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Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 29

Magnetic Fields - all with Video Answers

Educators


Chapter Questions

02:54

Problem 1

Alpha Particle An alpha particle travels at a velocity $\nabla$ of magnitude $550 \mathrm{~m} / \mathrm{s}$ through a uniform magnetic field $\vec{B}$ of magnitude $0.045 \mathrm{~T}$. (An alpha particle has a charge of $+3.2 \times 10^{-19} \mathrm{C}$ and a mass of $6.6 \times$ $10^{-27} \mathrm{~kg} .$ The angle between $\vec{V}$ and $\vec{B}$ is $52^{\circ} .$ What are the magnitudes of
(a) the force $\vec{F}^{\text {mag }}$ acting on the particle due to the field and (b) the acceleration of the particle due to $\vec{F}^{\mathrm{mag}} ?$ (c) Does the speed of the particle increase, decrease, or remain equal to $550 \mathrm{~m} / \mathrm{s}$ ?

Manish Kumar ( Iit K )
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03:53

Problem 2

TV Camera An electron in a TV camera tube is moving at $7.20 \times 10^{6} \mathrm{~m} / \mathrm{s}$ in a magnetic field of strength $83.0 \mathrm{mT}$. (a) Without knowing the direction of the field, what can you say about the greatest and least magnitudes of the force acting on the electron due to the field? (b) At one point the electron has an acceleration of magnitude $4.90 \times 10^{14} \mathrm{~m} / \mathrm{s}^{2} .$ What is the angle between the electron's velocity and the magnetic field?

Manish Kumar ( Iit K )
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03:52

Problem 3

Proton Traveling A proton traveling at $23.0^{\circ}$ with respect to the direction of a magnetic field of strength $2.60 \mathrm{mT}$ experiences a magnetic force of $6.50 \times 10^{-17} \mathrm{~N}$. Calculate (a) the proton's speed and
(b) its kinetic energy in electron-volts.

Isabel Ruffin
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01:52

Problem 4

Force on Charges An electron that has velocity
$$
\vec{v}=\left(2.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{i}}+\left(3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{j}}
$$
moves through the magnetic field $\vec{B}=(0.030 \mathrm{~T}) \hat{\mathrm{i}}-(0.15 \mathrm{~T}) \hat{\mathrm{j}}$.
(a) Find the force on the electron. (b) Repeat your calculation for a proton having the same velocity.

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04:49

Problem 5

Television Tube Each of the electrons in the beam of a television tube has a kinetic energy of $12.0 \mathrm{keV}$. The tube is oriented so that the electrons move horizontally from geomagnetic south to geomagnetic north. The vertical component of Earth's magnetic field points down and has a magnitude of $55.0 \mu \mathrm{T}$. (a) In what direction will the beam deflect? (b) What is the magnitude of the acceleration of a single electron due to the magnetic field? (c) How far will the beam deflect in moving $20.0 \mathrm{~cm}$ through the television tube?

Prabhu Ramji
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03:37

Problem 6

Accelerated from Rest An electron is accelerated from rest by a potential difference of $350 \mathrm{~V}$. It then enters a uniform magnetic field of magnitude $200 \mathrm{mT}$ with its velocity perpendicular to the field. Calculate (a) the speed of the electron and (b) the radius of its path in the magnetic field.

Shital Rijal
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00:53

Problem 7

Field Perpendicular to Beam A uniform magnetic field is applied perpendicular to a beam of electrons moving at $1.3 \times 10^{6} \mathrm{~m} / \mathrm{s}$. What is the magnitude of the field if the electrons travel in a circular arc of radius $0.35 \mathrm{~m}$ ?

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02:25

Problem 8

Heavy Ions Physicist S. A. Goudsmit devised a method for measuring the masses of heavy ions by timing their periods of revolution in a known magnetic field. A singly charged ion of iodine makes $7.00$ rev in a field of $45.0 \mathrm{mT}$ in $1.29 \mathrm{~ms}$. Calculate its mass, in atomic mass units. (Actually, the method allows mass measurements to be carried out to much greater accuracy than these approximate data suggest.)

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03:07

Problem 9

Knetic Energy An electron with kinetic energy $1.20 \mathrm{keV}$ circles in a plane perpendicular to a uniform magnetic field. The orbit radius is $25.0 \mathrm{~cm}$. Find (a) the speed of the electron, (b) the magnetic field, (c) the frequency, and (d) the period of the motion.

