathrm{V}$ battery has a 20 turn square coil
with sides of length 5.0 $\mathrm{cm}$ and total resistance 24$\Omega .$ When
spinning, the magnetic field felt by the wire in the coil is
0.020 $\mathrm{T} .$ What is the maximum torque on the motor?

Bruce Edelman

Numerade Educator

Problem 64

Estimate the approximate maximum deflection of the elec-
tron beam near the center of a CRT television screen due to
the Earth's $5.0 \times 10^{-5}$ Tield. Assume the screen is 18 $\mathrm{cm}$
from the electron gun, where the electrons are accelerated (a) by 2.0 $\mathrm{kV}$ , or $(b)$ by 28 $\mathrm{kV}$ . Note that in color TV sets, the
beam must be directed accurately to within less than 1 $\mathrm{mm}$
in order to strike the correct phosphor. Because the Earth's
field is significant here, mu-metal shields are used to reduce
the Earth's field in the CRT.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 65

The rectangular loop of wire shown in Fig. 22 has mass $m$ and carries current $I$ . Show
that if the loop is oriented
at an angle $\theta<1$ (in
radians), then when it is
released it will execute
simple harmonic motion
about $\theta=0 .$ Calculate the
period of the motion.

Bruce Edelman

Numerade Educator

Problem 66

FIGURE 22 Calculating the
torque on a current loop in a
magnetic field $\vec{\mathbf{B}}$ . (a) Loop face
parallel to $\vec{\mathbf{B}}$ field lines; (b) top
view; (c) loop makes an angle to
B, reducing the torque since the
lever arm is reduced.
The cyclotron (Fig. 50 ) is a device used to accelerate
elementary particles such as protons to high speeds. Parti-
cles starting at point A with some initial velocity travel in circular orbits in the magnetic field $B$ . The particles are
accelerated to higher speeds each time they pass in the
gap between the metal "dees," where there is an electric
field $E .$ (There is no electric field within the hollow metal
dees) The electric field changes direction each half-cycle, due to an ac voltage $V=V_{0} \sin 2 \pi f t,$ so that the particles
are increased in speed at each passage through the gap.
$(a)$ Show that the frequency $f$ of the voltage must be
$f=B q / 2 \pi m,$ where $q$ is the charge on the particles and $m$ their mass. (b) Show that the kinetic energy of the particles
increases by 2$q V_{0}$ each revolution, assuming that the gap is
small. (c) If the radius of the cyclotron is 0.50 $\mathrm{m}$ and the magnetic field strength is
$0.60 \mathrm{T},$ what will be the
maximum kinetic energy of
accelerated protons in
MeV?

Keshav Singh

Numerade Educator

Problem 67

Magnetic fields are very useful in particle accelerators for
"beam steering"; that is, magnetic fields can be used to
change the beam's direction without altering its speed
(Fig. 51). Show how this could work with a beam of protons.
What happens to protons that are not moving with the speed that the magnetic field is designed for? If the field
extends over a region 5.0 $\mathrm{cm}$ wide and has a magnitude of
$0.38 \mathrm{T},$ by approximately what angle will a
beam of protons
traveling at
$0.85 \times 10^{7} \mathrm{m} / \mathrm{s}$
be bent?

Brandy Heflin

Numerade Educator

Problem 68

A square loop of aluminum wire is 20.0 $\mathrm{cm}$ on a side. It is to
carry 15.0 $\mathrm{A}$ and rotate in a uniform $1.35-\mathrm{T}$ magnetic field as shown in Fig. $52 .(a)$ Determine
the minimum diameter of the
wire so that it will not fracture
from tension or shear. Assume a
safety factor of $10 .$ (b) What is
the resistance of a single loop of
this wire?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 69

A sort of "projectile launcher" is shown in Fig. $53 .$ A large
current moves in a closed loop composed of fixed rails, a
power supply, and a very light, almost frictionless bar touching the rails. A 1.8 $\mathrm{T}$ magnetic field is perpendicular to
the plane of the circuit. If the rails are a distance $d=24 \mathrm{cm}$
apart, and the bar has a mass of 1.5 $\mathrm{g}$ what constant current
flow is needed to accelerate the bar from rest to 25 $\mathrm{m} / \mathrm{s}$ in a
distance of 1.0 $\mathrm{m} ?$ In what direction must the field point?

Bruce Edelman

Numerade Educator

Problem 70

(a) What value of magnetic field would make a beam of elec-
trons, traveling to the right at a speed of $4.8 \times 10^{6} \mathrm{m} / \mathrm{s},$ go
undeflected through a region where there is a uniform electric
field of 8400 $\mathrm{V} / \mathrm{m}$ pointing vertically up? (b) What is the direction of the magnetic field if it is known to be perpendic-
ular to the electric field? (c) What is the frequency of the
circular orbit of the electrons if the electric field is turned off?

