Embed Size (px)
Citation preview
Prof. Anchordoqui
Problems set # 7 Physics 169 March 31, 2015
1. (i) Determine the initial direction of the deflection of charged particles as they enter the
magnetic fields as shown in Fig. 1. (ii) At the Equator near Earths surface, the magnetic field is
approximately 50.0 µT northward and the electric field is about 100 N/C downward in fair weather.
Find the gravitational, electric, and magnetic forces on an electron with an instantaneous velocity
of 6.00 × 106 m/s directed to the east in this environment. (iii) Consider an electron near the
Earths equator. In which direction does it tend to deflect if its velocity is directed: (a) downward,
(b) northward, (c) westward, or (d) south-eastward?
2. A particle of charge −e is moving with an initial velocity v when it enters midway between
two plates where there exists a uniform magnetic field pointing into the page, as shown in Fig. 2.
You may ignore effects of the gravitational force. (i) Is the trajectory of the particle deflected
upward or downward? (ii) What is the magnitude of the velocity of the particle if it just strikes
the end of the plate?
3. The entire x − y plane to the right of the origin O is filled with a uniform magnetic field
of magnitude B pointing out of the page, as shown in Fig. 3. Two charged particles travel along
the negative x axis in the positive x direction, each with velocity ~v, and enter the magnetic field
at the origin O. The two particles have the same mass m, but have different charges, q1 and q2.
When propagate thorugh the magnetic field, their trajectories both curve in the same direction (see
sketch in Fig. 3), but describe semi-circles with different radii. The radius of the semi-circle traced
out by particle 2 is exactly twice as big as the radius of the semi-circle traced out by particle 1.
(i) Are the charges of these particles positive or negative? Explain your reasoning. (ii) What is
the ratio q2/q1?
4. Shown in Fig. 4 are the essentials of a commercial mass spectrometer. This device is used to
measure the composition of gas samples, by measuring the abundance of species of different masses.
An ion of mass m and charge q = +e is produced in source S, a chamber in which a gas discharge is
taking place. The initially stationary ion leaves S, is accelerated by a potential difference ∆V > 0,
and then enters a selector chamber, S1, in which there is an adjustable magnetic field ~B1, pointing
out of the page and a deflecting electric field ~E, pointing from positive to negative plate. Only
particles of a uniform velocity ~v leave the selector. The emerging particles at S2, enter a second
magnetic field B2, also pointing out of the page. The particle then moves in a semicircle, striking
an electronic sensor at a distance x from the entry slit. Express your answers to the questions
below in terms of E ≡ | ~E|, e, x, m, B2 ≡ | ~B2|, and ∆V . (i) What magnetic field B1 in the selector
chamber is needed to insure that the particle travels straight through? (ii) Find an expression for
the mass of the particle after it has hit the electronic sensor at a distance x from the entry slit.
5. Electrons in a beam are accelerated from rest through a potential difference V . The beam
enters an experimental chamber through a small hole. As shown in Fig. 5, the electron velocity
vectors lie within a narrow cone of half angle φ oriented along the beam axis. We wish to use a
uniform magnetic field directed parallel to the axis to focus the beam, so that all of the electrons
can pass through a small exit port on the opposite side of the chamber after they travel the length
d of the chamber. What is the required magnitude of the magnetic field? [Hint: Because every
918 C H A P T E R 2 9 • Magnetic Fields
10. A current-carrying conductor experiences no magneticforce when placed in a certain manner in a uniform mag-netic field. Explain.
Is it possible to orient a current loop in a uniformmagnetic field such that the loop does not tend to rotate?Explain.
12. Explain why it is not possible to determine thecharge and the mass of a charged particle separatelyby measuring accelerations produced by electric andmagnetic forces on the particle.
How can a current loop be used to determine thepresence of a magnetic field in a given region of space?
14. Charged particles from outer space, called cosmic rays,strike the Earth more frequently near the poles than nearthe equator. Why?
15. What is the net force on a compass needle in a uniformmagnetic field?
16. What type of magnetic field is required to exert a resultantforce on a magnetic dipole? What is the direction of theresultant force?
17. A proton moving horizontally enters a uniform magneticfield perpendicular to the proton’s velocity, as shown inFigure Q29.17. Describe the subsequent motion of theproton. How would an electron behave under the samecircumstances?