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03:07

Problem 10

Circular Path An alpha particle $(q=+2 e, m=4.00 \mathrm{u})$ travels in a circular path of radius $4.50 \mathrm{~cm}$ in a uniform magnetic field with magnitude $B=1.20 \mathrm{~T}$. Calculate (a) its speed, (b) its period of revolution, (c) its kinetic energy in electron-volts, and (d) the potential difference through which it would have to be accelerated to achieve this energy.

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05:34

Problem 11

Frequency of Revolution (a) Find the frequency of revolution of an electron with an energy of $100 \mathrm{eV}$ in a uniform magnetic field of magnitude $35.0 \mu \mathrm{T}$. (b) Calculate the radius of the path of this electron if its velocity is perpendicular to the magnetic field.

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02:05

Problem 12

Source of Electrons A source injects an electron of speed $v=$ $1.5 \times 10^{7} \mathrm{~m} / \mathrm{s}$ into a uniform magnetic field of magnitude $B=$ $1.0 \times 10^{-3} \mathrm{~T}$. The velocity of the electron makes an angle $\theta=10^{\circ}$ with the direction of the magnetic field. Find the distance $d$ from the point of injection at which the electron next crosses the field line that passes through the injection point.

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02:34

Problem 13

Beam of Electrons A beam of electrons whose kinetic energy is $K$ emerges from a thin-foil "window" at the end of an accelerator tube.
There is a metal plate a distance $d$ from this window and perpendicular to the direction of the emerging beam (Fig. $29-28$ ). Show that we can prevent the beam from hitting the plate if we apply a uniform magnetic field $\vec{B}$ such that its magnitude is

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03:04

Problem 14

Proton, Deuteron, Alpha A proton, a deuteron $(q=+e, m=$ $2.0 \mathrm{u})$, and an alpha particle $(q=+2 e, m=4.0 \mathrm{u})$ with the same kinetic energies enter a region of uniform magnetic field $\vec{B}$, moving perpendicular to $\vec{B}$. Compare the radii of their circular paths.

Manish Kumar ( Iit K )
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04:24

Problem 15

Nuclear Experiment In a nuclear experiment a proton with kinetic energy $1.0 \mathrm{MeV}$ moves in a circular path in a uniform magnetic field. What energy must (a) an alpha particle $(q=+2 e, m=$ $4.0 \mathrm{u}$ ) and $(\mathrm{b})$ a deuteron $(q=+e, m=2.0 \mathrm{u})$ have if they are to circulate in the same circular path?

Shital Rijal
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02:36

Problem 16

Uniform Magnetic Field A proton of charge $+e$ and mass $m$ enters a uniform magnetic field $\vec{B}=B \hat{\mathrm{i}}$ with an initial velocity $\vec{v}=v_{1 x} \hat{i}+v_{1 y} \hat{j} .$ Find an expression in unit-vector notation for its velocity $\vec{v}$ at any later time $t$.

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05:08

Problem 17

Mass Spectrometer A certain commercial mass spectrometer (see Touchstone Example $29-2$ ) is used to separate uranium ions of mass $3.92 \times 10^{-25} \mathrm{~kg}$ and charge $3.20 \times 10^{-19} \mathrm{C}$ from related species. The ions are accelerated through a potential difference of $100 \mathrm{kV}$ and then pass into a uniform magnetic field, where they are bent in a path of radius $1.00 \mathrm{~m}$. After traveling through $180^{\circ}$ and passing through a slit of width $1.00 \mathrm{~mm}$ and height $1.00 \mathrm{~cm}$, they are collected in a cup. (a) What is the magnitude of the (perpendicular)
magnetic field in the separator? If the machine is used to separate out $100 \mathrm{mg}$ of material per hour, calculate (b) the current of the desired ions in the machine and (c) the thermal energy produced in the cup in $1.00 \mathrm{~h}$.

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04:30

Problem 18

Half Circle In Fig $29-29$, a charged particle moves into a region of uniform magnetic field $B$, goes through half a circle, and then exits that region. The particle is either a proton or an electron (you must decide which). It spends $130 \mathrm{~ns}$ within the region.
(a) What is the magnitude $|\vec{B}| ?$ (b) If the particle is sent back through the magnetic field (along the same initial path) but with $2.00$ times its previous kinetic energy, how much time does it spend within the field?

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05:28

Problem 19

Positron A positron with kinetic energy $2.0 \mathrm{keV}$ is projected into a uniform magnetic field $\vec{B}$ of magnitude $0.10 \mathrm{~T}$, with its velocity vector making an angle of $89^{\circ}$ with $\vec{B}$. Find (a) the period, (b) the pitch $p$, and (c) the radius $r$ of its helical path.

