Fundamentals of Physics

David Halliday, Robert Resnick , Jearl Walker

Chapter 28

Magnetic Fields - all with Video Answers

Educators

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Chapter Questions

Problem 1

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.

Alan Hendrick

Alan Hendrick

Numerade Educator

Problem 2

A particle of mass 10 $\mathrm{g}$ 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$\hat{0}$ is the
which is perpendicular to the magnetic ficld. What, then, is the
magnetic field?

Shital Rijal

Numerade Educator

Problem 3

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

Jack Hou

Numerade Educator

Problem 4

An alnha particle travels at a velocity $v$ of magnitude 550 $\mathrm{m} / \mathrm{s}$ through a uniform magnetic field $\vec{B}$ of magnitude 0.045 T. (An alpha particle has a charge of $+3.2 \times 10^{-19} \mathrm{C}$ and a mass of 6.6 $\mathrm{x}$ $10^{-27} \mathrm{kg} .$ ) 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 $F_{B} ?(\mathrm{c})$ Does the speed
of the particle increase, decrease, or remain the same?

Shital Rijal

Numerade Educator

Problem 5

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

Jack Hou

Numerade Educator

Problem 6

A nroton moves through a uniform magnetic field given by $\vec{B}=(10 \hat{\mathrm{i}}-20 \hat{\mathrm{j}}+30 \hat{\mathrm{k}}) \mathrm{mT}$ . At time $t_{1},$ the proton has a velocity given by $\vec{v}=v_{x} \hat{\mathrm{i}}+v_{y} \hat{\mathrm{j}}+(2.0 \mathrm{km} / \mathrm{s}) \hat{\mathrm{k}}$ and the magnetic force on the proton is $\vec{F}_{B}=\left(4.0 \times 10^{-17} \mathrm{N}\right) \hat{\mathrm{i}}+\left(2.0 \times 10^{-17} \mathrm{N}\right) \hat{\mathrm{j}}$ . At that instant, what are (a) $v_{x}$ and $(\mathrm{b}) v_{y} ?$

Shital Rijal

Numerade Educator

Problem 7

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

Jack Hou

Numerade Educator

Problem 8

An electric field of 1.50 $\mathrm{kV} / \mathrm{m}$ and a perpendicular magnetic ficld of 0.400 $\mathrm{T}$ act on a moving clectron to produce no net force.
What is the electron's speed?

Shital Rijal

Numerade Educator

Problem 9

In Fig. $28-32$ , an electron accelerated from rest through potential difference $V_{1}=1.00 \mathrm{kV}$ enters the gap between two parallel plates having separation $d=20.0 \mathrm{mm}$ 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. In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap?

Sachin Rao

Numerade Educator

Problem 10

A proton travels through uniform magnetic and electric fields. The magnetic field is $\vec{B}=-2.50 \hat{\mathrm{i}} \mathrm{mT} .$ 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 clectric ficld is (a) $4.00 \hat{k} \mathrm{V} / \mathrm{m},$ (b) $-4.00 \hat{\mathrm{k}} \mathrm{V} / \mathrm{m},$ and
(c) 4.00$\hat{\mathrm{i}} \mathrm{V} / \mathrm{m} ?$

Shital Rijal

Numerade Educator

Problem 11

An ion source is producing $^{6}$ Li ions, which have charge $+e$ and mass $9.99 \times 10^{-27} \mathrm{kg}$ . 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 $B=1.2 \mathrm{T}$ . Calculate the strength of the smallest electric field, to be set up over the same region, that will allow the $^{6} \mathrm{L}$ i ions to pass through
undeflected.

Nicholas Mogoi

Numerade Educator

Problem 12

At time $t_{1},$ an electron is sent along the positive direction of an $x$ axis, through both an clectric
field $\vec{E}$ and a magnetic field $\vec{B},$ with $\vec{E}$ directed parallel to the $y$ axis. Figure $28-33$ gives the $y$ component $F_{\text { net }, y}$ of the net force on the electron due to the two fields, as a function of the electron's speed $v$ at time $t_{1}$ . The scale of the velocity axis is set by
$v_{s}=100.0 \mathrm{m} / \mathrm{s}$ . The $x$ and $z$ components of the net force are zero at $t_{1 .}$ Assuming $B_{x}=0,$ find (a) the magnitude $E$ and $(b) \vec{B}$ in unit-vector notation.