13.
11.
18. In the cyclotron, why do particles having different speedstake the same amount of time to complete a one-half circletrip around one dee?
19. The bubble chamber is a device used for observing tracks ofparticles that pass through the chamber, which is immersedin a magnetic field. If some of the tracks are spirals andothers are straight lines, what can you say about theparticles?
20. Can a constant magnetic field set into motion an electroninitially at rest? Explain your answer.
21. You are designing a magnetic probe that uses the Halleffect to measure magnetic fields. Assume that you arerestricted to using a given material and that you havealready made the probe as thin as possible. What, ifanything, can be done to increase the Hall voltageproduced for a given magnitude of magnetic field?
v+
! ! !
! ! !
! ! !
! ! !
! !!
Figure Q29.17
Section 29.1 Magnetic Fields and Forces
Determine the initial direction of the deflection ofcharged particles as they enter the magnetic fields asshown in Figure P29.1.
1.
(a) downward, (b) northward, (c) westward, or (d) south-eastward?
3. An electron moving along the positive x axis perpendicu-lar to a magnetic field experiences a magnetic deflectionin the negative y direction. What is the direction of themagnetic field?
4. A proton travels with a speed of 3.00 ! 106 m/s at an angleof 37.0° with the direction of a magnetic field of 0.300 Tin the " y direction. What are (a) the magnitude of themagnetic force on the proton and (b) its acceleration?
A proton moves perpendicular to a uniform magnetic fieldB at 1.00 ! 107 m/s and experiences an acceleration of2.00 ! 1013 m/s2 in the " x direction when its velocity is inthe " z direction. Determine the magnitude and directionof the field.
6. An electron is accelerated through 2 400 V from rest andthen enters a uniform 1.70-T magnetic field. What are(a) the maximum and (b) the minimum values of themagnetic force this charge can experience?
A proton moving at 4.00 ! 106 m/s through a magneticfield of 1.70 T experiences a magnetic force of magnitude8.20 ! 10#13 N. What is the angle between the proton’svelocity and the field?
7.
5.
1, 2, 3 = straightforward, intermediate, challenging = full solution available in the Student Solutions Manual and Study Guide
= coached solution with hints available at http://www.pse6.com = computer useful in solving problem
= paired numerical and symbolic problems
P R O B L E M S
(a)
+
+
–
+
(c)
(b)
(d)
!!!!!!!!!!!!!!!!
45°
Bin
Bright
Bup
Bat 45°
Figure P29.1
2. Consider an electron near the Earth’s equator. In which di-rection does it tend to deflect if its velocity is directed
Figure 1: Problem 1.
electron passes through the same potential difference and the angle φ is small, they all require the
same time interval to travel the axial distance d.
6. A circular ring of radius R lying in the xy plane carries a steady current I, as shown in the
Fig. 6. What is the magnetic field at a point P on the axis of the loop, at a distance z from the
center?
7. Find the magnetic field at point P due to the current distribution shown in Fig. 7.
8. A thin uniform ring of radius R and mass M carrying a charge +Q rotates about its axis
with constant angular speed ω. Find the ratio of the magnitudes of its magnetic dipole moment to
its angular momentum. (This is called the gyromagnetic ratio.)
9. A wire ring lying in the xy-plane with its center at the origin carries a counterclockwise I.
There is a uniform magnetic field ~B = Bı in the +x-direction. The magnetic moment vector µ is
perpendicular to the plane of the loop and has magnitude µ = IA and the direction is given by
right-hand-rule with respect to the direction of the current. What is the torque on the loop?
10. A nonconducting sphere has mass 80.0 g and radius 20.0 cm. A flat compact coil of wire
with 5 turns is wrapped tightly around it, with each turn concentric with the sphere. As shown in
Fig. 8, the sphere is placed on an inclined plane that slopes downward to the left, making an angle
θ with the horizontal, so that the coil is parallel to the inclined plane. A uniform magnetic field of
0.350 T vertically upward exists in the region of the sphere. What current in the coil will enable
the sphere to rest in equilibrium on the inclined plane? Show that the result does not depend on
the value of θ.
