Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 28

Magnetic Fields - all with Video Answers

Educators

Chapter Questions

Problem 1

A conducting rectangular solid of dimensions $d_{x}=5.00 \mathrm{~m}, d_{y}=3.00 \mathrm{~m}$, and $d_{z}=2.00 \mathrm{~m}$ moves with a constant velocity $\vec{v}=(20.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$ through a uniform magnetic field $\vec{B}=(40.0 \mathrm{mT}) \hat{\mathrm{j}}$ (Fig. 28-22). What are the resulting (a) electric field within the solid, in unit-vector notation, and (b) potential difference across the solid? (c) Which face becomes negatively charged?

Joy Chugh

Joy Chugh

Numerade Educator

Problem 2

Figure 28-22 shows a metallic block, with its faces parallel to coordinate axes. The block is in a uniform magnetic field of magnitude $0.020 \mathrm{~T}$. One edge length of the block is $32 \mathrm{~cm}$; the block is not drawn to scale. The block is moved at $3.5 \mathrm{~m} / \mathrm{s}$ parallel to each axis, in turn, and the resulting potential difference $V$ that appears across the block is measured. With the motion parallel to the $y$ axis, $V=12 \mathrm{mV}$; with the motion parallel to the $z$ axis, $V=18 \mathrm{mV}$; with the motion parallel to the $x$ axis, $V=0$. What are the block lengths (a) $d_{x}$, (b) $d_{y}$, and (c) $d_{z} ?$

Joy Chugh

Numerade Educator

Problem 3

A particular type of fundamental particle decays by transforming into an electron $e^{-}$and a positron $e^{+} .$Suppose the decaying particle is at rest in a uniform magnetic field $\vec{B}$ of magnitude $9.57 \mathrm{mT}$ and the $\mathrm{e}^{-}$and $\mathrm{e}^{+}$move away from the decay point in paths lying in a plane perpendicular to $\vec{B}$. How long after the decay do the $\mathrm{e}^{-}$and $\mathrm{e}^{+}$collide?

Joy Chugh

Numerade Educator

Problem 4

An electron follows a helical path in a uniform magnetic field given by $\vec{B}=(20 \hat{\mathrm{i}}-50 \hat{\mathrm{j}}-30 \hat{\mathrm{k}}) \mathrm{mT}$. At time $t=0$, the electron's velocity is given by $\vec{v}=(40 \hat{\hat{i}}-30 \hat{j}+50 \hat{k}) \mathrm{m} / \mathrm{s}$. (a) What is the angle $\phi$ between $\vec{v}$ and $\vec{B}$ ? The electron's velocity changes with time. Do (b) its speed and (c) the angle $\phi$ change with time? (d) What is the radius of the helical path?

Rudra Singh

Numerade Educator

Problem 5

A wire $66.0 \mathrm{~cm}$ long carries a $0.750 \mathrm{~A}$ current in the positive direction of an $x$ axis through a magnetic field $\vec{B}=(3.00 \mathrm{mT}) \hat{j}+$ $(14.0 \mathrm{mT}) \hat{k}$. In unit-vector notation, what is the magnetic force on the wire?

Joy Chugh

Numerade Educator

Problem 6

A wire $2.30 \mathrm{~m}$ 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 magnetic force on the wire.

Joy Chugh

Numerade Educator

Problem 7

Figure $28-23$ shows a wood cylinder of mass $m=0.150 \mathrm{~kg}$ and length $L=0.100 \mathrm{~m}$, with $N=13.0$ turns of wire wrapped around it longitudinally, so that the plane of the wire coil contains the long central axis of the cylinder. The cylinder is released on a plane inclined at an angle $\theta$ to the horizontal, with the plane of the coil parallel to the incline plane. If there is a vertical uniform magnetic field of magnitude $0.92 \mathrm{~T}$, what is the least current $i$ through the coil that keeps the cylinder r from rolling down the plane?

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 8

In Fig. 28-24, a charged particle moves into a region of uniform magnetic field $\vec{B}$, 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 $160 \mathrm{~ns}$ in the region. (a) What is the magnitude of $B ?$ (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 in the field during this trip?

Joy Chugh

Numerade Educator

Problem 9

Prove that the relation $\tau=N i A B \sin \theta$ holds not only for the rectangular loop of Fig. 28-19 but also for a closed loop of any shape. (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.)