Shital Rijal

Numerade Educator

Problem 13

A strip of copper 150$\mu \mathrm{m}$ 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 $i=23 \mathrm{A}$ is then sent through the strip such that a Hall potential difference $V$ appcars 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/m $^{3}$ . $)$

Sachin Rao

Numerade Educator

Problem 14

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, $28-34 .$ A potential difference of 3.90$\mu V$ is measured between points $x$ and $y$ across the strip. Calculate the speed $v$ .

Shital Rijal

Numerade Educator

Problem 15

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 $\vec{v}=(20.0 \mathrm{m} / \mathrm{s})$ i through a uniform magnetic field $\vec{B}=(30.0 \mathrm{mT}) \hat{\mathrm{j}}($ Fig. $28-35) .$ What are the resulting (a) electric field within the solid, in unit-vector notation, and (b) potential difference across the solid?

Sachin Rao

Numerade Educator

Problem 16

Figurc $28-35$ shows a metallic block, with its faces parallel to
coordinate axes. The block is in a uniform magnctic ficld of magnitude 0.020 T. One edge length of the block is $25 \mathrm{cm} ;$ the block is not drawn to scale. The block is moved at 3.0 $\mathrm{m} / \mathrm{s}$ 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) $d_{x},\left($ b) $d_{y},$ and $(\mathrm{c}) d_{z} ?\right.$

Shital Rijal

Numerade Educator

Problem 17

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 u is the atomic mass unit, with $1 \mathrm{u}=1.661 \times 10^{-27} \mathrm{kg} .$ Suppose an alpha particle travcls 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.

Jack Hou

Numerade Educator

Problem 18

In Fig. $28-36$ , a particle moves along a circle in a region of uniform magnetic ficld of
magnitude $B=4.00 \mathrm{mT}$ . The particle is either a proton or an electron (you must decide which). It experiences a magnetic force of magnitude $3.20 \times 10^{-15} \mathrm{N}$ . What are (a) the particle's speed,
(b) the radius of the circle, and (c) the period of the motion?

Ummatul Choudary

Ummatul Choudary

Numerade Educator

Problem 19

A certain particle is sent into a uniform magnetic field, with the particle's velocity vector perpendicular to the direction of the field. Figure $28-37$ gives the period $T$ of the particle's motion versus the inverse of the field magnitude $B$ . The vertical axis scale is sct by $T_{s}=40.0 \mathrm{ns},$ and the horizontal axis scale is sct by $B_{s}^{-1}=5.0 \mathrm{T}^{-1} .$ What is the ratio $\mathrm{m} / \mathrm{q}$ of the particle's mass to the magnitude of its charge?

Supratim Pal

Numerade Educator

Problem 20

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 $28-38$ gives the radius $r$ of that
motion versus $V^{12}$ . The vertical axis scale is set by $r_{s}=3.0 \mathrm{mm},$ 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?

Shital Rijal

Numerade Educator

Problem 21

An electron of kinetic energy 1.20 $\mathrm{keV}$ circles in a plane per-
pendicular to a uniform magnctic field. The orbit radius is 25.0 $\mathrm{cm} .$ Find (a) the electron's speed, (b) the magnetic field magnitude, (c) the circling frequency, and (d) the period of the motion.

Jack Hou

Numerade Educator

Problem 22

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

Numerade Educator

Problem 23

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 0.350 $\mathrm{m}$ ?

Jack Hou

Numerade Educator

Problem 24

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

Numerade Educator

Problem 25

(a) Find the frequency of revolution of an electron with an energy of 100 $\mathrm{cV}$ in a uniform magnetic ficld 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.

Jack Hou

Numerade Educator

Problem 26

In Fig, $28-39,$ 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 130 ns in the region. (a) What is the magnitude of $\vec{B} ?$ (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 in the field during this trip?

Shital Rijal

Numerade Educator

Problem 27

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 differ-
ence 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 $\mathrm{h}$ .

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

A certain particle is sent into a uniform magnetic field, with the particle’s velocity vector perpendicular to the direction of the field. Figure 28-37 gives the period $T$ of the particle’s motion versus the inverse of the field magnitude $B$. The vertical axis scale is set by $T_{y}=40.0 \mathrm{~ns},$ and the horizontal axis scale is set by $B_{s}^{-1}=5.0 \mathrm{~T}^{-1} .$ What is the ratio $\mathrm{m} / q$ of the particle's mass to the magnitude of its charge?
(FIGURE CANT COPY)

Shital Rijal

Numerade Educator

Problem 29

An electron follows a helical path in a uniform magnetic field of magnitude 0.300 T. The pitch of the path is $6.00 \mu \mathrm{m},$ and the magnitude of the magnetic force on the electron is $2.00 \times 10^{-15} \mathrm{N}$ . What is the electron's speed?

