PHY131 HOMEWORK PROBLEMS 2011 JULY 21 - JULY 27
UNIT 3
1. An electric field of 1.50 kV/m and a magnetic field of
0.400 T act on a moving electron to produce no net force.
(a) Calculate the minimum speed v of the electron.
(b) Draw the vectors E, B, and v (E, B, and v are vector
symbols).
2. An alpha particle (q = +2e, m = 4.00 u) travels in a
circular path of radius 4.50 cm in a uniform magnetic
field with B = 1.20 T. Calculate (a) its speed, (b) its
period of revolution, (c) its kinetic energy in electron-
volts, and (d) the potential difference through which it
would have to be accelerated to achieve this energy.
3. An electron has an initial velocity of
(12.0 km/s)j + (15.0 km/s)k and a constant acceleration
of (2.00 Tm/s^2)i in a region in which uniform electric
and magnetic fields are present. If B = (400 microT)i,
find the electric field E (E and B are vectors).
4. A long, straight wire with linear mass density of
50 g/m is suspended by many fine threads; the
hanging part of the wire is perfectly horizontal.
A long section of the wire is within a constant
and uniform magnetic field; the rest of the wire
is in an ignorably small magnetic field. A 10 A
current in the wire experiences a horizontal
magnetic force; as a result, the wire is deflected
until the threads make a 10 degree angle with the
vertical. The length of the wire within the
magnetic field is not given. (a) Make a drawing
of the situation, showing the directions of the
current and the magnetic field. (b) Make a
free-body diagram for the current-carrying wire.
(c) What is the strength of the magnetic field?
5. 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 A in
the clockwise direction. The clock is located where
there is a constant, uniform external magnetic field
of magnitude 70 mT (but the clock still keeps perfect
time). At exactly 1:00 p.m., 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.
6. The figure below shows a wire of length L carrying
current fed by other wires that are not shown (the
current direction is to the right). Point A lies
on the perpendicular bisector, a distance y from the
wire. Show how to use the Biot-Savart Law to
demonstrate that the magnetic field at A due to the
straight wire alone has magnitude
(mu_0(I)L)/((2pi)y(sqrt(L^2+4y^2))).
What is the field direction?
_ . B . A
y
- ------------------------------ I -->
|<-- L -->|
7. Point B in the figure above lies a distance y
perpendicular to the end of the wire. Show how to
use the Biot-Savart Law to demonstrate that the
magnetic field at B due to the straight wire alone
has magnitude
(mu_0(I)L)/((4pi)y(sqrt(L^2+y^2))).
What is the field direction?
8. A square loop of wire of edge a carries a current i.
Show that the value of B at the center of the square
is given by
B =(2sqrt2)(mu_0)(i)/((pi)a)
Hint: You may use the result of Unit 3: 6.
9. A circular loop of radius 12 cm carries a current of
15 A. A flat coil of radius 0.82 cm, having 50 turns
and a current of 1.3 A, is concentric with the loop.
(a) What magnetic field strength B does the loop
produce at its center? (b) What is the magnitude of
the torque acting on the coil? Assume that the planes
of the loop and coil are perpendicular and that the
magnetic field due to the loop is essentially uniform
throughout the volume occupied by the coil.
10. A 5.0 cm by 10 cm rectangular wire loop is carrying
a current of 500 mA; the plane of the loop is
purely horizontal, with the 5.0 cm sides at the
north and south ends of the loop. A long straight
wire is carrying a current of 20 A due north; the
long straight wire is in the same plane as the loop
and is 2.0 cm to the west of the westernmost side of
the loop. Find the net magnetic force on the loop
by the current in the wire.
11. A conducting slab extends infinitely in the x and y
directions and has thickness h in the z direction.
It carries a uniform current density of magnitude J
in the positive x direction. Show how to use Ampere's
Law to find the magnetic field strength first (a) inside
and then (b) outside the slab, as functions of the
distance |z| from the central plane of the slab.
12. A solid conducting wire of radius R runs parallel
to the z axis and carries a current density of
magnitude J(r) = J_0(1-(r/R)) in the positive z
direction, where J_0 is a constant and r is the
radial distance from the wire axis.
Find expressions for (a) the total current in the
wire, (b) the magnetic field strength for r > R,
and (c) the magnetic field strength for r < R.
13. A long, hollow cylindrical conductor (inner radius
= 2.0 mm, outer radius = 4.0 mm) carries a current
of 24 A distributed uniformly across its cross
section. A long thin wire that is coaxial with
the cylinder carries a current of 24 A in the
opposite direction. What are the magnitudes of
the magnetic fields (a) 1.0 mm, (b) 3.0 mm, and
(c) 5.0 mm from the central axis of the wire and
cylinder?