Sachin Rao

Numerade Educator

Problem 10

The bent wire shown in Fig. 28-25 lies in a uniform magnetic field. Each straight section is $2.0 \mathrm{~m}$ long and makes an angle of $\theta=60^{\circ}$ with the $x$ axis, and the wire carries a current of $3.5 \mathrm{~A}$. What is the net magnetic force on the wire in unitvector notation if the magnetic field is given by (a) $4.0 \mathrm{k} \mathrm{T}$ and (b) 4.0î T?

Joy Chugh

Numerade Educator

Problem 11

Two concentric, circular wire loops, of radii $r_{1}=20.0 \mathrm{~cm}$ and $r_{2}=$ $40.0 \mathrm{~cm}$, are located in an $x y$ plane; each carries a clockwise current of 11.0 A (Fig. 28-26). (a) Find the magnitude of the net magnetic dipole moment of the system. (b) Repeat for reversed current in the inner loop.

Joy Chugh

Numerade Educator

Problem 12

A source injects an electron of speed $v=1.2 \times 10^{7} \mathrm{~m} / \mathrm{s}$ 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.

Joy Chugh

Numerade Educator

Problem 13

A $0.85 \mathrm{~kg}$ copper rod rests on two horizontal rails $1.0 \mathrm{~m}$ apart and carries a current of 65 A from one rail to the other. The coefficient of static friction between rod and rails is $0.50$. What are the (a) magnitude and (b) angle (relative to the vertical) of the smallest magnetic field that puts the rod on the verge of sliding?

Keshav Singh

Numerade Educator

Problem 14

Figure $28-27$ shows a wire ring of radius $a=1.8 \mathrm{~cm}$ that is perpendicular to the general direction of a radially symmetric, diverging magnetic fleld. The magnetic field at the ring is everywhere of the same magnitude $B=3.4 \mathrm{mT}$, and its direction at the ring everywhere makes an angle $\theta=15^{\circ}$ with a normal to the plane of the ring. The twisted lead wires have no effect on the problem. Find the magnitude of the force the field exerts on the ring if the ring carries a current $i=4.6 \mathrm{~mA}$.

Keshav Singh

Numerade Educator

Problem 15

(a) Find the frequency of revolution of an electron with an energy of $189 \mathrm{eV}$ in a uniform magnetic field of magnitude $70.0 \mu \mathrm{T}$. (b) Calculate the radius of the path of this electron if its velocity is perpendicular to the magnetic field.

Joy Chugh

Numerade Educator

Problem 16

Figure 28-28 gives the orientation energy $U$ of a magnetic dipole in an external magnetic field $\vec{B}$, as a function of angle $\phi$ between the directions of $\vec{B}$ and the dipole moment. The vertical axis scale is set by $U_{s}=2.0 \times 10^{-4} \mathrm{~J}$. The dipole can be rotated about an axle with negligible friction in order to change $\phi$. Counterclockwise rotation from $\phi=0$ yields positive vallues of $\phi$, and clockwise rotations yield negative values. The dipole is to be released at angle $\phi=0$ with a rotational kinetic energy of $9.0 \times 10^{-4} \mathrm{~J}$, so that it rotates counterclockwise. To what maximum value of $\phi$ will it rotate? (In the language of Module $8-3$, what value $\phi$ is the turning point in the potential well of Fig. $28-28 ?$ )

Keshav Singh

Numerade Educator

Problem 17

A current loop, carrying a current of $7.5 \mathrm{~A}$, 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 $120 \mathrm{mT}$ 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.

Joy Chugh

Numerade Educator

Problem 18

An electron is accelerated from rest by a potential difference of $380 \mathrm{~V}$. 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.

Joy Chugh

Numerade Educator

Problem 19

Figure $28-29$ shows a rectangular 28 -turn coil of wire, of dimensions $10 \mathrm{~cm}$ by $5.0 \mathrm{~cm}$. It carries a current of $0.80 \mathrm{~A}$ and is hinged along one long side. It is mounted in the $x y$ plane, at angle $\theta=25^{\circ}$ to the direction of a uniform magnetic field of magnitude $0.50 \mathrm{~T}$. In unitvector notation, what is the torque acting on the coil about the hinge line?

