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 6.00x106 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,
~ 1 , pointing
and then enters a selector chamber, S1 , in which there is an adjustable magnetic field B
~ pointing from positive to negative plate. Only
out of the page and a deflecting electric field E,
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
~ e, x, m, B2 ≡ |B
~ 2 |, and ∆V . (i) What magnetic field B1 in the selector
below in terms of E ≡ |E|,
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
Section 29.1
1.
Magnetic Fields and Forces
Determine the initial direction of the deflection of
charged particles as they enter the magnetic fields as
shown in Figure P29.1.
(a)
+
(c)
(b)
Bin
×
×
×
×
×
×
×
×
×
×
×
×
Bright
Bup
×
×
×
×
–
(d)
Bat 45°
+
45°
+
(a) downward
eastward?
3. An electron m
lar to a magn
in the negati
magnetic field
4. A proton trave
of 37.0° with
in the " y dir
magnetic forc
5. A proton mov
B at 1.00 ! 1
2.00 ! 1013 m
the " z direct
of the field.
6. An electron i
then enters a
(a) the maxi
magnetic forc
Figure P29.1
7. A proton mo
field of 1.70 T
2. Consider an electron near the Earth’s equator. In which di8.20 ! 10#13
electron passes rection
through the
sameitpotential
φ is small,
they all require the
does
tend todifference
deflectand
if the
its angle
velocity
is directed
velocity and th
Figure 1: Problem 1.
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.
~ = Bı̂ in the +x-direction. The magnetic moment vector µ is
There is a uniform magnetic field B
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.
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, q1 and q2 . 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.
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
!
(a)
Are the
charges
of positive
these particles
positive
negative?
Explain
negative
x axis
in the
x direction,
eachorwith
enter the magnetic
v , andyour
Figure
2: Problem
2. velocity
reasoning.
field at the origin O. The two particles have the same mass m , but have different
charges, q1 and q2 . When
!
! magnetic field, their trajectories both curve in the same
!in the
Solution:
Because
charges of these
POSITIVE.
FB but
= qvdescribe
! B , thesemi-circles
direction (see
sketch),
with particles
different are
radii.
The radius of the
semi-circle traced out by particle 2 is exactly twice as big as the radius of the semi-circle
(b)
What
ratio q1.2 / q1 ?
traced
out isbythe
particle
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
v 2 / R . Therefore applying Newton’s Second Law yields
mv 2
qvB =
.
R
We can solve this for the radius of the circular orbit
R=
mv
qB
Therefore
charged
ratio particles positive or negative? Explain your
(a) Are thethe
charges
of these
Figure 3: Problem 3.
q2 ! mv " ! mv " R1
reasoning.
.
=#
$ #
$=
R
B
R
B
q
R
%
2
&
%
1
&
1
2
!
! !
Solution: Because FB = qv ! B , the charges of these particles are POSITIVE.
(b) What is the ratio q2 / q1 ?
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
!
B 2 , 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 ! B 2 , and !V .
58. Review Problem. A wire having a linear mass density of
1.00 g/cm is placed on a horizontal surface that has a
coefficient of kinetic friction of 0.200. The wire carries a
current of 1.50 A toward the east and slides horizontally to
the north. What are the magnitude and direction of the
smallest magnetic field that enables the wire to move in
this fashion?
long. The springs
wire and the circ
a magnetic field i
springs stretch an
tude of the magn
62. A hand-held ele
Model the motor
ing electric curre
59. Electrons in a beam are accelerated from rest through a
produced by an e
potential difference !V. The beam enters an experimental
sider only one in
chamber through a small hole. As shown in Figure P29.59,
will consider mot
the electron velocity vectors lie within a narrow cone of
because the mag
half angle " oriented along the beam axis. We wish to use
described in Sec
a uniform magnetic field directed parallel to the axis to
mates of the ma
focus the beam, so that all of the electrons can pass
!
Figure
4:
Problem
4.
current in it, its ar
through a small exit port on the opposite side of the
a) What magnetic
field
in
the
selector
chamber
is
needed
to
insure tha
B
that they are relat
chamber after 1they travel the length d of the chamber.
the input power t
What
is the through?
required magnitude of the magnetic field?
particle travels
straight
and the useful ou
Hint: Because every electron passes through the same
potential difference and the angle " is small, they all
63. A nonconductin
Solution: We first find
an expression
for thetospeed
ofaxial
the distance
particled. after it20.0
is accelerate
require
the same time interval
travel the
cm. A flat co
around
the potential difference !V , in terms of m , e , and !V . The change intightly
kinetic
energi
sphere. As shown
o
!K = (1/ 2)mv 2 . The change in potential energyd is !U = "e!V From conservation
an inclined plane
an angle ' with th
energy, !K = "!U , we have that
the inclined plan
(1/ 2)mvφ 2 = e!V .
vertically upward
Exit
current in the co
So the speed is
port
rium on the incli
2e!V
depend on the va
v
=
Entrance
∆V
m
port
Figure P29.59
Figure 5: Problem 5.
Inside the selector the force on the charge is given by
60. Review Problem. A proton is at rest at the plane vertical
! containing
! !a uniform
!
boundary of a region
vertical magFe particle
= e(E +moving
v ! Bhorizontally
netic field B. An alpha
makes
1) .
a head-on elastic collision with the proton. Immediately
after the collision, both particles enter the magnetic field,
moving perpendicular to the direction of the field. The
em 5: Magnetic Field of a Ring of Current
cular ring of radius R lying in the xy plane carries a steady current I , as sho
gure below.
Magnetic Fields
Problem 6.
is the magnetic field at a point PFigure
on 6:the
axis of the loop, at a distance z from
r?
gnetic field at point P due to the following current distribution.
ion:
) We shall use the Biot-Savart law to find the magnetic field on the symmetr
Figure 7: Problem 7.
r
ue to the straight wire segments are zero at P because d s and
ource point:
amber.
field?
same
hey all
ance d.
Exit
port
vertical
al magmakes
diately
c field,
d. The
of the
rticle is
that of
he top
eft and
d. The
.00 cm
that they are related according to Equation 29.11. Note that
the 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 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 Figure P29.63, the sphere is placed on
an inclined plane that slopes downward to the left, making
an angle ' 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 '.
B
θ
Figure P29.63
Figure 8: Problem 10.
64. A metal rod having a mass per unit length ( carries a
current I. The rod hangs from two vertical wires in a
uniform vertical magnetic field as shown in Figure P29.64.
The wires make an angle ' with the vertical when in equilibrium. Determine the magnitude of the magnetic field.
θ