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02:40

Problem 20

Neutral Particle A neutral particle is at rest in a uniform magnetic field $\vec{B}$. At time $t=0$ it decays into two charged particles, each of mass $m$. (a) If the charge of one of the particles is $+q$, what is the charge of the other? (b) The two particles move off in separate paths, both of which lie in the plane perpendicular to $\vec{B}$. At a later time the particles collide. Express the time from decay until collision in terms of $m,|\vec{B}|$, and $|q|$

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05:55

Problem 21

Horizontal Motion An electron with kinetic energy $2.5 \mathrm{keV}$ moves horizontally into a region of space in which there is a downward-directed uniform electric field of magnitude $10 \mathrm{kV} / \mathrm{m}$.
(a) What are the magnitude and direction of the (smallest) uniform magnetic field that will cause the electron to continue to move horizontally? Ignore the gravitational force, which is small. (b) Is it possible for a proton to pass through the combination of fields undeflected? If so, under what circumstances?

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03:21

Problem 22

At One Instant A proton travels through uniform magnetic and electric fields. The magnetic field is $\vec{B}=(-2.5 \mathrm{mT}) \hat{\mathrm{i}}$. At one instant the velocity of the proton is $\vec{v}=(2000 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}} .$ At that instant, what is the magnitude of the net force acting on the proton if the electric field is (a) $(4.0 \mathrm{~V} / \mathrm{m}) \hat{\mathrm{k}}$ and $(\mathrm{b})(4.0 \mathrm{~V} / \mathrm{m}) \mathrm{i}$ ?

Manish Kumar ( Iit K )
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03:43

Problem 23

Potential Difference An electron is accelerated through a potential difference of $1.0 \mathrm{kV}$ and directed into a region between two parallel plates separated by $20 \mathrm{~mm}$ with a potential difference of $100 \mathrm{~V}$ between them. The electron is moving perpendicular to the electric field of the plates when it enters the region between the plates. What magnitude of uniform magnetic field, applied perpendicular to both the electron path and the electric field, will allow the electron to travel in a straight line?

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02:05

Problem 24

Electric and Magnetic Field An electric field of magnitude $1.50 \mathrm{kV} / \mathrm{m}$ and a magnetic field of $0.400 \mathrm{~T}$ act on a moving electron to produce no net force. (a) Calculate the minimum speed $|\vec{v}|$ of the electron. (b) Draw a set of vectors $\vec{E}, \vec{B}$, and $\vec{v}$ that could yield the net force.

Manish Kumar ( Iit K )
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02:52

Problem 25

Ion Source An ion source is producing ions of ${ }^{6} \mathrm{Li}$ (mass = $6.0 \mathrm{u}$ ), each with a charge of $+e$. The ions are accelerated by a
potential difference of $10 \mathrm{kV}$ and pass horizontally into a region in which there is a uniform vertical magnetic field of magnitude $|\vec{B}|=$ 1.2 T. Calculate the strength of the smallest electric field, to be set up over the same region, that will allow the ${ }^{6} \mathrm{Li}$ ions to pass through undeflected.

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04:27

Problem 26

Initial Velocity An electron has an initial velocity of $(12.0 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{j}}+(15.0 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{k}}$ and a constant acceleration of $(2.00 \times$
$\left.10^{12} \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}$ in a region in which uniform electric and magnetic fields are present. If $\vec{B}=(400 \mu \mathrm{T}) \hat{\mathrm{i}}$, find the electric field $\vec{E}$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:33

Problem 27

Field Ratio (a) In Fig $29-14$, show that the ratio of the magnitudes of the Hall electric field $\vec{E}$ to the electric field $\vec{E}^{\text {curr }}$ responsible for moving charge (the current) along the length of the strip is
$$
\frac{E}{E^{\text {curr }}}=\frac{B}{n e \rho}
$$
where $\rho$ is the resistivity of the material and $n$ is the number density of the charge carriers and $e$ is the amount of charge on the electron. (b) Compute this ratio numerically for Problem 28. (See Table 26-2.)