Problem 3: Particle Orbits in a Uniform Magnetic Field The entire x-y plane to the right of the origin O is filled with a uniform magnetic field of magnitude B pointing out of the page, as shown. Two charged particles travel along the negative x axis in the positive x direction, each with velocity v! , and enter the magnetic field at the origin O. The two particles have the same mass m , but have different charges, 1q and 2q . When in the magnetic field, their trajectories both curve in the same direction (see sketch), but describe semi-circles with different radii. The radius of the semi-circle traced out by particle 2 is exactly twice as big as the radius of the semi-circle traced out by particle 1. (a) Are the charges of these particles positive or negative? Explain your reasoning. Solution: Because BF qv B= !
! !! , the charges of these particles are POSITIVE. (b) What is the ratio 2 1/q q ? Solution: We first find an expression for the radius R of the semi-circle traced out by a particle with charge q in terms of q , v v! ! , B , and m . The magnitude of the force on the charged particle is qvB and the magnitude of the acceleration for the circular orbit is 2 /v R . Therefore applying Newton’s Second Law yields
2mvqvBR
= .
We can solve this for the radius of the circular orbit
mvRqB
=
Therefore the charged ratio
2 1
2 11 2
q Rmv mvR B R Bq R
! " ! "= =# $ # $% & % &
.
Figure 2: Problem 2.
Problem 3: Particle Orbits in a Uniform Magnetic Field The entire x-y plane to the right of the origin O is filled with a uniform magnetic field of magnitude B pointing out of the page, as shown. Two charged particles travel along the negative x axis in the positive x direction, each with velocity v! , and enter the magnetic field at the origin O. The two particles have the same mass m , but have different charges, 1q and 2q . When in the magnetic field, their trajectories both curve in the same direction (see sketch), but describe semi-circles with different radii. The radius of the semi-circle traced out by particle 2 is exactly twice as big as the radius of the semi-circle traced out by particle 1. (a) Are the charges of these particles positive or negative? Explain your reasoning. Solution: Because BF qv B= !
! !! , the charges of these particles are POSITIVE. (b) What is the ratio 2 1/q q ? Solution: We first find an expression for the radius R of the semi-circle traced out by a particle with charge q in terms of q , v v! ! , B , and m . The magnitude of the force on the charged particle is qvB and the magnitude of the acceleration for the circular orbit is 2 /v R . Therefore applying Newton’s Second Law yields
2mvqvBR
= .
We can solve this for the radius of the circular orbit
mvRqB
=
Therefore the charged ratio
2 1
2 11 2
q Rmv mvR B R Bq R
! " ! "= =# $ # $% & % &
.
Figure 3: Problem 3.
Problem 4 Mass Spectrometer Shown below are the essentials of a commercial mass spectrometer. This device is used to measure the composition of gas samples, by measuring the abundance of species of different masses. An ion of mass m and charge q = +e is produced in source S , a chamber in which a gas discharge is taking place. The initially stationary ion leaves S , is accelerated by a potential difference 0V! > , and then enters a selector chamber, S1 , in
which there is an adjustable magnetic field 1B!
, pointing out of the page and a deflecting
electric field E!
, pointing from positive to negative plate. Only particles of a uniform velocity v! leave the selector. The emerging particles at S2 , enter a second magnetic field
2B!
, also pointing out of the page. The particle then moves in a semicircle, striking an electronic sensor at a distance x from the entry slit. Express your answers to the questions below in terms of E ! E
!, e , x , m , 2 2B ! B
!, and V! .
a) What magnetic field 1B!
in the selector chamber is needed to insure that the particle travels straight through?
Solution: We first find an expression for the speed of the particle after it is accelerated by the potential difference !V , in terms of m , e , and !V . The change in kinetic energy is
2(1/ 2)K mv! = . The change in potential energy is U e V! = " ! From conservation of energy, K U! = "! , we have that
2(1/ 2)mv e V= ! . So the speed is
2e Vvm!=
Inside the selector the force on the charge is given by
1( )e e= + !F E v B! ! !! .
Figure 4: Problem 4.
Problems 923
58. Review Problem. A wire having a linear mass density of1.00 g/cm is placed on a horizontal surface that has acoefficient of kinetic friction of 0.200. The wire carries acurrent of 1.50 A toward the east and slides horizontally tothe north. What are the magnitude and direction of thesmallest magnetic field that enables the wire to move inthis fashion?
