Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 27

Magnetism - all with Video Answers

Educators

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Chapter Questions

Problem 1

(a) What is the force per meter of length on a straight $\begin{array}{llll}\text { 0.90-T } & \text { uniform } & \text { m agnetic } & \text { field? }\end{array}$
(b) What if the angle between the wire and field is $35.0^{\circ}$ ?

James Kiss

James Kiss

Numerade Educator

Problem 2

$\begin{array}{llllll}\text { } & \text { Calculate the magnitude } & \text { of the magnetic force } & \text { on }\end{array}$
$\begin{array}{llllll}\text { carrying a } & \text { 150-A } & \text { current. The Earth's magnetic field } & \text { of }\end{array}$ $5.0 \times 10-5 \mathrm{~T}$ makes an angle of $68^{\circ}$ with the wire.

James Kiss

Numerade Educator

Problem 3

A 1.6-m length of wire carrying 4.5 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 $5.5 \times 105 \mathrm{~T}$ at this point.

Prashant Bana

Numerade Educator

Problem 4

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.

James Kiss

Numerade Educator

Problem 5

The force on a wire is a maximum of $7.50 \times 10-2 \mathrm{~N}$ when placed betw een 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?

James Kiss

Numerade Educator

Problem 6

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 $\overrightarrow{\mathbf{B}},$ which is horizontal, perpendicular 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.

James Kiss

Numerade Educator

Problem 7

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 $\overrightarrow{\mathbf{B}}=(0.318 \hat{\mathbf{i}})$ T. Determine the magnitude and direction of the total force on the wire.

Jacob Shpiece

Numerade Educator

Problem 8

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

James Kiss

Numerade Educator

Problem 9

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. $27-39 .$ 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 $B=0 .)$

Eduard Sanchez

Numerade Educator

Problem 10

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

Eduard Sanchez

Numerade Educator

Problem 11

A curved wire, connecting two points a and $b$, lies in a plane perpendicular to a uniform magnetic field $\overrightarrow{\mathbf{B}}$ 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. $27-40$

Jacob Shpiece

Numerade Educator

Problem 12

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. (That is, the field makes an angle $\theta$ with the loop at all points, Fig. $27-41$ where $\tan \theta=r / d .$ ) Deter-
mine the force on the loop.

James Kiss

Numerade Educator

Problem 13

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 $0.45 \mathrm{~T}$.

Prashant Bana

Numerade Educator

Problem 14

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.

James Kiss

Numerade Educator

Problem 15

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 $r=0.18 \mathrm{~m} ?$

James Kiss

Numerade Educator

Problem 16

Find the direction of the force on a negative charge for each diagram shown in Fig. $27-42,$ where $\overrightarrow{\mathbf{v}}$ (green) is the velocity of the charge and $\overrightarrow{\mathbf{B}}$ (blue) is the direction of the magnetic field. ( $\otimes$ means the vector points inward. $\odot$ means it points outward, toward you.)

James Kiss

Numerade Educator

Problem 17

Determine the direction of $\overrightarrow{\mathbf{B}}$ for each case in Fig. $27-43,$ where $\overrightarrow{\mathbf{F}}$ represents the maximum magnetic force on a positively charged particle moving with velocity $\overrightarrow{\mathbf{v}}$.

Eduard Sanchez

Numerade Educator

Problem 18

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 electron orbit if the electric field is turned off?

James Kiss

Numerade Educator

Problem 19

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?

Eduard Sanchez

Numerade Educator

Problem 20

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 $\overrightarrow{\mathbf{B}},$ where they move in circular paths perpendicular to $\overrightarrow{\mathbf{B}}$. Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton.

Eduard Sanchez

Numerade Educator

Problem 21

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.

Zhaojie Xu

Numerade Educator

Problem 22

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

Zhaojie Xu

Numerade Educator

Problem 23

A 6.0-MeV (kinetic energy) proton enters a 0.20-T field, in a plane perpendicular to the field. What is the radius of its path? See Section $23-8$.

James Kiss

Numerade Educator

Problem 24

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 northward. 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?

Prashant Bana

Numerade Educator

Problem 25

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 26

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

James Kiss

Numerade Educator

Problem 28

An electron enters a uniform magnetic field $B=0.28 \mathrm{~T}$ at a $45^{\circ}$ angle to
$\overrightarrow{\text { B. }}$ Determine the radius $r$ and pitch $p$ (distance between loops) of the electron's helical path assuming its speed is $3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}$.

James Kiss

Numerade Educator

Problem 29

A particle with charge $q$ and momentum $p$, initially moving along the $x$ axis, enters a region where a uniform magnetic field $\overrightarrow{\mathbf{B}}=B_{0} \hat{\mathbf{k}}$ extends over a width $x=\ell$ as shown in Fig. $27-45$. The particle is deflected
a distance $d$ in the $+y$ as it traverses the field. Determine (a) whether $q$ is positive or negative, and (b) the magnitude of its momentum $p$.