14. A long solenoid with n turns per unit length carries
a current I. The current returns to its driving
battery along a wire of radius R that passes through
the solenoid, along its axis. Find expressions for
(a) the magnetic field strength at the surface of
the wire, and (b) the angle the field at the wire
surface makes with the solenoid axis.
15. Consider a toroidal solenoid with a square cross
section, each side of which has length 3 cm.
The inner wall of the torus forms a cylinder of
radius 12 cm (figure 28.25(a) on page 976 could
roughly be a toroid of this description -- for a
cutaway view see figure 30.8 on page 1037). The
torus is wound evenly with 200 turns of 0.3 mm-
DIAMETER copper wire. The wire is connected to a
3.0 V battery with negligible internal resistance.
(a) Calculate the largest and smallest magnetic
field across the cross section of the toroid.
(b) Calculate the absolute value of the magnetic
flux through one turn of the toroidal solenoid.
(c) Do you need to cool the solenoid? (Calculate
the heat created per second when current is flowing.)
16. The figure for this problem is Y&F figure 29.36 on
page 1022. The distance between the rails is 10 cm,
the magnetic field strength is 0.50 T, the resistance
is 4.0 Ohms, and the bar is being pulled to the right
at a constant speed of 2.0 m/s. (a) Show carefully
how to use the principle of motional EMF to find the
absolute value of the EMF induced between the points
a and b. Find (b) the current in the resistor and
(c) the magnetic force on the bar. (d) Calculate
directly the rate of energy dissipation in the
resistor. (e) Calculate directly the rate at which
work is being done by the agent pulling the bar.
17. An elastic conducting material is stretched into a
circular loop of 12.0 cm radius. It is placed with
its plane perpendicular to a uniform 0.800 T
magnetic field. When released, the radius of the
loop starts to shrink at an instantaneous rate of
75.0 cm/s. Assume that you are facing the loop and
that the magnetic field points into the loop. In
your work, use a cylindrical coordinate system
(r,theta,z), with the origin at the center of the
loop and the magnetic field in the negative
z-direction. Your task is to find the induced EMF
in the shrinking loop at the instant described above,
both using (a) motional EMF and (b) by using Faraday's
Law. In showing your work, use r(t) for the
time-dependent radius of the shrinking loop. (When
doing part (b), remember to use the chain rule
properly).
18. Consider a rectangular loop which lies in the xy
plane with corners at (+a,h/2), (+b,h/2), (+b,-h/2),
and (+a,-h/2) with b>a. The loop lies in a magnetic
field which is in the +z direction; for parts (a)-(e)
of this problem, the magnetic field is uniform in
space and has strength B. Make a drawing of the loop
and field with the field pointing out of the page;
give all directions for current in reference to this
drawing. (a) Find the magnetic flux through the loop.
What is the size of the induced EMF, and the direction
of the induced current, while: (b) the magnetic field
decreases steadily from B to zero during time interval
deltaT; (c) the magnetic field increases steadily from
B to 5*B during time interval deltaT; (d) the loop is
pulled in the +x direction with speed v; (e) the loop
is pulled in the +z direction with speed v. Instead
of a uniform field, the field is being produced by a
long straight current I on the y-axis, so that the
field strength is mu_0*I/(2pi*|x|). (f) What is the
magnetic flux through the loop? (g) As a function of
time t, what is the size of the induced EMF, and the
direction of the induced current, while the loop is
pulled in the +x direction with speed v?
19. At a certain place, Earth's magnetic field has
magnitude B = 0.590 gauss and is inclined downward at
an angle of 70.0 degrees to the horizontal. A flat
horizontal circular coil with a radius of 10.0 cm has
1000 turns and a total resistance of 85.0 Ohms. It
is connected to a meter with 140 Ohm resistance. The
coil is flipped through a half-revolution about a
diameter, so that it is again horizontal. (a) What
is the absolute value of the change in magnetic flux
through the coil as a result of the flip? (b) How
much charge flows through the meter during the flip?
(HINT: Use Faraday's Law; you aren't given enough
information to find the induced current, and you don't
know the amount of time required for the flip; however,
you can solve for the product of current -- assumed
constant -- and time.)
20. The figure shows a graph of the output voltage
versus time for an AC generator, and also the
current versus time for an attached resistor.