Joy Chugh

Numerade Educator

Problem 20

A circular wire loop of radius
$15.0 \mathrm{~cm}$ carries a current of $3.20 \mathrm{~A}$. 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 magnetic dipole moment of the loop. (b) What is the magnitude of the torque acting on the loop?

Joy Chugh

Numerade Educator

Problem 21

21 A $6.75 \mathrm{~g}$ wire of length $L=15.0 \mathrm{~cm}$ is suspended by a pair of flexible leads in a uniform magnetic field of magnitude $0.440 \mathrm{~T}$ (Fig. $28-30$ ). What are the (a) magnitude and (b) direction (left or right) of the current required to ing leads?

Sheh Lit Chang

University of Washington

Problem 22

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 $B=1.20 \mathrm{mT}$ directed perpendicular to the strip, as shown in Fig. 28-31. A potential difference of $3.30 \mu \mathrm{V}$ is measured between points $x$ and $y$ across the strip. Calculate the speed $v$.

Joy Chugh

Numerade Educator

Problem 23

A wire of length $12.5 \mathrm{~cm}$ carrying a current of $2.33 \mathrm{~mA}$ is to be formed into a circular coil and placed in a uniform magnetic field $B$ of magnitude $5.71 \mathrm{~m} 1$. If the torque on the coil from the field is maximized, what are (a) the angle between $\vec{B}$ and the coil's magnetic dipole moment and (b) the number of turns in the coil? (c) What is the magnitude of that maximum torque?

Joy Chugh

Numerade Educator

Problem 24

An electron moves in a circle of radius $r=5.29 \times 10^{-11} \mathrm{~m}$ with speed $4.12 \times 10^{6} \mathrm{~m} / \mathrm{s}$. Treat the circular path as a current loop with a constant current equal to the ratio of the electron's charge magnitude to the period of the motion. If the circle lies in a uniform magnetic field of magnitude $B=7.10 \mathrm{mT}$, what is the maximum possible magnitude of the torque produced on the loop by the field?

Joy Chugh

Numerade Educator

Problem 25

A proton circulates in a cyclotron, beginning approximately at rest at the center. Whenever it passes through the gap between dees, the electric potential difference between the dees is $350 \mathrm{~V}$. (a) By how much does its kinetic energy increase with each passage through the gap? (b) What is its kinetic energy as it completes 100 passes through the gap? Let $r_{100}$ be the radius of the proton's circular path as it completes those 100 passes and enters a dee, and let $r_{101}$ be its next radius, as it enters a dee the next time. (c) By what percentage does the radius increase when it changes from $r_{100}$ to $r_{101}$ ? That is, what is
percentage increase $=\frac{r_{101}-r_{100}}{r_{100}} 100 \% ?$

Joy Chugh

Numerade Educator

Problem 26

In Fig. 28-32, a rectangular loop carrying current lies in the plane of a uniform magnetic field of magnitude $0.050 \mathrm{~T}$. The loop consists of a single turn of flexible conducting wire that is wrapped around a flexible mount such that the dimensions of the rectangle can be changed. (The total length of the wire is not changed.) As edge length $x$ is varied from approximately zero to its maximum value of approximately $4.0 \mathrm{~cm}$, the magnitude $\tau$ of the torque on the loop changes. The maximum value of $\tau$ is $4.80 \times 10^{-8} \mathrm{~N} \cdot \mathrm{m}$. What is the current in the loop?

Joy Chugh

Numerade Educator

Problem 27

A positron with kinetic energy $950 \mathrm{eV}$ is projected into a uniform magnetic field $\vec{B}$ of magnitude $0.732 \mathrm{~T}$, with its velocity vector making an angle of $89.0^{\circ}$ with $\vec{B}$.
Find (a) the period, (b) the pitch $p$, and
(c) the radius $r$ of its helical path.

Sheh Lit Chang

University of Washington

Problem 28

Figure 28-33 shows a current loop $A B C D E F A$ carrying a current $i=3.00 \mathrm{~A}$. The sides of the loop are parallel to the coordinate axes shown, with $A B=20.0 \mathrm{~cm}$, $B C=30.0 \mathrm{~cm}$, and $F A=10.0 \mathrm{~cm}$. In unit-vector notation, what is 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 $A D E F A$.)