Eduard Sanchez

Numerade Educator

Problem 30

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 $\overrightarrow{\mathbf{B}}$ in the bending magnet, which we assume is uniform and can extend over an area $d \times d,$ must have magnitude $B \geq\left(2 m K / e^{2} d^{2}\right)^{\frac{1}{2}},$ where $m$ is the mass of a proton and $K$ is its kinetic energy.

Eduard Sanchez

Numerade Educator

Problem 31

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? [Ignore relativity.]

Eduard Sanchez

Numerade Educator

Problem 32

A 3.40-g bullet moves with a speed of $155 \mathrm{~m} / \mathrm{s}$ perpendicular to the Earth's magnetic field of $5.00 \times 10^{-5} \mathrm{~T}$. 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 $1.00 \mathrm{~km} ?$

James Kiss

Numerade Educator

Problem 32

A 3.40-g bullet moves with a speed of $155 \mathrm{~m} / \mathrm{s}$ perpendicular to the Earth's magnetic field of $5.00 \times 10^{-5} \mathrm{~T}$. 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 $1.00 \mathrm{~km} ?$

Eduard Sanchez

Numerade Educator

Problem 33

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 $B=0.850 \mathrm{~T}(\overrightarrow{\mathbf{B}} \perp \overrightarrow{\mathbf{v}}) .$ If the proton enters the magnetic field region at a $45^{\circ}$ angle as shown in Fig. $27-46,(a)$ at what angle does it leave, and $(b)$ at what distance $x$ does it exit the field?

Eduard Sanchez

Numerade Educator

Problem 34

A particle with charge $+q$ and mass $m$ travels in a uniform magnetic field $\overrightarrow{\overrightarrow{\mathbf{B}}}=B_{0} \hat{\mathbf{k}}$. 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. $27-47 .$ At a later time $t=t_{\alpha},$ the particle will cross the $x$ axis at $x=\alpha .$ In terms of $q, m, v_{0},$ and $B_{0}$, determine $(a) \alpha,$ and $(b) t_{\alpha}$

Eduard Sanchez

Numerade Educator

Problem 35

How much work is required to rotate the current loop (Fig. $27-22)$ in a uniform magnetic field $\overrightarrow{\mathbf{B}}$ from
(a) $\theta=0^{\circ}(\overrightarrow{\mathbf{\mu}} \| \overrightarrow{\mathbf{B}})$ to $\theta=180^{\circ},$ (b) $\theta=90^{\circ}$ to $\theta=-90^{\circ} ?$

James Kiss

Numerade Educator

Problem 36

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?

Eduard Sanchez

Numerade Educator

Problem 37

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 $5.50 \times 10^{-5} \mathrm{~T}$ and 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.

Eduard Sanchez

Numerade Educator

Problem 38

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 $\mu=\frac{e}{2 m} L$

Zhaojie Xu

Numerade Educator

Problem 39

A 15-loop circular coil $22 \mathrm{~cm}$ in diameter lies in the $x y$ plane. The current in each loop of the coil is 7.6 A clockwise, and an external magnetic field $\overrightarrow{\mathbf{B}}=(0.55 \hat{\mathbf{i}}+0.60 \hat{\mathbf{j}}-0.65 \hat{\mathbf{k}}) \mathrm{T}$
passes through the coil. Determine $(a)$ the magnetic moment of the coil, $\overrightarrow{\boldsymbol{\mu}} ;(b)$ the torque on the coil due to the external magnetic field; $(c)$ the potential energy $U$ of the coil in the field (take the same zero for $U$ as we did in our discussion of Fig. $27-22$ ).

Vishal Gupta

Numerade Educator

Problem 40

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. $27-48 .$ Show that the magnetic dipole moment of this rod is $\frac{1}{6} Q \omega d^{2}$. [Hint: Consider the motion of each infinitesimal length of the rod.

James Kiss

Numerade Educator

Problem 41

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

James Kiss

Numerade Educator

Problem 42

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?

James Kiss

Numerade Educator

Problem 43

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

Krystal K

Numerade Educator

Problem 44

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?

James Kiss

Numerade Educator

Problem 45

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. $27-31 .$ If the potential difference between the plates is $340 \mathrm{~V}$, how many excess electrons does this drop have?