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01:57

Problem 28

Strip of Copper A strip of copper $150 \mu \mathrm{m}$ wide is placed in a uniform magnetic field $\vec{B}$ of magnitude $0.65 \mathrm{~T}$, with $\vec{B}$ perpendicular to the strip. A current $i=23 \mathrm{~A}$ is then sent through the strip such that a Hall potential difference $\Delta V$ appears across the width of the strip. Calculate $\Delta V$.(The number of charge carries per unit volume for copper is $8.47 \times 10^{2 \mathrm{~s}}$ electrons/m $^{3}$.)

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02:28

Problem 29

Metal Strip A metal strip $6.50 \mathrm{~cm}$ long, $0.850 \mathrm{~cm}$ wide, and $0.760 \mathrm{~mm}$ thick moves with constant velocity $\vec{v}$ through a uniform magnetic field of magnitude $|\vec{B}|=1.20 \mathrm{mT}$ directed perpendicular to the strip, as shown in Fig. 29-30. A potential difference of $3.90 \mu \mathrm{V}$ is measured between points $x$ and $y$ across the strip. Calculate the speed $|\vec{v}|$.

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01:48

Problem 30

A Wire Carries a Current A wire $1.80 \mathrm{~m}$ long carries a current of $13.0 \mathrm{~A}$ and makes an angle of $35.0^{\circ}$ with a uniform magnetic field of magnitude $B=1.50 \mathrm{~T}$. Calculate the magnitude of the magnetic force on the wire.

Shital Rijal
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02:02

Problem 31

Horizontal Conductor A horizontal conductor that is part of a power line carries a current of $5000 \mathrm{~A}$ from south to north. The magnitude of the Earth's magnetic field is $60.0 \mu \mathrm{T}$. The field is directed toward the north and is inclined downward at $70^{\circ}$ to the horizontal. Find the magnitude and direction of the magnetic force on $100 \mathrm{~m}$ of the conductor due to Earth's field.

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03:13

Problem 32

Along the $x$ Axis A wire $50 \mathrm{~cm}$ long lying along the $x$ axis carries a current of $0.50 \mathrm{~A}$ in the positive $x$ direction. It passes through a magnetic field $\vec{B}=(0.0030 \mathrm{~T}) \hat{\mathrm{j}}+(0.0100 \mathrm{~T}) \hat{\mathrm{k}} .$ Find the magnetic
force on the wire.

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02:20

Problem 33

A Wire of Length A wire of $62.0 \mathrm{~cm}$ length and $13.0 \mathrm{~g}$ mass is suspended by a pair of flexible leads in a uniform magnetic field of magnitude $0.440 \mathrm{~T}$ (Fig. $29-31$ ). What are the magnitude and direction of the current required to remove the tension in the supporting leads?

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03:33

Problem 34

Electric Train Consider the possibility of a new design for an electric train. The engine is driven by the force on a conducting axle due to the vertical component of Earth's magnetic field. To produce the force, current is maintained down one rail, through a conducting wheel, through the axle. through another conducting wheel, and then back to the source via the other rail. (a) What amount of current is needed to provide a modest force of magnitude $10 \mathrm{kN}$ ? Take the vertical component of Earth's field to be $10 \mu \mathrm{T}$ and the length of the axle to be $3.0 \mathrm{~m}$.
(b) At what rate would electric energy be lost for each ohm of resistance in the rails? (c) Is such a train totally or just marginally unrealistic?

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Problem 35

Copper Rod A $1.0 \mathrm{~kg}$ copper rod rests on two horizontal rails $1.0 \mathrm{~m}$ apart and carries a current of $50 \mathrm{~A}$ from one rail to the other. The coefficient of static friction between rod and rails is $0.60 .$ What is the magnitude of the smallest magnetic field (not necessarily vertical) that would cause the rod to slide?

Manish Kumar ( Iit K )
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04:11

Problem 36

Current Loop A single-turn current loop, carrying a current of $4.00 \mathrm{~A}$, is in the shape of a right triangle with sides $50.0,120$, and $130 \mathrm{~cm}$. The loop is in a uniform magnetic field of magnitude $75.0 \mathrm{mT}$ whose direction is parallel to the current in the $130 \mathrm{~cm}$ side of the loop. (a) Find the magnitude of the magnetic force on each of the three sides of the loop. (b) Show that the total magnetic force on the loop is zero.

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02:45

Problem 37

Rectangular Coil Figure $29-32$ shows a rectangular 20-turn coil of wire, of dimensions $10 \mathrm{~cm}$ by $5.0 \mathrm{~cm}$. It carries a current of $0.10 \mathrm{~A}$ and is hinged along one long side. It is mounted in the $x y$ plane, at $30^{\circ}$ to the direction of a uniform magnetic field of magnitude $0.50 \mathrm{~T}$. Find the magnitude and direction of the torque acting on the coil about the hinge line.