Sachin Rao

Numerade Educator

Problem 30

In Fig. $28-40,$ an electron with an initial kinetic energy of 4.0 keV enters region 1 at time $t=0 .$ That region contains a uniform magnetic field directed into the page, with magnitude 0.010 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 T.The electron goes through a half-circle and then leaves region $2 .$ At what time $t$ does it leave?

Shital Rijal

Numerade Educator

Problem 31

A particular type of fundamental particle decays by transforming into an electron $\mathrm{e}^{-}$ and a positron $\mathrm{e}^{+} .$ Suppose the decaying particle is at rest in a uniform magnetic field $\vec{B}$ of magnitude 3.53 $\mathrm{mT}$ and the $\mathrm{e}^{-}$ and $\mathrm{e}^{\text { t }}$ 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 $\mathrm{e}^{\prime}$ collide?

Sachin Rao

Numerade Educator

Problem 32

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.

Shital Rijal

Numerade Educator

Problem 33

A positron with kinctic encrgy 2.00 $\mathrm{keV}$ is projected into a uniform magnetic field $\vec{B}$ of magnitude 0.100 'I, 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.

Sachin Rao

Numerade Educator

Problem 34

An electron follows a helical path in a uniform magnetic field given by $\vec{B}=(20 \hat{\mathrm{i}}-50 \hat{\mathrm{j}}-30 \hat{\mathrm{k}})$ mT. At time $t=0,$ the electron's velocity is given by $V=(20 \hat{\mathrm{i}}-30 \hat{\mathrm{j}}+50 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$ . (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?

Shital Rijal

Numerade Educator

Problem 35

A proton circulates in a cyclotron, beginning approximately at rest at the center. Whenever it passes through the gap between decs, the clectric potential difference between the decs is 200 $\mathrm{V}$ .
(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 $r_{1000}$ be the radius of the proton's
circular path as it completes those 100 passes and enters a dec,
and let $r_{101}$ be its next radius, as it enters a dee the next time. (c) By
what percentage docs the radius increase when it changes from
$r_{100}$ to $r_{101} ?$ That is, what is percentage increase $=\frac{r_{101}-r_{100}}{r_{100}} 100 \% ?$

Sachin Rao

Numerade Educator

Problem 36

A cyclotron with dee radius 53.0 $\mathrm{cm}$ 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?

Shital Rijal

Numerade Educator

Problem 37

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 80 $\mathrm{kV}$ .

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

In a certain cyclotron a proton moves in a circle of radius 0.500 $\mathrm{m}$ . The magnitude of the magnetic field is 1.20 $\mathrm{T}$ . (a) What is
the oscillator frequency? (b) What is the kinetic energy of the proton, in electron-volts?

Shital Rijal

Numerade Educator

Problem 39

A horizontal power line carries a current of 5000 $\mathrm{A}$ 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.

Jack Hou

Numerade Educator

Problem 40

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 magnetic force on the wire.

Shital Rijal

Numerade Educator

Problem 41

A 13.0 g wire of length $L=62.0 \mathrm{cm}$ is suspended by a pair
of flexible leads in a uniform mag-
nctic ficld of magnitude 0.440 $\mathrm{T}$ Fig. $28-41$ . What are the (a) magni-
tude and (b) direction (left or right)
of the current required to remove
the tension in the supporting leads?

Sachin Rao

Numerade Educator

Problem 42

The bent wire shown in Fig. 28 42 lies in a uniform magnctic ficld.
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 2.0 $\mathrm{A}$ . What is the net magnetic force on the wire in unit-vector notation if the magnetic field is
given by (a) 4.0$\hat{\mathrm{k}} \mathrm{T}$ and (b) 4.0$\hat{\mathrm{i}} \mathrm{T}$ ?