59. Electrons in a beam are accelerated from rest through apotential difference !V. The beam enters an experimentalchamber through a small hole. As shown in Figure P29.59,the electron velocity vectors lie within a narrow cone ofhalf angle " oriented along the beam axis. We wish to usea uniform magnetic field directed parallel to the axis tofocus the beam, so that all of the electrons can passthrough a small exit port on the opposite side of thechamber after they travel the length d of the chamber.What is the required magnitude of the magnetic field?Hint: Because every electron passes through the samepotential difference and the angle " is small, they allrequire the same time interval to travel the axial distance d.
long. The springs stretch 0.500 cm under the weight of thewire and the circuit has a total resistance of 12.0 #. Whena magnetic field is turned on, directed out of the page, thesprings stretch an additional 0.300 cm. What is the magni-tude of the magnetic field?
62. A hand-held electric mixer contains an electric motor.Model the motor as a single flat compact circular coil carry-ing electric current in a region where a magnetic field isproduced by an external permanent magnet. You need con-sider only one instant in the operation of the motor. (Wewill consider motors again in Chapter 31.) The coil movesbecause the magnetic field exerts torque on the coil, asdescribed in Section 29.3. Make order-of-magnitude esti-mates of the magnetic field, the torque on the coil, thecurrent in it, its area, and the number of turns in the coil, sothat they are related according to Equation 29.11. Note thatthe input power to the motor is electric, given by ! $ I !V,and the useful output power is mechanical, ! $ %&.
63. A nonconducting sphere has mass 80.0 g and radius20.0 cm. A flat compact coil of wire with 5 turns is wrappedtightly around it, with each turn concentric with thesphere. As shown in Figure P29.63, the sphere is placed onan inclined plane that slopes downward to the left, makingan angle ' with the horizontal, so that the coil is parallel tothe inclined plane. A uniform magnetic field of 0.350 Tvertically upward exists in the region of the sphere. Whatcurrent in the coil will enable the sphere to rest in equilib-rium on the inclined plane? Show that the result does notdepend on the value of '.
Exitport
d
Entranceport
!V
"
Figure P29.59
24.0 V
5.00 cm
Figure P29.61
#
B
Figure P29.63
#
B
I
g#
Figure P29.64
60. Review Problem. A proton is at rest at the plane verticalboundary of a region containing a uniform vertical mag-netic field B. An alpha particle moving horizontally makesa head-on elastic collision with the proton. Immediatelyafter the collision, both particles enter the magnetic field,moving perpendicular to the direction of the field. Theradius of the proton’s trajectory is R . Find the radius of thealpha particle’s trajectory. The mass of the alpha particle isfour times that of the proton, and its charge is twice that ofthe proton.
61. The circuit in Figure P29.61 consists of wires at the topand bottom and identical metal springs in the left andright sides. The upper portion of the circuit is fixed. Thewire at the bottom has a mass of 10.0 g and is 5.00 cm
64. A metal rod having a mass per unit length ( carries acurrent I. The rod hangs from two vertical wires in auniform vertical magnetic field as shown in Figure P29.64.The wires make an angle ' with the vertical when in equi-librium. Determine the magnitude of the magnetic field.
Figure 5: Problem 5.
Problem 5: Magnetic Field of a Ring of Current A circular ring of radius R lying in the xy plane carries a steady current I , as shown in the figure below.
What is the magnetic field at a point P on the axis of the loop, at a distance z from the center? Solution:
(a) We shall use the Biot-Savart law to find the magnetic field on the symmetry axis.
(1) Source point: In Cartesian coordinates, the differential current element located at
ˆ ˆ' (cos ' sin ' )R ! != +r i j!can be written as ˆ ˆ( '/ ') ' '( sin ' cos ' )Id I d d d IRd! ! ! ! != = " +s r i j! !
. (2) Field point: Since the field point P is on the axis of the loop at a distance z from the center, its position vector is given by
!r = zk . (3) Relative position vector
!r ! !r ' :
Figure 6: Problem 6.
Problem 6: Magnetic Fields Find the magnetic field at point P due to the following current distribution.