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02:36

Problem 38

Arbitrarily Shaped Coil Prove that the relation $\tau=N|i| A B \sin \phi$ (Eq. 29-31) holds for closed loops of arbitary shape and not only for rectangular loops as in Fig. 29-22. (Hint: Replace the loop of arbitrary shape with an assembly of adjacent long, thin, approximately rectangular loops that are nearly equivalent to the loop of arbitrary shape as far as the distribution of current is concerned.)

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02:21

Problem 39

Show That A length $L$ of wire carries a current $i$. Show that if the wire is formed into a circular coil, then the magnitude of the maximum torque in a given magnetic field is developed when the coil has one turn only. Also show that maximum torque has the magnitude $\tau=L^{2} i B / 4 \pi$

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01:30

Problem 40

Zero Total Force A closed wire loop with current $i$ is in a uni. form magnetic field $\vec{B}$, with the plane of the loop at angle $\theta$ to the direction of $\vec{B}$. Show that the total magnetic force on the loop is zero. Does your proof also hold for a nonuniform magnetic field?

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03:20

Problem 41

Wire Ring Figure $29-33$ shows a wire ring of radius $a$ that is perpendicular to the general direction of 3 radially symmetric, diverging magnetic field. The magnetic field at the ring is everywhere of the same mag. nitude $|\vec{B}|$, and its direction at the ring everywhere makes an angle $\bar{\theta}$ with a normal to the plane of the ring. The twisted lead wires have no
effect on the problem. Find the magnitude and direction of the force the field exerts on the ring if the ring carries a positive current $i$

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00:56

Problem 42

Maximum Torque A particle of charge $q$ moves in a circular wire loop of radius $a$ with speed $|\vec{\nu}| .$ Find the maximum torque exerted on the loop by a uniform magnetic field of magnitude $|\vec{B}|$.

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02:22

Problem 43

Wooden Cylinder Figure $29-34$ shows a wooden cylinder with mass $m=0.250 \mathrm{~kg}$ and length $L=0.100$
$\mathrm{m}$, with $N=10.0$ turns of wire wrapped around it longitudinally, so that the plane of the wire coil contains the axis of the cylinder. Also the plane of the coil is parallel to the inclined plane. There is a vertical, uniform magnetic field of magnitude $0.500 \mathrm{~T}$. What is the least amount of current $|i|$ through the coil that will prevent the cylinder from rolling down a plane inclined at an angle $\theta$ to the horizontal?

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01:18

Problem 44

Earth's Moment The magnitude of magnetic dipole moment of Earth is $8.00 \times 10^{22} \mathrm{~J} / \mathrm{T}$. Assume that this is produced by charges flowing in Earth's molten outer core. If the radius of their circular path is $3500 \mathrm{~km}$, calculate the amount of current associated with each moving charge.

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02:13

Problem 45

Calculate the Current A circular coil of 160 turns has a radius of $1.90 \mathrm{~cm} .$ (a) Calculate the current that results in a magnetic dipole moment of $2.30 \mathrm{~A} \cdot \mathrm{m}^{2}$. (b) Find the maximum magnitude of torque that the coil, carrying this current, can experience in a uniform $35.0 \mathrm{mT}$ magnetic field.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:30

Problem 46

Moment and Torque A circular wire loop whose radius is $15.0 \mathrm{~cm}$ carries an amount of current of $2.60 \mathrm{~A} .$ It is placed so that the normal to its plane makes an angle of $41.0^{\circ}$ with a uniform magnetic field of magnitude $12.0 \mathrm{~T}$. (a) Calculate the magnitude of the

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
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01:54

Problem 47

Right Triangle A current loop, carrying an amount of current of $5.0 \mathrm{~A}$, is in the shape of a right triangle with sides 30,40 , and $50 \mathrm{~cm} .$ The loop is in a uniform magnetic field of magnitude $80 \mathrm{mT}$ whose direction is parallel to the current in the $50 \mathrm{~cm}$ side of the loop. Find the magnitude of (a) the magnetic dipole moment of the loop and (b) the torque on the loop.