(a) Assuming the generator consists of a single
coil of many turns being rotated in a uniform
magnetic field, how many times per second is
the coil being rotated? (b) If the coil is
rectangular with sides 7.5 cm and 13 cm, and
if the magnetic field strength is 14 mT, at
least how many turns must the coil contain?
(c) What is the resistance of the circuit
containing the generator and the resistor?
(d) What is the maximum possible flux through
the generator coil as it is spinning? (e) For
the times shown on the graph, at which times is
the flux through the coil a maximum? Here is
a pdf version of the figure.
21. A stiff wire bent into a semicircle of radius 0.700 m
is rotated with frequency 80 Hz in a uniform magnetic
field of 0.400 T, as suggested in the figure
(pdf version). By means of the two pivots on which it
rotates, the stiff wire is connected to a long wire
of resistance 3.4 Ohms; the stiff wire combined with
the long wire forms a conducting loop. When the stiff
wire is in the position shown in the figure, the
absolute value of the magnetic flux through the
non-circular part of the loop is 1.20 Wb. (a) Find
the absolute value of the total flux through the loop
when the stiff wire is in the position shown in the
figure. (b) When the stiff wire has been rotated
first by 90 degrees, and then by 180 degrees from the
position shown in the figure, what is the absolute
value of the total flux through the loop? (c) With
respect to the position shown in the figure, at what
point or points during the rotation of the stiff wire
is the absolute value of the flux through the loop a
maximum? (d) With respect to the position shown in
the figure, at what point or points during the
rotation of the stiff wire is the absolute value of
the flux changing with time at a maximum rate?
(e) What is the maximum rate at which the flux
through the loop is changing with time?
(f) What is the amplitude of the varying EMF induced
in the loop? (You must clearly show your logic to
get full credit.)
22. A rectangular loop of length L and width w is
located a distance a from a long, straight wire.
What is the mutual inductance of this arrangement?
The picture for this problem is Y&F Figure 29.27 on
page 1021, with b-a in that figure equal to w in
this problem statement.
23. A coil with 150 turns, a radius of 5.0 cm, and a
resistance of 12 Ohms surrounds a solenoid with
200 turns/cm and a radius of 4.4 cm; see the figure.
The current in the solenoid changes at a constant
rate from 0 to 2.1 A in 0.10 s. Ignore any helicity,
or pitch, in the turns; i.e. you may assume that each
turn is circular and in a single plane. (a) Calculate
the absolute value of the induced EMF in each turn of
the surrounding coil. (b) Assuming that each turn of
the surrounding coil is a circle centered on the axis
of the solenoid, what is the induced electric field
strength within the wire of the surrrounding coil?
(c) Calculate the absolute value of the induced
current in the 150-turn coil. (d) What is the
direction of the induced current as it passes through
the resistor of the 150-turn coil? (e) What is the
mutual inductance of this two-coil geometry? (HINT:
Part (e) is easily calculated from your answer to
part (a).) Here is a pdf version of the figure.
Young and Freedman - 12th Ed - CHAPTER 27
76. Please add the following part:
(d) For each of the above magnetic fields, what
will be the maximum kinetic energy of the loop if
it is released from rest from the position shown
in Figure 27.66 (you may consider the loop to be
pivoted about the y-axis and you may ignore all
friction)?
Young and Freedman - 12th Ed - CHAPTER 28
69. To get credit for this problem, you must show
carefully how to use the Biot-Savart Law to arrive
at the answer.
Young and Freedman - 12th Ed - CHAPTER 29
28. Please add the following parts:
(c) What is the magnitude of the induced electric
field at a point just outside the center of the
solenoid (lengthwise) and 4.00 cm from the axis of
the solenoid? (d) Using Figure 29.54 (p. 1027),
and applying that figure to the situation of this
problem, what would be the direction of the induced
electric field in part (a) of this problem?
63. The rod in this problem is intended to be the moving
part in a simple DC generator. Here is a figure
of such a generator; the connecting wires are not
shown. This problem is most straightforwardly done
by integrating vcrossBdotdl over the length of the
moving rod (note that v in this integral will not
be constant). Here is a pdf version of the figure.
77. Note that the metal rails in the figure for this
problem are drawn in gray; the thin black lines
represent a non-metal frame. For part (b), remember
that the terminal speed is the maximum speed, i.e.
the speed at which the acceleration becomes zero.
Young and Freedman - 12th Ed - CHAPTER 30
73. There is a misprint in this problem in my version of
the text. The power series expansion for
ln(1+z) = z - z^2/2 + higher order terms.
Y&F mistakenly have a plus sign between the first
and second terms of the expansion.