Shital Rijal

Numerade Educator

Problem 43

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. What is
the magnitude of the magnetic force on (a) the 130 $\mathrm{cm}$ side, (b) the
50.0 $\mathrm{cm}$ side, and $(\mathrm{c})$ the 120 $\mathrm{cm}$ side? (d) What is the magnitude of the net force on the loop?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 44

Figure $28-43$ shows a wire ring of radius $a=1.8 \mathrm{cm}$ that is perpendicular to the general direction of a radially symmetric, diverging magnetic field. The magnetic field at
the ring is cverywhere of the same magnitude $B=3.4 \mathrm{mT},$ and its direction at the ring everywhere makes an angle $\theta=20^{\circ}$ 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 $i=4.6 \mathrm{mA} .$

Eduard Sanchez

Numerade Educator

Problem 45

A wire 50.0 $\mathrm{cm}$ long carries a 0.500 A current in the positive direction of an $x$ axis through a magnetic field $\vec{B}=$ $(3.00 \mathrm{mT}) \hat{\mathrm{j}}+(10.0 \mathrm{mi} 1) \hat{\mathrm{k}}$ . In unit-vector notation, what is the magnetic force on the wire?

Jack Hou

Numerade Educator

Problem 46

In Fig. $28-44,$ 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 56.3 $\mathrm{mT}$ : 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)?

Shital Rijal

Numerade Educator

Problem 47

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
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 48

A long, rigid conductor, lying along an $x$ axis, carries a current of 5.0 $\mathrm{A}$ in the negative $x$ direction. A magnetic field
$\vec{B}$ is present, given by $\vec{B}=3.0 \hat{\mathrm{i}}+8.0 \mathrm{x}^{2} \mathrm{j},$ 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 $x=3.0 \mathrm{m}$ .

Shital Rijal

Numerade Educator

Problem 49

Figure $28-45$ 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 angle $\theta=30^{\circ}$ to the direction of a uniform magnetic field
of magnitude 0.50 $\mathrm{T}$ . In unit-vector notation, what is the torque acting
on the coil about the hinge line?

Sachin Rao

Numerade Educator

Problem 50

An electron moves in a circle of radius $r=5.29 \times 10^{-11} \mathrm{m}$ with
speed $2.19 \times 10^{6} \mathrm{m} / \mathrm{s}$ . 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?

Shital Rijal

Numerade Educator

Problem 51

Figure $28-46$ shows a wood cylinder of 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 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
0.500 $\mathrm{T}$ , what is the least current $i$ through the coil that keeps the cylinder from rolling down the plane?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

In Fig. $28-47,$ a rectangular loop carrying current lies in the plane of a uniform
magnetic field of magnitude 0.040 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?

Shital Rijal

Numerade Educator

Problem 53

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 54

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 52 $\mathrm{mT}$ . 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?

Shital Rijal

Numerade Educator

Problem 55

Two concentric, circular wire loops, of radii $r_{1}=20.0 \mathrm{cm}$ and $r_{2}=30.0 \mathrm{cm},$ are located in an $x y$ plane; each carries a clockwise current of 7.00 $\mathrm{A}$ (Fig. $28-48 )$ . (a) Find the magnitude of the net magnetic dipole moment of the system.
(b) Repeat for reversed current in the inner loop.

Sachin Rao

Numerade Educator

Problem 56

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

Shital Rijal

Numerade Educator

Problem 57

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 magnitude 2.30 $\mathrm{A} \cdot \mathrm{m}^{2} .$ (b) Find the maximum magnitude of the
torque that the coil, carrying this current, can experience in a uni-
form 35.0 mT magnetic field.

Sachin Rao

Numerade Educator

Problem 58

The magnetic dipole moment of Earth has magnitude $8.00 \times$ $10^{22} \mathrm{JT}$ . 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 current they produce.

Shital Rijal

Numerade Educator

Problem 59

A current loop, carrying a 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.

Sachin Rao

Numerade Educator

Problem 60

Figure $28-49$ shows a current loop $A B C D E F A$ carrying a current $i=$
5.00 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 magnctic 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 $A D E F A . )$

Shital Rijal

Numerade Educator

Problem 61

The coil in Fig. $28-50$ carries current $i=2.00 \mathrm{A}$ in the direction indicated, is parallel to an $x z$ plane, has 3.00 turns and an area of $4.00 \times 10^{-3} \mathrm{m}^{2},$ and lies in a uniform magnetic field $\vec{B}=(2.00 \hat{\mathrm{i}}-3.00 \mathrm{k}) \mathrm{mT}$ . What
are (a) the oricntation energy of the coil in the magnetic field and
(b) the torque (in unit-vector notation) on the coil due to the
magnetic field?