Joy Chugh

Numerade Educator

Problem 29

A circular coil of 500 turns has a radius of $1.90 \mathrm{~cm}$. (a) Calculate the current that results in a magnetic dipole moment of magnitude $1.90 \mathrm{~A} \cdot \mathrm{m}^{2}$. (b) Find the maximum magnitude of the torque that the coil, carrying this current, can experience in a uniform $35.0 \mathrm{mT}$ magnetic field.

Joy Chugh

Numerade Educator

Problem 30

A long, rigid conductor, lying along an $x$ axis, carries a current of $7.0 \mathrm{~A}$ in the negative $x$ direction. A magnetic field $\vec{B}$ is present, given by $\vec{B}=3.0 \mathrm{i}+8.0 x^{2} \hat{\mathrm{j}}$, with $x$ in meters and $\vec{B}$ in milliteslas. Find, in unit-vector notation, the force on the $2.0 \mathrm{~m}$ segment of the conductor that lies between $x=1.0 \mathrm{~m}$ and $x=3.0 \mathrm{~m}$.

Joy Chugh

Numerade Educator

Problem 31

In Fig. 28-34, an electron accelerated from rest through potential difference $V_{1}=2.50 \mathrm{kV}$ enters the gap between two parallel plates having separation $d=16.0 \mathrm{~mm}$ and potential difference $V_{2}=100 \mathrm{~V}$. The lower plate is at the lower potential. Neglect fringing and assume that the electron's velocity vector is perpendicular to the electric field vector between the plates. (a) In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap? (b) If the potential difference is increased slightly, in what direction does the electron veer from straight-line motion.

Joy Chugh

Numerade Educator

Problem 32

In Fig. 28-35, a metal wire of mass $m=24.1 \mathrm{mg}$ can slide with negligible friction on two horizontal parallel rails separated by distance $d=2.56 \mathrm{~cm}$. The track lies in a vertical uniform magnetic field of magnitude $73.5 \mathrm{mT}$. At time $t=0$, device $G$ is connected to the rails, producing a constant current $i=9.13 \mathrm{~mA}$ in the wire and rails (even as the wire moves). At $t=61.1 \mathrm{~ms}$, what are the wire's (a) speed and (b) direction of motion (left or right)?

Joy Chugh

Numerade Educator

Problem 33

A horizontal power line carries a current of 7000 A from south to north. Earth's magnetic field $(60.0 \mu \mathrm{T})$ is directed toward the north and inclined downward at $70.0^{\circ}$ to the horizontal. Find the (a) magnitude and (b) direction of the magnetic force on $100 \mathrm{~m}$ of the line due to Earth's field.

Joy Chugh

Numerade Educator

Problem 34

A horizontal power line carries a current of 7000 A from south to north. Earth's magnetic field $(60.0 \mu \mathrm{T})$ is directed toward the north and inclined downward at $70.0^{\circ}$ to the horizontal. Find the (a) magnitude and (b) direction of the magnetic force on $100 \mathrm{~m}$ of the line due to Earth's field. Assuming $B_{x}=0$, find
(a) the magnitude $E$ and (b) $B$ in unit-vector notation.

Joy Chugh

Numerade Educator

Problem 35

A circular loop of wire having a radius of $8.0 \mathrm{~cm}$ carries a current of $0.20 \mathrm{~A}$. A vector of unit length and parallel to the dipole moment $\vec{\mu}$ of the loop is given by $0.60 \hat{\mathrm{i}}-0.80 \mathrm{j}$. (This unit vector gives the orientation of the magnetic dipole moment vector.) If the loop is located in a uniform magnetic field given by $\vec{B}=(0.50 \mathrm{~T}) \hat{\mathrm{i}}+(0.20 \mathrm{~T}) \hat{\mathrm{k}}$, find (a) the torque on the loop (in unit-vector notation) and (b) the orientation energy of the loop.

Joy Chugh

Numerade Educator

Problem 36

An electron is accelerated from rest through potential difference $V$ and then enters a region of uniform magnetic field, where it undergoes uniform circular motion. Figure 28-37 gives the radius $r$ of that motion versus $V^{1 / 2}$. The vertical axis scale is set by $r_{s}=9.0 \mathrm{~mm}$, and the horizontal axis scale is set by $V_{s}^{1 / 2}=40.0 \mathrm{~V}^{1 / 2}$. What is the magnitude of the magnetic field?

