ADDITIONS TO Y&F PROBLEMS AND PROBLEMS FROM OTHER TEXTS
JULY 6 - JULY 14
YOUNG - CHAPTER 21
60. The potassium chloride molecule (KCl) has a dipole
moment of 8.9 x 10^-30 Cm. (a) Assuming that this
dipole moment arises from two charges, each of
magnitude 1.6 x 10^-19 C, separated by distance d,
calculate d. (b) What is the maximum magnitude of
the torque that a uniform electric field with
magnitude 6.0 x 10^5 N/C can exert on a KCl
molecule? Sketch the relative orientations of the
electric dipole moment p (p is a vector) and the
electric field E (E is a vector) when the torque
is a maximum. (c) Suppose that the electric field
of part (b) points in the +y direction. If the
dipole swings from an initial orientation pointing
in a direction 20 degrees CCW away from the -x axis
to a final orientation pointing in a direction 20
degrees CW away from the -x axis, what is the
change in the dipole's electric potential energy?
78. (a) Suppose all the electrons in 20.0 g of carbon
atoms were located at the North Pole of the earth and
all the protons at the South Pole. What would be the
total force of attraction exerted on each group of
charges by the other? The atomic number of carbon
is 6, and the atomic mass of carbon is 12 g/mol.
(b) What would be the magnitude and direction of
the force exerted by the charges in part (a) on a
third charge that is equal to the charge at the
South Pole, and located at a point of the surface
of the earth at the equator? Draw a diagram showing
the locations of the charges and the forces on the
charge at the equator.
Young and Freedman - 12th Ed - CHAPTER 22
30. Do not use Gauss' Law for any part of this problem.
We have already found the result for the E field due
to a infinitely thin sheet of charge in two different
ways; you may simply use that result here. First
consider each of the four infinitely thin sheets of
charge separately. Then at each of the locations
A, B, and C, add the vector contributions to the
field from each of the four infinitely thin sheets.
Show your work carefully. GBA
54. Omit part (a) of this problem. To find the E field
strength within the slab, use a Gaussian pillbox
which is centered on the center of the slab, and which
has its parallel faces equidistant from the central
plane of the slab. With such a Gaussian pillbox, the
symmetry argument in part (a) is not required; instead
a simpler symmetry argument about the E field direction
within the slab will suffice. You must give your
symmetry argument clearly for this problem. GBA
YOUNG - CHAPTER 22
30. This is essentially problem 22:32 in Y&F 12e but
with a nonuniform electric field. The x-component
of the E field is given by (-5.00 N/(Cm))x. The
y-component of the E field is zero. The z-component
of the E field is given by (+3.00 N/(Cm))z.
The value of L is 0.30 m.
(a) Find the electric flux through each of the six
cube faces S_1 through S_6. (b) In the uniform
field case, the electric flux through the entire
cube would have been zero (what goes in must come
out); but, in this case you will find that the
electric flux through the entire cube is not zero.
Find the total electric charge inside the cube.
44. A small, INSULATING, spherical shell with inner
radius a and outer radius b is concentric with a
larger INSULATING spherical shell with inner
radius c and outer radius d (Figure 22.39 in
Y&F 12e). The inner shell has total charge +q
distributed uniformly over its volume, and the
outer shell has charge -q distributed uniformly
over its volume. (a) Calculate the charge
densities in the inner shell and the outer shell.
(b) Calculate the electric field (magnitude and
direction) in terms of q and the distance r from
the common center of the two shells for (i) r<a;
(ii) a<r<b; (iii) b<r<c; (iv) c<r<d; (v) r>d.
(c) Show your results in a graph of the radial
component of E (E is a vector) as a function of
the distance r.
45. A long coaxial cable consists of an inner cylindrical
conductor with radius a and an outer coaxial cylinder
(also conducting) with inner radius b and outer
radius c. The outer cylinder is mounted on insulating
supports and has no net charge. The inner cylinder
has a uniform positive charge per unit length lambda.
Calculate the electric field (a) at any point between
the cylinders, a distance r from the axis; (b) at
any point outside the outer cylinder. (c) Sketch the
magnitude of the electric field as a function of
the distance r from the axis of the cable, from
r=0 to r=2c. (d) Find the charge per unit length
on the inner surface and on the outer surface of
the outer cylinder.