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03:12

Problem 48

Wall Clock A stationary circular wall clock has a face with a radius of $15 \mathrm{~cm} .$ Six turns of wire are wound around its perimeter; the wire carries a current of $2.0 \mathrm{~A}$ in the clockwise direction. The clock is located where there is a constant, uniform external magnetic field of magnitude $70 \mathrm{mT}$ (but the clock still keeps perfect

Prabhu Ramji
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02:09

Problem 49

Concentric Loops Two concentric, circular wire loops, of radii $20.0$ and $30.0 \mathrm{~cm}$, are located in the $x y$ plane; each carries a clockwise current of $7.00 \mathrm{~A}$ (Fig. $29-35$ ). (a) Find the magnitude of the net magnetic dipole moment of this system. (b) Repeat for reversed current in the inner loop.

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03:21

Problem 50

ABCDEFA Figure $\quad 29-36$ $\begin{array}{l}\text { shows } a & \text { current } \text { loop }\end{array}$ $A B C D E F A$ carrying a current $i$ $=5.00 \mathrm{~A}$. The sides of the loop are parallel to the coordinate axes, with $A B=20.0 \mathrm{~cm}, B C=$
$30.0 \mathrm{~cm}$, and $F A=10.0 \mathrm{~cm}$.
Calculate the magnitude and direction of the magnetic dipole moment of this loop. (Hint: Imagine equal and opposite currents $i$ in the line segment $A D$; then treat the two
rectangular loops $A B C D A$ and $A D E F A .)$

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03:09

Problem 51

Circular Loop A circular loop of wire having a radius of $8.0 \mathrm{~cm}$ carries a current of $0.20$ A. A vector of unit length and parallel to the dipole moment $\vec{\mu}$ of the loop is given by $0.60 \hat{i}-0.80 \hat{j}$. If the loop is located in a uniform magnetic field given by $\vec{B}=$ $(0.25 \mathrm{~T}) \hat{\mathrm{i}}+(0.30 \mathrm{~T}) \mathrm{k}$, find (a) the torque on the loop (in unitvector notation) and (b) the magnetic potential energy of the loop.

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03:56

Problem 52

Permanent Magnet You can observe that a permanent magnet
a force on a moving charge, would the magnet experience any can exert forces on moving charges or currents (a) If a magnet exerts forces? Explain. (b) In the case of the gravitational or electrostatic
interaction between two objects, each object has a common property, such as mass in the case of gravitational interaction or excess charge in the case of the electrostatic interaction. A permanent magnet and a moving electron seem very different. Can you think of any way that they might have a common property? Explain.

Manish Jain
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02:55

Problem 53

U-Shaped $\quad$ Magnet An electron having a velocity of magnitude $v$ enters a region between the poles of a Ushaped magnet. This region has a uniform magnetic field, $\vec{B}$, pointing out of the paper in the positive $z$ direction as shown in Fig. $29-37$.

Manish Jain
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03:26

Problem 54

A Velocity Selector A group of physicists at Argonne National Laboratory in Illinois wants to bombard metals with monoenergetic beams of alpha particles to study radiation damage. (Alpha particles are helium nuclei, which consist of two neutrons and two protons and thus have a net charge of $+2 e$ where $e$ is the amount of the charge on the electron.) They have managed to create a beam of alpha particles from the decay of radioactive elements, but some of the alpha particles lose energy as they collide with other atoms in the source. As a new physicist assigned to the group you have been asked to use a velocity selector to select only the alpha particles in the beam that are close to one velocity and get rid of the others. The velocity selector consists of: (1) a power supply capable of de-
livering large potential differences between capacitor plates and (2) a large permanent magnet that has a uniform magnetic field perpendicular to the beam. The setup for the velocity selector is shown in Fig. 29-38. The direction of the $\mathrm{B}$ -field is out of the paper. Your magnet has a field of $0.22 \mathrm{~T}$ and the capacitor plates have a spacing of $2.5 \mathrm{~cm}$. You are asked to figure out how the velocity selector works and then tell your group what voltage to put across the capacitor plates to select a velocity of $4.2 \times 10^{6} \mathrm{~m} / \mathrm{s} .$
This is your first job and you feel overwhelmed by the assignment, but you calm down and begin to analyze the situation one step at a time. You come up with the following:
(a) The magnet is oriented so its magnetic field is out of the paper in the diagram you are given, so you use the right-hand rule to determine the direction of the magnetic force on an alpha particle passing from left to right into the magnetic field. What direction did you come up with for the force?
(b) You realize that by using $\vec{F}^{\prime} \mathrm{mag}=q \vec{v} \times \vec{B}$, you can calculate the magnitude of force on an alpha particle moving at speed $v$ just as it enters the uniform magnetic field as a function of the charge on the alpha particle and the magnitude of the magnetic field $B$. What is the expression for the magnitude of the force in terms of $e$, $v$, and $B ?$
(c) You realize that you might be able to put just the right voltage across the two capacitor plates so that the electrical force on a given alpha particle will be equal in magnitude and opposite in direction to the Lorentz magnetic force. Then any alpha particles with just the right velocity will pass straight through the poles of the magnet without being deflected. First you think about whether the voltage on the upper capacitor plate should be positive or negative to give a canceling force. What do you decide?
(d) Next you realize that if you know the electric field between the plates and the charge on the alpha particle then you can compute the electrical force on it. What is the relationship between the electrical force $\vec{F}^{\text {elec }}$, charge, $q$, and electric field $\vec{E} ?$
(e) Finally, you use the fact that the magnitude of the electric field between capacitor plates is given by $E=|\Delta V| / d$ where $d$ is the spacing between the plates. Show that the voltage needed to have the electrical force and the magnetic force be "equal and opposite" can be calculated using the equation $|\Delta V|=v B d$. Calculate the voltage needed.