Sachin Rao

Numerade Educator

Problem 62

In Fig. $28-51 a,$ 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 $28-51 b$ gives the net magnetic moment of the two-coil system as a function
of $i_{2}$ . The vertical axis scale is set by $\mu_{\text { nect }}=2.0 \times 10^{-5} \mathrm{A} \cdot \mathrm{m}^{2},$ and the horizontal axis scale is set by $i_{2 s}=10.0 \mathrm{mA}$ . If the current in
coil 2 is then reversed, what is the magnitude of the net magnetic
moment of the two-coil system when $i_{2}=7.0 \mathrm{mA} ?$

Shital Rijal

Numerade Educator

Problem 63

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 $\overrightarrow{\mu}$ of the loop is given by $0.60 \hat{\mathrm{i}}-0.80 \hat{\mathrm{j}}$ . (This unit vector gives the orientation of the magnetic dipole moment vector. $)$ If the loop is located in a uniform magnctic field given by $\vec{B}=$ $(0.25 \mathrm{T}) \hat{\mathrm{i}}+(0.30 \mathrm{T}) \hat{\mathrm{k}}$ , find (a) the torque on the loop (in unit-vector notation) and (b) the orientation energy of the loop.

Sachin Rao

Numerade Educator

Problem 64

Figurc $28-52$ gives the orientation encrgy $U$ 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 sct by $U_{x}=2.0 \times 10^{-4} \mathrm{J}$ . The dipolc can be rotated about an axle with negligible friction in order to change $\phi$ .
Counterclockwise rotation from $\phi=0$ yields positive values of $\phi$ ,and clockwise rotations yield negative values. The dipole is to be released at angle $\phi=0$ with a rotational kinetic energy of $6.7 \times$ $10^{-4} \mathrm{J},$ so that it rotates counterclock 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. $28-52$ ?

Shital Rijal

Numerade Educator

Problem 65

\Lambda wire of length 25.0 $\mathrm{cm}$ carrying a current of 4.51 $\mathrm{mA}$ is to be formed into a circular coil and placed in a uniform
magnetic field $\vec{B}$ of magnitude 5.71 $\mathrm{mT}$ . If the torque on the coil
from the field is maximitude 5.71 $\mathrm{mT}$ . If the angle between $\vec{B}$ and
the coil's magnetic dipole moment are ( b) the number of turns in
the coil? (c) What is the magnitude of that maximum torque?

Sachin Rao

Numerade Educator

Problem 66

A proton of charge $+e$ and mass $m$ enters a uniform magnetic field $\vec{B}=B \hat{\mathrm{i}}$ with an initial velocity $v=v_{0.1}$ $+v_{\mathrm{lv}} \hat{\mathrm{j}}$ Find an expression in unit-vector notation for its velocity $\vec{v}$ at any later time $t .$

Shital Rijal

Numerade Educator

Problem 67

A stationary circular wall clock has a face with a radius of 15 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 time). At
exactly $1 : 00 \mathrm{eM},$ the hour hand of the clock points in the direction
of the external magnetic field. (a) After how many minutes will the minute hand point in the direction of the torque on the winding
due to the magnetic field? (b) Find the torque magnitude.

Sachin Rao

Numerade Educator

Problem 68

A wire lying along a $y$ axis from $y=0$ to $y=0.250 \mathrm{m}$ carries a current of 2.00 $\mathrm{mA}$ in the negative direction of the axis. The wire
fully lies in a nonuniform magnetic field that is given by
$B=(0.300 \mathrm{T} / \mathrm{m}) y \hat{\mathrm{i}}+(0.400 \mathrm{T} / \mathrm{m})$ yj. In unit-vector notation, what is the magnetic force on the wire?

Shital Rijal

Numerade Educator

Problem 69

Atom 1 of mass 35 $\mathrm{u}$ and atom 2 of mass 37 $\mathrm{u}$ are both singly ionized with a charge of $t$ . After being introduced into a mass spectrometer $($ Fig. $28-12)$ and accelerated from rest through a potential difference $V=7.3 \mathrm{kV},$ each ion follows a circular path in a
uniform magnetic field of magnitude $B=0.50 \mathrm{T}$ . What is the distance $\Delta x$ between the points where the ions strike the detector?