48. A very long, solid cylinder with radius R has
positive charge uniformly distributed throughout it,
with charge per unit volume rho. (a) Derive the
expression for the electric field inside the volume
at a distance r from the axis of the cylinder in
terms of the charge density rho. (b) What is the
electric field at a point outside the volume in
terms of the charge per unit length labmda in the
cylinder? (c) Compare the answers to parts (a)
and (b) for r=R. (d) Graph the electric field
magnitude as a function of r from r=0 to r=3R.
(P prefixes) HRW Problem Supplement #1 - CHAPTER 22
26. Calculate the number of coulombs of positive charge
in 250 cm^3 of (neutral) water (about a glassful).
Young and Freedman - 12th Ed - CHAPTER 23
50. To get any credit for this problem, you MUST prove
any result that you take from problem 23.49. Please
do so by first using Gauss' Law to find the electric
field at all points in space, then integrating the
electric field over the appropriate displacement
to find the desired potential or potential difference.
74. In part (a) of this question, Y&F ask "What will a
voltmeter read ....". This is not a serious question
as asked. The shell is insulating and all voltmeters
measure current (as you will learn in section 26.3).
No current will flow in this case, so the voltmeter
will always read zero. So please change the wording
of this phrase to "What is the absolute value of the
potential difference between the following points?"
For parts (b) and (c), disregard the book and follow
the instructions here; there is an additional part (d)
here. (b) Sketch a graph of electric potential versus
radial position, setting V=0 at infinity, and labeling
the positions a, b, and c on your position axis.
(c) Redraw the graph for the case when the surface
charge is negative instead of positive. (d) Redraw
the graph for the case when the insulating material is
a solid ball (instead of a shell) of radius 60 cm, and
the +150 microC is distributed uniformly throughout
the volume of the insulating material rather than only
on the outer surface.
YOUNG - CHAPTER 23
48. Three small spheres with charge 2.00 microC are
arranged in a line, with sphere 2 in the middle.
Adjacent spheres are initially 8.0 cm apart. The
spheres have masses m1=20.0 g, m2=85.0 g, and
m3=20.0 g, and their radii are much smaller than
their separation. The three spheres are released
from rest. (a) What is the acceleration of sphere 1
just after it is released? (b) What is the speed of
each sphere when they are far apart?
WOLFSON - CHAPTER 23
16. A charge 3q is at the origin, and a charge -2q is on
the positive x axis at x = a. Where would you place
a third charge so it would experience no net electric
force?
22. Three identical charges +q and a fourth charge -q form
a square of side a. (a) Find the magnitude of the
electric force on a charge Q placed at the center of
the square. (b) Describe the direction of this force.
32. A 1.0 microC charge and a 2.0 microC charge are 10 cm
apart. (a) Find a point where the net electric field
is zero. (b) Sketch the net electric field lines
qualitatively.
34. A positive charge +2q lies on the x axis at x = -a,
and a charge -q lies at x = +a. (a) Find an expression
for the electric field as a function of x for points to
the right of the charge -q. (b) Taking q = 1.0 microC
and a = 1.0 m, plot the field as a function of position
for x = 5.0 m to x = 25 m.
50. A seimcircular loop of radius a carries positive charge
Q distributed uniformly over its length (the loop begins
at the point (0,a) and extends counterclockwise to the
point (0,-a)). Find the electric field at the center of
the loop. Show all the steps in your work.
54. A thin rod of length L carries charge Q distributed
uniformly over its length (the rod begins at the point
(-L/2,0) and extends rightward to the point (+L/2,0)).
(a) What is the line charge density on the rod?
(b) What must be the electric field direction on the
rod's perpendicular bisector (i.e. the y axis).
(c) Find an expression for the electric field at a
point P at the location (0,+y). (d) Show that your
result for (c) reduces to the field of a point charge
Q for y >> L.
56. How strong an electric field is needed to accelerate
electrons in an old-style TV tube from rest to one-tenth
the speed of light in a distance of 5.0 cm?