Joy Chugh

Numerade Educator

Problem 37

An electron moves through a uniform magnetic field given by $\vec{B}=B_{x} \hat{\mathrm{i}}+\left(-3.0 B_{x}\right) \hat{\mathrm{j}}$. At a particular instant, the electron has velocity $\vec{v}=(2.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ and the magnetic force acting on it is $\left(6.4 \times 10^{-19} \mathrm{~N}\right) \hat{\mathrm{k}}$. Find $B_{x^{-}}$

Isabel Ruffin

Numerade Educator

Problem 38

A cyclotron with dee radius $47.0 \mathrm{~cm}$ is operated at an oscillator frequency of $12.0 \mathrm{MHz}$ to accelerate protons. (a) What magnitude $B$ of magnetic field is required to achieve resonance? (b) At that field magnitude, what is the kinetic energy of a proton emerging from the cyclotron? Suppose, instead, that $B=1.57 \mathrm{~T}$. (c) What oscillator frequency is required to achieve resonance now? (d) At that frequency, what is the kinetic energy of an emerging proton?

Joy Chugh

Numerade Educator

Problem 39

(a) What uniform magnetic field, applied perpendicular to a beam of electrons moving at $1.30 \times 10^{6} \mathrm{~m} / \mathrm{s}$, is required to make the electrons travel in a circular arc of radius $0.500 \mathrm{~m} ?(\mathrm{~b})$ What is the period of the motion?

Joy Chugh

Numerade Educator

Problem 40

The magnetic dipole moment of Earth has magnitude $8.00 \times 10^{22} \mathrm{~J} / \mathrm{T}$. Assume that this is produced by charges flowing in Earth's molten outer core. If the radius of their circular path is $3700 \mathrm{~km}$, calculate the current they produce.

Joy Chugh

Numerade Educator

Problem 41

An electron follows a helical path in a uniform magnetic field of magnitude $1.30 \mathrm{~T}$. The pitch of the path is $6.00 \mu \mathrm{m}$, and the magnitude of the magnetic force on the electron is $2.00 \times 10^{-14} \mathrm{~N}$. What is the electron's speed?

Rudra Singh

Numerade Educator

Problem 42

A particle of mass $12 \mathrm{~g}$ and charge $80 \mu \mathrm{C}$ moves through a uniform magnetic field, in a region where the free-fall acceleration is $-9.8 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}^{2}$. The velocity of the particle is a constant $20 \hat{\mathrm{i}} \mathrm{km} / \mathrm{s}$, which is perpendicular to the magnetic field. What, then, is the magnetic field?

Sheh Lit Chang

University of Washington

Problem 43

An electron has an initial velocity of $(12.0 \hat{\mathrm{j}}-15.0 \mathrm{k}) \mathrm{km} / \mathrm{s}$ and a constant acceleration of $\left(2.00 \times 10^{12} \mathrm{~m} / \mathrm{s}^{2}\right) \mathrm{i}$ in a region in which uniform electric and magnetic fields are present. If $\vec{B}=(300 \mu \mathrm{T}) \hat{\mathrm{i}}$, find the electric field $\vec{E}$.

Sheh Lit Chang

University of Washington

Problem 44

A particle undergoes uniform circular motion of radius $28.7 \mu \mathrm{m}$ in a uniform magnetic field. The magnetic force on the particle has a magnitude of $1.60 \times 10^{-17} \mathrm{~N}$. What is the kinetic energy of the particle?

Rudra Singh

Numerade Educator

Problem 45

A certain particle is sent into a uniform magnetic field, with the particle's velocity vector perpendicular to the direction of the field. Figure 28-38 gives the period $T$ of the particle's motion versus the inverse of the field magnitude $B$. The vertical axis scale is set by $T_{s}=80.0 \mathrm{~ns}$, and the horizontal axis scale is set by $B_{s}^{-1}=10.0 \mathrm{~T}^{-1}$. What is the ratio $m / q$ of the particle's mass to the magnitude of its charge?