James Kiss

Numerade Educator

Problem 46

A Hall probe, consisting of a rectangular slab of current-carrying material, is calibrated by placing it in a known magnetic field of magnitude $0.10 \mathrm{~T}$. 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 $(b) \theta=60^{\circ} .$

Eduard Sanchez

Numerade Educator

Problem 47

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 $K_{\mathrm{H}}=\mathscr{E}_{\mathrm{H}} / I B,$ indicates the magnitude of the Hall emf achieved for a given applied magnetic field and current. A slab with a large $K_{\mathrm{H}}$ is a good candidate for use as a Hall probe. (a) Show that $K_{\mathrm{H}}=1$ /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
(i) a typical metal $\left(n \approx 1 \times 10^{29} / \mathrm{m}^{3}\right)$ and
(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 (nm) should a metal slab be to yield a $K_{H}$ 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 $B=0.1 \mathrm{~T} ?$

Eduard Sanchez

Numerade Educator

Problem 48

A rectangular sample of a metal is $3.0 \mathrm{~cm}$ wide and $680 \mu \mathrm{m}$ thick. When it carries a 42-A current and is placed in a 0.80 -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.

Eduard Sanchez

Numerade Educator

Problem 49

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 $1.86 \mu \mathrm{V},$ what is the magnitude of the magnetic field (take it perpendicular to the flat face of the strip)? Assume one free electron per atom of $\mathrm{Na}$, and take its specific gravity to be 0.971 .

Eduard Sanchez

Numerade Educator

Problem 50

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.)

Eduard Sanchez

Numerade Educator

Problem 51

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?

James Kiss

Numerade Educator

Problem 52

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.

Eduard Sanchez

Numerade Educator

Problem 53

Suppose the electric field between the electric plates in the mass spectrometer of Fig. $27-33$ 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 $\left.1.66 \times 10^{-27} \mathrm{~kg} .\right)$ 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?

Prashant Bana

Numerade Educator

Problem 54

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 spectrometer have if these two molecules are to be separated at the film or detectors by $0.65 \mathrm{~mm} ?$

Eduard Sanchez

Numerade Educator

Problem 55

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?

Eduard Sanchez

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?

Victor Salazar

Numerade Educator

Problem 57

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

Eduard Sanchez

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. $27-49 .$ A magnetic field $\overrightarrow{\mathbf{B}}$ 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 coefficient of friction is $\mu_{\mathrm{k}} .(c)$ In which direction does the rod move, east or west, if the current through it heads north?

Eduard Sanchez

Numerade Educator

Problem 60

Suppose the rod in Fig. $27-49$ (Problem 59) has mass $m=0.40 \mathrm{~kg}$ and length $22 \mathrm{~cm}$ and the current through it is $I=36$ A. If the coefficient of static friction is $\mu_{\mathrm{s}}=0.50,$ determine the minimum magnetic field $\overrightarrow{\mathbf{B}}$ (not necessarily vertical) that will just cause the rod to slide. Give the magnitude of $\overrightarrow{\mathbf{B}}$ 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?

Bruce Edelman

Numerade Educator

Problem 62

Calculate the magnetic force on an airplane which has acquired a net charge of $1850 \mu \mathrm{C}$ 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-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 electron beam near the center of a CRT television screen due to the Earth's $5.0 \times 10^{-5} \mathrm{~T}$ field. 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. (See Section $23-9 .)$

Eduard Sanchez

Numerade Educator

Problem 66

The cyclotron (Fig. $27-50$ ) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point $\mathrm{A}$ 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 $\mathrm{MeV} ?$

Eduard Sanchez

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. $27-51)$. 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 $\quad$ at $0.85 \times 10^{7} \mathrm{~m} / \mathrm{s}$ be bent?

Bruce Edelman

Numerade Educator

Problem 68

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. 27-51). 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 $0.85 \times 10^{7} \mathrm{~m} / \mathrm{s}$ be bent?

Bruce Edelman

Numerade Educator

Problem 69

A sort of "projectile launcher" is shown in Fig. $27-53 .$ 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 electrons, 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 perpendicular to the electric field? $(c)$ What is the frequency of the circular orbit of the electrons if the electric field is turned off?

Eduard Sanchez

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 $\overrightarrow{\mathbf{B}}$, the rotation frequency of electrons rotating in a plane perpendicular to $\overrightarrow{\mathbf{B}}$ 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 $\overrightarrow{\mathbf{B}}$ 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. $27-54 .$ In two successive loops, at points $P$ and $\mathrm{Q},$ 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 $P$ to $Q$

Eduard Sanchez

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. 27-55). A magnetic field is directed along $z$, with a magnitude 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.

Eduard Sanchez

Numerade Educator

Problem 75

The power cable for an electric trolley (Fig. $27-56$ ) 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. $27-57$ ). 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 $d$ exert negligible force on the rod are not shown.)

Jayashree Behera

Jayashree Behera

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. $27-58) .(a)$ Is the current flowing upward (toward the pivot) or downward in Fig. $27-58 ? \quad(b)$ If $\theta=13^{\circ},$ determine $B$ (c) What is the largest magnetic field magnitude that can be measured using this device?

Bruce Edelman

Numerade Educator