Solution: The fields due to the straight wire segments are zero at P because d sr and r are parallel or anti-parallel. For the field due to the arc segment, the magnitude of the magnetic field due to a differential current carrying element is given in this case by
d!B =
µ0I4!
d !s " rR2 =
µ0
4!IRd#(sin# i $ cos# j)" ($cos# i $ sin# j)
R2
= $µ0
4!I(sin2# + cos2# )d#
Rk = $
µ0
4!Id#R
k.
Therefore,
!B = !
µ0I4"R
d#0
" /2
$ k = !µ0I4"R
"2
%&'
()*
k = !µ0I8R
%&'
()*
k (into the plane of the figure).
Figure 7: Problem 7.
Problems 923
58. Review Problem. A wire having a linear mass density of1.00 g/cm is placed on a horizontal surface that has acoefficient of kinetic friction of 0.200. The wire carries acurrent of 1.50 A toward the east and slides horizontally tothe north. What are the magnitude and direction of thesmallest magnetic field that enables the wire to move inthis fashion?
59. Electrons in a beam are accelerated from rest through apotential difference !V. The beam enters an experimentalchamber through a small hole. As shown in Figure P29.59,the electron velocity vectors lie within a narrow cone ofhalf angle " oriented along the beam axis. We wish to usea uniform magnetic field directed parallel to the axis tofocus the beam, so that all of the electrons can passthrough a small exit port on the opposite side of thechamber after they travel the length d of the chamber.What is the required magnitude of the magnetic field?Hint: Because every electron passes through the samepotential difference and the angle " is small, they allrequire the same time interval to travel the axial distance d.
long. The springs stretch 0.500 cm under the weight of thewire and the circuit has a total resistance of 12.0 #. Whena magnetic field is turned on, directed out of the page, thesprings stretch an additional 0.300 cm. What is the magni-tude of the magnetic field?
62. A hand-held electric mixer contains an electric motor.Model the motor as a single flat compact circular coil carry-ing electric current in a region where a magnetic field isproduced by an external permanent magnet. You need con-sider only one instant in the operation of the motor. (Wewill consider motors again in Chapter 31.) The coil movesbecause the magnetic field exerts torque on the coil, asdescribed in Section 29.3. Make order-of-magnitude esti-mates of the magnetic field, the torque on the coil, thecurrent in it, its area, and the number of turns in the coil, sothat they are related according to Equation 29.11. Note thatthe input power to the motor is electric, given by ! $ I !V,and the useful output power is mechanical, ! $ %&.
63. A nonconducting sphere has mass 80.0 g and radius20.0 cm. A flat compact coil of wire with 5 turns is wrappedtightly around it, with each turn concentric with thesphere. As shown in Figure P29.63, the sphere is placed onan inclined plane that slopes downward to the left, makingan angle ' with the horizontal, so that the coil is parallel tothe inclined plane. A uniform magnetic field of 0.350 Tvertically upward exists in the region of the sphere. Whatcurrent in the coil will enable the sphere to rest in equilib-rium on the inclined plane? Show that the result does notdepend on the value of '.
Exitport
d
Entranceport
!V
"
Figure P29.59
24.0 V
5.00 cm
Figure P29.61
#
B
Figure P29.63
#
B
I
g#
Figure P29.64
60. Review Problem. A proton is at rest at the plane verticalboundary of a region containing a uniform vertical mag-netic field B. An alpha particle moving horizontally makesa head-on elastic collision with the proton. Immediatelyafter the collision, both particles enter the magnetic field,moving perpendicular to the direction of the field. Theradius of the proton’s trajectory is R . Find the radius of thealpha particle’s trajectory. The mass of the alpha particle isfour times that of the proton, and its charge is twice that ofthe proton.
61. The circuit in Figure P29.61 consists of wires at the topand bottom and identical metal springs in the left andright sides. The upper portion of the circuit is fixed. Thewire at the bottom has a mass of 10.0 g and is 5.00 cm
64. A metal rod having a mass per unit length ( carries acurrent I. The rod hangs from two vertical wires in auniform vertical magnetic field as shown in Figure P29.64.The wires make an angle ' with the vertical when in equi-librium. Determine the magnitude of the magnetic field.
Figure 8: Problem 10.