Sheh Lit Chang

University of Washington

Problem 46

A magnetic dipole with a dipole moment of magnitude $0.020 \mathrm{~J} / \mathrm{T}$ is released from rest in a uniform magnetic field of magnitude $46 \mathrm{mT}$. The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is $0.80 \mathrm{~mJ}$. (a) What is the initial angle between the dipole moment and the magnetic field? (b) What is the angle when the dipole is next (momentarily) at rest?

Rudra Singh

Numerade Educator

Problem 47

A strip of copper $75.0 \mu \mathrm{m}$ thick and $4.5 \mathrm{~mm}$ 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 $i=57 \mathrm{~A}$ is then sent through the strip such that a Hall potential difference $V$ appears across the width of the strip. Calculate $V$. (The number of charge carriers per unit volume for copper is $8.47 \times 10^{28}$ electrons $/ \mathrm{m}^{3}$.)

Rudra Singh

Numerade Educator

Problem 48

An alpha particle travels at a velocity $\vec{v}$ of magnitude $620 \mathrm{~m} / \mathrm{s}$ 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 is the magnitude of (a) the force $\vec{F}_{B}$ acting on the particle due to the field and $(b)$ the acceleration of the particle due to $\vec{F}_{B} ?$ (c) Does the speed of the particle increase, decrease, or remain the same?

Rudra Singh

Numerade Educator

Problem 49

An alpha particle travels at a velocity $\vec{v}$ of magnitude $620 \mathrm{~m} / \mathrm{s}$ 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 is the magnitude of (a) the force $\vec{F}_{B}$ acting on the particle due to the field and $(b)$ the acceleration of the particle due to $\vec{F}_{B} ?$ (c) Does the speed of the particle increase, decrease, or remain the same?

Rudra Singh

Numerade Educator

Problem 50

A proton travels through uniform magnetic and electric fields. The magnetic field is $\vec{B}=-3.25 \hat{\mathrm{i}} \mathrm{mT}$. At one instant the velocity of the proton is $\vec{v}=2000 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}$. At that instant and in unit-vector notation, what is the net force acting on the proton if the electric field is (a) $4.00 \mathrm{k} \mathrm{V} / \mathrm{m}$, (b) $-4.00 \mathrm{k} \mathrm{V} / \mathrm{m}$, and (c) $4.00 \hat{\mathrm{i}} \mathrm{V} / \mathrm{m}$ ?

Sheh Lit Chang

University of Washington

Problem 51

A proton traveling at $42.0^{\circ}$ with respect to the direction of a magnetic field of strength $2.60 \mathrm{mT}$ experiences a magnetic force of $1.17 \times 10^{-17} \mathrm{~N}$. Calculate (a) the proton's speed, (b) its kinetic energy in electron-volts, and (c) its momentum.

Joy Chugh

Numerade Educator

Problem 53

An electron of kinetic energy $600 \mathrm{eV}$ circles in a plane perpendicular to a uniform magnetic field. The orbit radius is $12.5 \mathrm{~cm}$. Find (a) the electron's speed, (b) the magnetic field magnitude, (c) the circling frequency, and (d) the period of the motion. (e) Through what potential difference would the electron have to be accelerated from rest to reach this kinetic energy?

Joy Chugh

Numerade Educator

Problem 54

In a certain cyclotron a proton moves in a circle of radius $0.500 \mathrm{~m}$. The magnitude of the magnetic field is $1.00 \mathrm{~T}$. (a) What is the oscillator frequency? (b) What is the kinetic energy of the proton, in electron-volts?

Joy Chugh

Numerade Educator

Problem 55

In a certain cyclotron a proton moves in a circle of radius $0.500 \mathrm{~m}$. The magnitude of the magnetic field is $1.00 \mathrm{~T}$. (a) What is the oscillator frequency? (b) What is the kinetic energy of the proton, in electron-volts?

Joy Chugh

Numerade Educator

Problem 56

Estimate the total path length traveled by a deuteron in a cyclotron of radius $53 \mathrm{~cm}$ and operating frequency $12 \mathrm{MHz}$ during the (entire) acceleration process. Assume that the accelerating potential between the dees is $120 \mathrm{kV}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

A single-turn current loop, carrying a current of $8.00 \mathrm{~A}$, 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. What is the magnitude of the magnetic force on (a) the $130 \mathrm{~cm}$ side, (b) the $50.0 \mathrm{~cm}$ side, and (c) the $120 \mathrm{~cm}$ side? (d) What is the magnitude of the net force on the loop?