58. An oscilloscope display requires that a beam of
electrons moving at 8.2 Mm/s be deflected through
an angle of 22 degrees by a uniform electric field
that occupies a region 5.0 cm long. What should be
the field strength?
(P prefixes) HRW Problem Supplement #1 - CHAPTER 23
25. A semi-infinite nonconducting rod (i.e. infinite in one
direction only) has uniform positive linear charge density
lambda. Show that the electric field at point P makes an
angle of 45 degrees with the rod and that this result is
independent of the distance R. (HINT: Separately find
the parallel and perpendicular (to the rod) components
of the electric field at P, and then compare those
components.)
rod
starts
here
_ +++++++++++++++++++++++++ =====> very long
| |
|
R |
|
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- P <== this point is a distance R from the end
of the rod
37. In Millikan's experiment, an oil drop of radius
1.64 microns and density 0.851 g/cm^3 is suspended in
the experimental chamber when a downward-directed
electric field of 1.92 X 10^5 N/C is applied. Find
the charge on the drop, in terms of e.
55. A charge (uniform linear density = 9.0 nC/m) lies on a
string that is stretched along an x axis from x = 0 to
x = 3.0 m. Determine the magnitude of the electric
field at x = 4.0 m on the x axis.
(P prefixes) HRW Problem Supplement #1 - CHAPTER 24
24. A charge of uniform linear density 2.0 nC/m is distributed
along a long, thin, nonconducting rod. The rod is coaxial
with a long, hollow, conducting cylinder (inner radius =
5.0 cm, outer radius = 10 cm). The net charge on the
conductor is zero. (a) What is the magnitude of the
electric field 15 cm from the axis of the cylinder? What
is the surface charge density on (b) the inner surface and
(c) the outer surface of the conductor?
66. A metallic spherical shell of inner radius a1 and outer
radius a2 has charge Qa. Concentric with it is another
metallic spherical shell of inner radius b1 (where b1 > a2)
and outer radius b2; shell b has charge Qb. Both Qa and Qb
are positive. Find the electric field strength at points a
distance r from the common center, where (a) r < a1,
(b) a2 < r < b1, and (c) r > b2. (d) Determine how the
charges are distributed on the inner and outer surfaces of
the shells (give the surface charge densities). (e) Sketch
the electric field strength as a function of r over the
range r=0 to r=2*b2.
GIANCOLI - CHAPTER 23
11. A uniform electric field of strength 4200 N/C points
in the negative x direction. Point A is at location
(4.00 m, 1.00 m), B is at (4.00 m, 4,00 m), and C is
at (-3.00 m, 4.00 m). A charge of 28.0 nC is moving
within this uniform field. What work is done by the
electric force when the charge moves (a) from A to B,
(b) from B to C, (c) from C to A. (d) If point A is
assigned as the location where the electric potential
is zero, what is the electric potential at location
B and at location C? (e) Location D is 3.68 m from
location A in a direction of 210 degrees measured
counterclockwise from the positive x direction. What
work is done by the electric force when the 28.0 nC
charge moves from A to D, and what is the electric
potential at point D?
WOLFSON - CHAPTER 24
24. A thin spherical shell of radius 15 cm carries 4.8 microC,
distributed uniformly over its surface. At the center of
the shell is a point charge. (a) If the electric field at
the surface of the sphere is 750 kN/C and points outward,
what is the charge of the point charge? (b) What is the
field just inside the shell?
56. A point charge -q is at the center of a insulating thin
spherical shell carrying charge -(3/2)q. That shell, in
turn, is concentric with a larger shell (also insulating
and also thin) carrying charge +2q. (a) Draw a cross section
of this structure, and sketch the electric field lines
using the convention that 8 lines correspond to a charge
of magnitude q. (b) If the shells were metallic instead
of insulating, how would the field lines that you have
drawn be different, and what would be the total charge on
each of the inner and outer surfaces of each of the two
shells?
FISHBANE - CHAPTER 24
38. Two large, thin, metallic plates are placed parallel to
each other, separated by 15 cm. The top plate carries a
uniform charge density of 24 microC/m^2, while the bottom
plate carries a uniform charge density of -38 microC/m^2.
What is the electric field (magnitude and direction) (a)
halfway between the plates? (b) above the two plates?