Manish Jain
Manish Jain
Numerade Educator
02:53

Problem 55

Region A-Region B Figure $29-39$ shows a charged particle that is moving in the positive $x$ direction when it encounters region A with a uniform magnetic field. Its path is bent in a halfcircle and then moves into region $\mathrm{B}$ also with a uniform magnetic field. The particle undergoes another half revolution.
Finally it passes between two charged capacitor plates and is deflected downward in the negative $y$ direction.
(a) Is the charge positive or negative? Explain.
(b) What is the direction of the magnetic field in region $\mathrm{A}$ ? Explain.
(c) What is the direction of the magnetic field in region $\mathrm{B}$ ? Explain.
(d) Which region has the larger magnetic field, $A$ or $B$ ? Explain.

Manish Jain
Manish Jain
Numerade Educator
02:44

Problem 56

A Mass Spectrometer It is possible to accelerate ions to a known kinetic energy in an electric field. Sometimes chemists and physicists do this as part of a method to identify the chemical elements present in a beam of ions by determining the mass of each ion. This can be done by bending the ion beam in a uniform magnetic field and measuring the radius of the semicircular path each ion takes. A device that does this is called a mass spectrometer. $\mathrm{A}$ schematic of a mass spectrometer is shown in Fig. $29-40$.

Manish Jain
Manish Jain
Numerade Educator
04:06

Problem 57

Bubble Chamber Tracks Energetic gamma rays like those coming from outer space can disappear near a heavy nucleus producing a rapidly moving pair of particles consisting of an electron and a positron. (A positron is a small positively charged particle that has the same mass and amount of charge as an electron). This process is called pair production. A device called a bubble chamber allows one to observe the path taken by electron-positron pairs produced by gamma rays. The study of bubble chamber tracks in the presence of magnetic fields has revealed a great deal about high-energy gamma rays, the processes of pair production, and the loss of
energy by electrons and positrons. A sample bubble chamber track is shown in Fig. $29-41$.
(a) If the magnetic field is uniform pointing into the paper, which trajectory (the upper one or the lower one) shows the motion of the positron? Explain your reasoning. (b) In which part of the spiral does the positron have the greatest energy $-$ the large radius part or the small radius part? Explain the reasons for your answer. (c) Is the electron moving faster, slower, or at the same speed as the positron at the point in time when the two particles are created? Cite the evidence for your answer. (d) Suppose the bubble
chamber photograph in Fig. $29-41$ is an enlargement of the actual event so that the length $L$ is actually only $2.4 \times 10^{-3} \mathrm{~m} .$ Show that the radius of curvature of the electron path just after the electron is created is approximately $0.8 \times 10^{-3} \mathrm{~m} .$ Hint: Measure $L$ in picture units to find a scale factor and then measure the appropriate feature of the electron path in picture units and use the scale factor to find $R$ in meters. (e) Use the Lorentz force law and the expression for centripetal force to find the equation relating the speed of the electron to $B, R, e$, and $m .$ (f) Suppose the magnitude of the magnetic field in the bubble chamber is $B=0.54 \mathrm{~T}$. Calculate the approximate speed of the electron when it is first created in the bubble chamber.

Manish Jain
Manish Jain
Numerade Educator
04:39

Problem 58

Three Force Fields We have studied three long-range forces:
gravity, electricity, and magnetism. Compare and contrast these three forces giving at least one feature that all three forces have in common, and at least one feature that distinguishes each force.