Sheh Lit Chang

University of Washington

Problem 58

Fig. 28-41a, two concentric coils, lying in the same plane, carry currents in opposite directions. The current in the larger coil 1 is fixed. Current $i_{2}$ in coil 2 can be varied. Figure $28-41 b$ gives the net magnetic moment of the two-coil system as a function of $i_{2}$. The vertical axis scale is set by $\mu_{\text {net }, x}=2.0 \times 10^{-5} \mathrm{~A} \cdot \mathrm{m}^{2}$, and the horizontal axis scale is set by $i_{2 s}=20.0 \mathrm{~mA}$. If the current in coil 2 is then reversed, what is the magnitude of the net magnetic moment of the two-coil system when $i_{2}=7.0 \mathrm{~mA}$ ?

Sheh Lit Chang

University of Washington

Problem 59

A mass spectrometer (Fig. 28-12) 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 $180 \mathrm{kV}$ 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 $1.00 \mathrm{~h}$.

Sheh Lit Chang

University of Washington

Problem 60

An electric field of $1.50 \mathrm{kV} / \mathrm{m}$ and a perpendicular magnetic field of $0.350 \mathrm{~T}$ act on a moving electron to produce no net force. What is the electron's speed?

Joy Chugh

Numerade Educator

Problem 61

An ion source is producing ${ }^{6} \mathrm{Li}$ ions, which have charge $+e$ and mass $9.99 \times 10^{-27} \mathrm{~kg}$. The ions are accelerated by a potential difference of $25 \mathrm{kV}$ and pass horizontally into a region in which there is a uniform vertical magnetic field of magnitude $B=1.2 \mathrm{~T}$. Calculate the strength of the electric field, to be set up over the same region, that will allow the ${ }^{6} \mathrm{Li}$ ions to pass through without any deflection.

Keshav Singh

Numerade Educator

Problem 62

In a nuclear experiment a proton with kinetic energy $1.2 \mathrm{MeV}$ 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 (b) a deuteron $(q=+e, m=2.0 \mathrm{u})$ have if they are to circulate in the same circular path?

Shital Rijal

Numerade Educator

Problem 63

An electron that has an instantaneous velocity of
$$
\vec{v}=\left(-5.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{i}}+\left(3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{j}}
$$
is moving through the uniform magnetic field $\vec{B}=(0.030 \mathrm{~T}) \hat{\mathrm{i}}-$ $(0.15 \mathrm{~T}) \hat{j}$. (a) Find the force on the electron due to the magnetic field. (b) Repeat your calculation for a proton having the same velocity.

Keshav Singh

Numerade Educator

Problem 64

In Fig. 28-42, an electron with an initial kinetic energy of $5.0 \mathrm{keV}$ enters region 1 at time $t=0$. That region contains a uniform magnetic field directed into the page, with magnitude $0.010 \mathrm{~T}$. The electron goes through a half-circle and then exits region 1, headed toward region 2 across a gap of $25.0 \mathrm{~cm}$. There is an electric potential difference $\Delta V=2000 \mathrm{~V}$ across the gap, with a polarity such that the electron's speed increases uniformly as it traverses the gap. Region 2 contains a uniform magnetic field directed out of the page, with magnitude $0.020 \mathrm{~T}$. The electron goes through a half-circle and then leaves region 2. At what time $t$ does it leave?

Sheh Lit Chang

University of Washington

Problem 65

An alpha particle can be produced in certain radioactive decays of nuclei and consists of two protons and two neutrons. The particle has a charge of $q=+2 e$ and a mass of $4.00 \mathrm{u}$, where $\mathrm{u}$ is the atomic mass unit, with $1 \mathrm{u}=1.661 \times 10^{-27} \mathrm{~kg}$. Suppose an alpha particle travels in a circular path of radius $4.50 \mathrm{~cm}$ in a uniform magnetic field with $B=1.20 \mathrm{~T}$. Calculate (a) its speed, (b) its period of revolution, (c) its kinetic energy, and (d) the potential difference through which it would have to be accelerated to achieve this energy. (e) If the field magnitude is doubled, what is the ratio of the new value of kinetic energy to the initial value?

Sheh Lit Chang

University of Washington