(c) below the two plates? (d) What are the surface charge
densities on the top and bottom surfaces of both plates?
45. A metal sphere of radius a is surrounded by a metal shell
of inner radius b and outer radius R. The flux through a
spherical Gaussian surface located between a and b is
Q/epsilon0, and the flux through a spherical Gaussian
surface just outside radius R is 2Q/epsilon0. (a) What
are the total charges on the inner sphere and on the shell?
(b) Where are the charges located, and (c) what are the
charge densities?
Halliday and Resnick - 2nd Edition - CHAPTER 25
1. Water in an irrigation ditch of width w = 3.22 m and
depth d = 1.04 m flows with a speed of 0.207 m/s. The
mass flux of the flowing water through an imaginary
surface is the product of the water's density (1000 kg/m^3)
and its volume flux through that surface. Find the mass
flux through the following imaginary surfaces: (a) a
surface of area wd, entirely in the water, perpendicular
to the flow; (b) a surface with area 3wd/2, of which wd
is in the water, perpendicular to the flow; (c) a surface
of area wd/2, entirely in the water, perpendicular to the
flow; (d) a surface of area wd, half in the water and
half out, perpendicular to the flow; (e) a surface of
area wd, entirely in the water, with its normal 34 degrees
from the direction of flow.
(P prefixes) HRW Problem Supplement #1 - CHAPTER 25
58. Consider a flat, nonconducting ring of outer
radius R and inner radius r = 0.200R; the ring
has a uniform charge per unit area of sigma.
With V = 0 at infinity, find an expression for
the electric potential at point P on the central
axis of the ring, at a distance z = 2.00R from
the center of the ring.
88. Three particles with the same charge q and same mass m are
initially fixed in place to form an equilateral triangle
with edge lengths d. (a) If the particles are released
simultaneously, what are their speeds when they have
traveled a large distance (effectively an infinite distance)
from each other? (Measure the speeds in the original rest
frame of the particles.)
Suppose, instead, the particles are released one at a time:
The first one is released, and then, when the first one is
at a large distance, a second one is released, and then, when
that second one is at a large distance, the last one is
released. What then are the final speeds of (b) the first
particle, (c) the second particle, and (d) the last particle?
WOLFSON - CHAPTER 25
26. A solid insulating sphere of radius R carries a net
charge Q distributed uniformly throughout its volume.
(a) Show how to use Gauss' Law to find the electric
field strength (as a function of radius) inside the
sphere. (b) Find the absolute value of the potential
difference from the sphere's surface to its center.
(c) Which is at the higher potential, the surface or
the center, if Q is positive? If Q is negative?
(d) If Q is positive, what is the potential at the
surface if zero potential is chosen to be at the
center of the sphere? If zero potential is chosen
to be at infinity?
35. A hollow, spherical conducting shell of inner radius
b and outer radius c surrounds and is concentric with
a solid conducting sphere of radius a (a < b). The
sphere carries a net charge -Q and the shell carries
a net charge +3Q. Both conductors are in electrostatic
equilibrium. (a) In terms of the radial distance r as
measured from the center of the sphere, find the
electric potential at all points in space. Set V=0
at r=infinity. (b) Make a graph of the electric
potential versus r.
50. A charge +4q is located at the origin and a charge -q
is on the x axis at x = a. (a) Write an expression
for the potential on the x axis for x > a. (b) Find
a point in this region where V = 0. (c) Use the result
of (a) to find the electric field on the x axis for
x > a, and (d) find a point where E = 0.
58. Two small metal spheres are located 2.0 m apart. One
has radius 0.50 cm and carries 0.20 microC. The other
has radius 1.0 cm and carries 0.080 microC. (a) What
is the potential difference between the spheres?
(b) If they were connected by a thin wire, how much
charge would move along it, and in what direction?
WOLFSON - CHAPTER 26
3. Four 50 microC charges are brought from far apart onto a
line where they are spaced at 2.0 cm intervals. How much
work does it take to assemble this charge distribution?
Halliday and Resnick - 2nd Ed - CHAPTER 26
19. A positive charge per unit length lambda is distributed
uniformly along a straight-line segment of length L.