Manish Jain
Manish Jain
Numerade Educator
03:34

Problem 59

Comparing $\vec{E}$ and $\vec{B}$ Fields We have studied two fields: electric and magnetic. Explain why we introduce the idea of field, and compare and contrast the electric and magnetic fields. In your comparison, be certain to discuss at least one similarity and one difference.

Manish Jain
Manish Jain
Numerade Educator
02:32

Problem 60

Anti-matter Ion Cosmic Rays An international consortium is presently building a device to look for anti-matter nuclei in cosmic rays to help us decide whether there are galaxies made of anti-matter. Anti-matter is just like ordinary matter except the basic particles (anti-protons and anti-electrons) have opposite charge from ordinary matter counterparts. Anti-protons are negative, and anti-electrons (positrons) are positive.
A schematic of the device is shown in Fig. $29-42 .$ A cosmic ray-say, a carbon nucleus or an anti-carbon nucleus-enters the device at the left where its position and velocity are measured. It then passes through a (reasonably uniform) magnetic field. Its path is bent in one direction if its charge is positive and in the opposite direction it its charge is negative. Its deflection is measured as it goes out of the device.
(a) In Fig. $29-42$, what is the direction of the magnetic field? How do you know?
(b) Which path is followed by each particle in the device? How do you know?
(c) If you were given the magnetic field, $B$, the size of the device,
$D$, the amount of charge on the incoming particle, $q$, and the mass of the incoming particle, $M$, would this be enough to calculate the displacement of the charge, $d ?$ If so, describe briefly how you would
do it (but don't do it). If not, explain what additional information you would need (but don't estimate it).

Manish Jain
Manish Jain
Numerade Educator
03:03

Problem 61

Magnets and Charge A bar magnet is hung from a string through its center as shown in Fig. $29-43 .$ A charged rod is brought up slowly into the position shown. In what direction will the magnet tend to rotate? Suppose the charged rod is replaced by a bar magnet with the north pole on top. In what direction will the magnet tend to rotate? Is there a difference between what happens to the hanging magnet in the two situations? Explain why you either do or do not think so.
C14 is a radioactive isotope of carbon that behaves chemically almost identically to its more common but
slightly lighter sibling, C12. The amount of C14 in the atmosphere stays about constant since it is being produced continually by cosmic rays. Once carbon from the air is bound into an organic substance, the $\mathrm{C} 14$ will decay with half of them vanishing every 5730 years. The ratio of $\mathrm{C} 14$ to $\mathrm{C} 12 \mathrm{in}$ an organic substance therefore tells how long ago it died.

Manish Jain
Manish Jain
Numerade Educator
03:11

Problem 62

Buying a Mass Spectrometer You are assigned the task of working with a desktop-sized magnetic spectrometer for the purpose of measuring the ratio of $\mathrm{C}^{12}$ to $\mathrm{C}^{14}$ atoms in a sample in order to determine the sample's age. For this problem, let's concentrate on the magnet that will perform the separation of masses. Suppose you have burned and vaporized the sample so that the carbon atoms are in a gas. You now pass this gas through an "ionizer" that on the average strips one electron from each atom. You then accelerate the ions by putting them through an electrostatic accelerator - two capacitor plates with small holes that permit the ions to enter and leave. (From the University of Washington Physics Education Group)
The two plates are charged so that they are at a voltage difference of $\Delta V$ volts. The electric field produced by charges on the capacitor plates accelerates the ions to an energy of $q \Delta V$. These are then introduced into a nearly constant, vertical magnetic field. If we ignore gravity, the magnetic field will cause the charged particles to follow a circular path in a horizontal plane. The radius of the circle will depend on the atom's mass. (Assume the whole device will be placed inside a vacuum chamber.) Answer these three questions about how the device works.
(a) We want to keep the voltage at a moderate level. If $\Delta V$ is 1000 volts, how big of a magnetic field would we require to have a plausible tabletop-sized instrument? Is this a reasonable magnetic field to have with a tabletop-sized magnet?
(b) Do the $\mathrm{C}^{12}$ and $\mathrm{C}^{14}$ atoms hit the collection plate far enough apart? (If they are not separated by at least a few millimeters at the end of their path, we will have trouble collecting the atoms in separate bins.)
(c) Can we get away with ignoring gravity? (Hint: Calculate the time it would take the atom to travel its semicircle and calculate how far it would fall in that time.)

Manish Jain
Manish Jain
Numerade Educator