Keshav Singh

Numerade Educator

Problem 71

In a certain cathode ray tube, electrons are accelerated
horizontally by 25 $\mathrm{kV}$ . They then pass through a uniform
magnetic field $B$ for a distance of $3.5 \mathrm{cm},$ which deflects
them upward so they reach the top of the screen 22 $\mathrm{cm}$
away, 11 $\mathrm{cm}$ above the center. Estimate the value of $B$ .

Bruce Edelman

Numerade Educator

Problem 72

Zeeman effect. In the Bohr model of the hydrogen atom,
the electron is held in its circular orbit of radius $r$ about its
proton nucleus by electrostatic attraction. If the atoms are
placed in a weak magnetic field $\vec{\mathbf{B}}$ , the rotation frequency of
electrons rotating in a plane perpendicular to $\vec{\mathbf{B}}$ is changed
by an amount
$$
\Delta f=\pm \frac{e B}{4 \pi m}
$$
where $e$ and $m$ are the charge and mass of an electron.
(a) Derive this result, assuming the force due to $\vec{\mathbf{B}}$ is much
less than that due to electrostatic attraction of the nucleus.
(b) What does the $\pm$ sign indicate?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 73

A proton follows a spiral path through a gas in a magnetic field
of 0.018 $\mathrm{T}$ , perpendicular to the plane of the spiral, as shown in Fig. $54 .$ In two successive loops, at
points $P$ and $O,$ the radii are
10.0 $\mathrm{mm}$ and 8.5 $\mathrm{mm}$ , respectively.
Calculate the change in the kinetic
energy of the proton as it travels
from $\mathrm{P}$ to $\mathrm{Q}$ .

Bruce Edelman

Numerade Educator

Problem 74

The net force on a current loop whose face is perpendicular
to a uniform magnetic field is zero, since contributions to
the net force from opposite sides of the loop cancel.
However, if the field varies in magnitude from one side of
the loop to the other, then there can be a net force on the loop. Consider a square loop with sides whose length is $a$
located with one side at $x=b$ in the $x y$ plane
(Fig. 55). A magnetic field is directed along z, with a magni-
tude that varies with $x$ according to
$B=B_{0}\left(1-\frac{x}{b}\right)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 74

The net force on a current loop whose face is perpendicular
to a uniform magnetic field is zero, since contributions to
the net force from opposite sides of the loop cancel.
However, if the field varies in magnitude from one side of
the loop to the other, then there can be a net force on the loop. Consider a square loop with sides whose length is $a$
located with one side at $x=b$ in the $x y$ plane
(Fig. 55). A magnetic field is directed along z, with a magni-
tude that varies with $x$ according to
$B=B_{0}\left(1-\frac{x}{b}\right)$
If the current in the loop circulates counterclockwise (that
is, the magnetic dipole moment of the loop is
along the $z$ axis $),$ find
an expression for the
net force on the loop.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 75

The power cable for an electric trolley (Fig. 56$)$ carries a
horizontal current of 330 $\mathrm{A}$ toward the east. The Earth's magnetic field has a strength
$5.0 \times 10^{-5} \mathrm{T}$ and makes an
angle of dip of $22^{\circ}$ at this
location. Calculate the
magnitude and direction of
the magnetic force on a $5.0-$
m length of this cable.

Bruce Edelman

Numerade Educator

Problem 76

A uniform conducting rod of length $d$ and mass $m$ sits
atop a fulcrum, which is placed a distance $d / 4$ from the
rod's left-hand end and is immersed in a uniform magnetic
field of magnitude $B$ directed into the page (Fig. $57 ) .$ An object whose mass $M$ is 8.0 times greater than the rod's
mass is hung from the rod's left-hand end. What current
(direction and magnitude) should flow through the rod in
order for it to be "balanced" (i.e., be at rest horizontally) on the fulcrum? (Flexible
connecting wires which
exert negligible force on
the rod are not shown.)

Keshav Singh

Numerade Educator

Problem 77

In a simple device for measuring the magnitude $B$ of a
magnetic field, a conducting rod (length $d=1.0 \mathrm{m},$ mass
$m=150 \mathrm{g} )$ hangs from a friction-free pivot and is oriented
so that its axis of rotation is aligned with the direction of the magnetic field to be measured. Thin flexible wires (which
exert negligible force on the rod ) carry a current $I=12 \mathrm{A}$ , which causes the rod to
deflect an angle $\theta$ with
respect to the vertical,
where it remains at rest
(Fig, 58). (a) Is the current
flowing upward (toward
the pivot) or downward in Fig. 58$?$ (b) If $\theta=13^{\circ}$ ,
determine $B$ . (c) What is
the largest magnetic field
magnitude that can be
measured that can be
device?

Penny Riley

Numerade Educator