Sachin Rao

Numerade Educator

Problem 70

An electron with kinetic energy 2.5 $\mathrm{keV}$ moving along the positive direction of an $x$ axis enters a region in which a uniform
clectric ficld of magnitude 10 $\mathrm{kV} / \mathrm{m}$ is in the negative dircction of
the $y$ axis. $\mathrm{A}$ uniform magnetic field $\vec{B}$ is to be set up to keep the electron moving along the $x$ axis, and the direction of $\vec{B}$ is to be chosen to minimize the required magnitude of $\vec{B}$ . In unit-vector notation, what $\vec{B}$ should be set up?

Shital Rijal

Numerade Educator

Problem 71

Physicist S. A. Goudsmit devised a method for measuring the mass of heavy ions by timing their period of revolution in a known
magnetic field. A singly charged ion of iodine makes 7.00 rev in a
45.0 mT field in 1.29 ms. Calculate its mass in atomic mass units.

Jack Hou

Numerade Educator

Problem 72

A beam of electrons whose kinetic energy is $K$ emerges from a thin-foil "window" at the end of an
accelerator tube. A metal plate at distance $d$ from this window is perpendicular to the direction of the emerging bcam (Fig, $28-53 )$ . (a) Show that we can prevent the beam from hitting
the plate if we apply a uniform magnetic ficld such that $$B \geqq \sqrt{\frac{2 m K}{e^{2} d^{2}}}$$ $$
\begin{array}{l}{\text { in which } m \text { and } e \text { are the electron mass and charge. (b) How should }} \\ {\vec{B} \text { be oriented? }}\end{array}$$

Satyam Gupta

Numerade Educator

Problem 73

$$t=0, \text { an electron with kinetic energy } 12$$ $$
\begin{array}{l}{\text { moves through } x=0 \text { in the positive direction of an } x \text { axis that is }} \\ {\text { parallel to the horizontal component of Earth's magnetic field } \vec{B}} \\ {\text { The field's vertical component is downward and has magnitude }}\end{array}$$
$$\begin{array}{l}{55.0 \mu \mathrm{T} \text { (a) What is the magnitude of the electron's acceleration }} \\ {\text { due to } \vec{B} ? \text { (b) What is the electron's distance from the } x \text { axis when }} \\ {\text { the electron reaches coordinate } x=20 \mathrm{cm} ?}\end{array}$$

Sachin Rao

Numerade Educator

Problem 74

A particle with charge 2.0 $\mathrm{Cmoves}$ through a uniform magnetic field. At one instant the velocity of the particle is
$(2.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}+6.0 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$ and the magnetic force on the particle is
$(4.0 \hat{\mathrm{i}}-20 \mathrm{j}+12 \hat{\mathrm{k}}) \mathrm{N}$ . The $x$ and $y$ components of the magnetic
field are equal. What is $\vec{B}$ ?

Shital Rijal

Numerade Educator

Problem 75

A proton, a deuteron $(q=+e, m=2.0 \mathrm{u}),$ and an alpha particle $(q=+2 e, m=4.0 \mathrm{u})$ all having the same kinetic energy enter a region of uniform magnetic field $\vec{B}$ , moving perpendicular to $\vec{B}$ . What is the ratio of (a) the radius $r_{d}$ of the deuteron path to the radius $r_{p}$ of the proton path and (b) the radius $r_{\alpha}$ of the alpha particle path to $r_{p} ?$

Jack Hou

Numerade Educator

Problem 76

Bainbridge's mass spectrometer, shown in Fig. $28-54$ , separates ions
having the same velocity. The ions, after entering through slits, $S_{1}$ and $S_{2}$ pass through a velocity selector composed of an electric field produced by the charged plates $P$ and $P^{\prime},$ and a magnetic field $\vec{B}$ perpendicular to the electric ficld and the ion path. The
ions that then pass undeviated through the crossed $\vec{E}$ and $\vec{B}$ fields enter into a region where a second magnetic field $\vec{B}$ ' exists, where they are made to follow circular
paths. A photographic plate (or a modern detector) registers their
arrival. Show that, for the ions, $q / m=E / r B B^{\prime},$ where $r$ is the radius
of the circular orbit.

Shital Rijal

Numerade Educator

Problem 77

In Fig. $28-55,$ an electron moves at spced $v=100 \mathrm{m} / \mathrm{s}$ along an
$x$ axis through uniform electric and magnetic fields. The magnetic field $\vec{B}$ is directed into the page and has magnitude 5.00 $\mathrm{T}$ . In unit-vector notation, what is the electric field?

Jack Hou

Numerade Educator

Problem 78

(a) In Fig. $28-8$ show that the ratio of the Hall clectric ficld magnitude $E$ to the magnitude $E_{C}$ of the electric field responsible for moving charge (the current) along the length of the strip is
$$\frac{E}{E_{C}}=\frac{B}{\operatorname{ne} \rho}$$
$\begin{array}{l}{\text { where } \rho \text { is the resistivity of the material and } n \text { is the number den- }} \\ {\text { sity of the charge carriers. (b) Compute this ratio numerically for }} \\ {\text { Problem } 13 . \text { (Sce Table } 26-1 . )}\end{array}$

Shital Rijal

Numerade Educator

Problem 79

A proton, a deuteron $(q=+e, m=2.0 \mathrm{u}),$ and an alpha particle $(q=+2 e, m=4.0 \mathrm{u})$ are accelerated through the same
potential difference and then enter the same region of uniform
magnetic field $\vec{B}$ , moving perpendicular to $\vec{B}$ . What is the ratio of (a) the proton's kinetic energy $K_{p}$ to the alpha particle's kinetic
cncrey $K_{\alpha}$ and $(b)$ the deutcron's kinctic cnergy $K_{d}$ to $K_{a}$ ? If the
radius of the proton's circular path is $10 \mathrm{cm},$ what is the radius of
(c) the deuteron's path and (d) the alpha particle's path?

Jack Hou

Numerade Educator

Problem 80

An elcctron is moving at $7.20 \times 10^{6} \mathrm{m} / \mathrm{s}$ in a magnctic ficld of strength 83.0 $\mathrm{mT}$ . What is the (a) maximum and (b) minimum
magnitude of the force acting on the electron due to the field? (c) 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?

Shital Rijal

Numerade Educator

Problem 81

A 5.0$\mu \mathrm{C}$ particle moves through a region containing the uniform magnetic field $-20 \hat{\mathrm{i}} \mathrm{mT}$ and the uniform electric ficld
300$\hat{\mathrm{j}} \mathrm{V} / \mathrm{m} .$ At a certain instant the velocity of the particle is
$(17 \hat{\mathrm{i}}-11 \hat{\mathrm{j}}+7.0 \hat{\mathrm{k}}) \mathrm{km} / \mathrm{s}$ . At that instant and in unit-vector notation, what is the net electromagnetic force (the sum of the electric
and magnetic forces on the particle?

Jack Hou

Numerade Educator

Problem 82

In a Hall-effect experiment, a current of 3.0 $\mathrm{A}$ sent lengthwise through a conductor 1.0 $\mathrm{cm}$ wide, 4.0 $\mathrm{cm}$ long, and 10 $\mathrm{\mum}$
thick produces a transverse (across the width) Hall potential dif-
ference of 10$\mu \mathrm{V}$ when a magnetic field of 1.5 $\mathrm{T}$ is passed perpendicularly through the thickness of the conductor. From these data, find (a) the drift velocity of the charge carriers and (b) the number density of charge carriers. (c) Show on a diagram the polarity of the Hall potential difference with assumed current and magnetic field directions, assuming also that the charge carriers
are electrons.

Shital Rijal

Numerade Educator

Problem 83

A particle of mass 6.0 $\mathrm{g}$ moves at 4.0 $\mathrm{km} / \mathrm{s}$ in an $x y$ plane,
in a region with a uniform magnetic field given by 5.0$\hat{\mathrm{i}} \mathrm{mT}$ . At one
instant, when the particle's velocity is directed $37^{\circ}$ counterclockwise from the positive direction of the $x$ axis, the magnetic force on
the particle is 0.48 $\mathrm{k}$ N. What is the particle's charge?

Jack Hou

Numerade Educator

Problem 84

A wire lying along an $x$ axis from $x=0$ to $x=1.00 \mathrm{m}$ carries a current of 3.00 $\mathrm{A}$ in the positive $x$ direction. The wire is immersed in a nonuniform magnetic field that is given by $\vec{B}=$ $\left(4.00 \mathrm{T} / \mathrm{m}^{2}\right) x^{2} \mathrm{x}^{2} \mathrm{i}-\left(0.600 \mathrm{T} / \mathrm{m}^{2} \mathrm{x}^{2} \mathrm{j}$ . In unit-vector notation, what is \right. the magnetic force on the wire?

Shital Rijal

Numerade Educator

Problem 85

At one instant, $\vec{v}=(-2.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}}-6.00 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$ is the velocity of a proton in a uniform magnetic field $\vec{B}=(2.00 \hat{\mathrm{i}}-$ $4.00 \mathrm{j}+8.00 \mathrm{k} ) \mathrm{mT} .$ At that instant, what are (a) the magnetic force $\vec{F}$ acting on the proton, in unit-vector notation, (b) the angle between $\vec{v}$ and $\vec{F},$ and $(\mathrm{c})$ the angle between $\vec{v}$ and $\vec{B} ?$

Jack Hou

Numerade Educator

Problem 86

An electron has velocity $\vec{v}=(32 \hat{\mathrm{i}}+40 \hat{\mathrm{j}}) \mathrm{km} / \mathrm{s}$ as it enters a uniform magnetic field $\vec{B}=60 \hat{\mathrm{i}} \mu \mathrm{T}$ . What are (a) the radius of the
helical path taken by the electron and (b) the pitch of that path?
(c) To an obscrver looking into the magnctic ficld region from the
entrance point of the electron, does the electron spiral clockwise or
counterclockwise as it moves?

Shital Rijal

Numerade Educator

Problem 87

Figure $28-56$ shows a homopolar generator. which has a solid conducting disk as rotor and which is rotated by a motor (not
shown). Conducting brushes connect this emf device to a circuit
through which the device drives current. The device can produce a
greater emf than wire loop rotors because they can spin at a much higher angular speed without rupturing. The disk has radius $R=$ 0.250 $\mathrm{m}$ and rotation frequency $f=4000 \mathrm{Hz}$ and the device is in a uniform magnetic field of magnitude $B=60.0 \mathrm{mT}$ that is perpendicular to the disk. As the disk is rotated, conduction electrons along the conducting path (dashed line) are forced to move through the
magnetic ficld.(a) For the indicated rotation, is the magnetic force
on those electrons up or down in the figure? (b) Is the magnitude
of that force greater at the rim or near the center of the disk?
(c) What is the work per unit charge done by that force in moving
charge along the radial line, between the rim and the center?
(d) What, then, is the emf of the device? (c) If the current is $50.0 \mathrm{A},$
what is the power at which electrical energy is being produced?

Sachin Rao

Numerade Educator

Problem 88

In Fig. $28-57$ , the two ends of a U-shaped wire of mass $m=$ 10.0 $\mathrm{g}$ and length $L=20.0 \mathrm{cm}$ are immersed in mercury (which is a
conductor). The wire is in a uniform field of magnitude $B=0.100 \mathrm{T}$ . A switch (unshown) is rapidly closed and then reopened, sending a
pulse of current through the wire, which causes the wire to jump up-
ward. If jump height $h=3.00 \mathrm{m}$ , how much charge was in the pulse?
Assume that the duration of the pulse is much less than the time
of flight. Consider the definition of impulse (Eq. $9-30$ ) and its
relationship with momentum (Eq. $9-31 )$ . Also consider the
relationship between charge and current (Eq. $26-2 )$ .

Shital Rijal

Numerade Educator

Problem 89

In Fig. $28-58,$ an electron of mass $m,$ charge $-e,$ and low (ncgligible) speed enters the region between two plates of potential difference $V$ and plate separation $d,$ initially headed directly toward the top plate. A uniform magnetic field of magnitude $B$ is normal to the plane of the figure. Find the minimum value of $B$ such that the electron will not strike the top plate.

Jack Hou

Numerade Educator

Problem 90

A particle of charge $q$ moves in a circle of radius $r$ with spced $v$ Treating the circular path as a current loop with an average current,
find the maximum torque exerted on the loop by a uniform field of
magnitude $B$ .

Shital Rijal

Numerade Educator

Problem 91

In a Hall-effect experiment, express the number density of charge carriers in terms of the Hall-effect electric field magnitude
$E,$ the current density magnitude $J,$ and the magnetic field magnitude $B .$

Sachin Rao

Numerade Educator

Problem 92

An electron that is moving through a uniform magnetic field has velocity $\vec{v}=(40 \mathrm{km} / \mathrm{s}) \hat{\mathrm{i}}+(35 \mathrm{km} / \mathrm{s}) \hat{\mathrm{j}}$ when it experiences
a force $\vec{F}=-(4.2 \mathrm{fN}) \hat{\mathrm{i}}+(4.8 \mathrm{fN}) \hat{\mathrm{j}}$ due to the magnetic field. If $B_{x}=0,$ calculate the magnetic field $\vec{B}$ .

Shital Rijal

Numerade Educator