(a) Determine the potential (chosen to be zero at
infinity) at a point P a distance y from one end of
the charged segment and in line with it. (b) Use the
result of (a) to compute the component of the electric
field at P in the y-direction (along the line). (c)
Determine the component of the electric field at P in
a direction perpendicular to the straight line.
- . P
|
|
y
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|
- + Figure for H 26:19
| +
| +
| + the line of plus marks
L + stands for the line of
| + positive charge
| +
| +
- +
20. On a thin rod of length L lying along the x-axis with one
end at the origin (x=0), there is distributed a positive
charge per unit length given by lambda = cx, where c is a
constant. (a) Taking the electrostatic potential at
infinity to be zero, find V at the point P on the y-axis.
(b) Determine the vertical component E_y of the electric
field at P from the result of part (a). (c) Why cannot
E_x, the horizontal component of the electric field at P,
be found using the result of part (a)?
y axis
|
| Figure for H 26:20
P is on the |
y axis, a . P
distance y |
from the | the line of plus marks
origin | stands for the line of
| positive charge
|
--------+++++++++++++++++++------ x axis
(0,0)|<------ L ------>
|
|
A Prefixes
A1: An isolated copper sphere of radius 16 cm is found to have
an electric field at its surface which is radially inward
(i.e. pointing towards the center of the sphere) and which
has a strength 1150 N/C. (a) How many excess electrons are
on this copper sphere? (b) A square made up of nine copper
atoms (3 atoms by 3 atoms) takes up an area of about one
square nanometer. Approximately how many copper atoms are
on the surface of this copper sphere? (c) On the surface
of the copper sphere, there are about how many copper atoms
per excess electron?
A2: Consider a thin disk, of negligible thickness, of radius R
which is oriented perpendicular to the x axis such that the
x axis runs through the center of the disk. The disk is
centered at x=0, and has positive charge density sigma.
(a) Show that the electric field on the positive x axis is
given by
E_x = (sigma/(2epsilon0))*(1-1/sqrt(1+(R/x)^2))
(b) Show how to apply the binomial expansion to your result
for (a) in order to demonstrate that the commonly-used
expression, E=sigma/(2epsilon0), is valid to within
1.0% for all values of x which are 1.0% of R or less.
A3: Assume that water flowing through a pipe with a circular
cross section flows most rapidly at the center of the pipe
and least rapidly near the pipe walls (this might be
reasonable because of friction between the water and the
pipe walls). Assume that the water speed at the walls is
half as great as the speed at the center, and assume that
the decrease is linear (i.e. the graph of water speed
versus radius is a straight line). If the radius of the
pipe is 10 cm and if the volume flux of water is 5.0 liters
per second, find the speed of the water at the center of
the pipe. (One liter is 1000 cubic centimeters or 0.001
cubic meters). HINT: Draw the graph of water speed versus
distance from the centerline of the pipe; you must do
the integral for volume flux symbolically.
A4: A thin wooden stick of length 12 cm has a tiny metal sphere
glued to each end. A charge of +3 microC is placed on one
sphere and a charge of -2 microC is placed on the other.
The center of mass is located 7 cm from the positively-
charged sphere. The system is mounted on a fixed
horizontal E-W axle passing through the center of mass
about which the system is free to rotate with no friction.
When the system is then placed in a horizontal uniform
southward electric field of 800 N/C, the resulting
equilibrium position of the system is horizontal with
the positive charge due S of the negative charge.
(a) When the system is in its equilibrium position,
what is the horizontal force on the system by the axle?
(b) What amount of torque (about the axle) is required
to hold the system at an angular displacement of 25
degrees away from the equilibrium position?
(c) What minimum amount of work must be done to move
the system from its equilibrium position to an angular
displacement of 25 degrees? (HINT: write the needed
torque as a function of displacement, then integrate
from zero to 25 degrees to find the needed work.)
(d) What work would be done by the electric field
during the displacement described in part (c)?
A5: Two conductors, A and B, are each in the shape of a
tetrahedron, but of different sizes. They are charged
in the following manner:
1. Tetrahedron A is charged from an electrostatic
generator to charge q.
2. Tetrahedron A is briefly touched to tetrahedron B.
3. Steps 1 and 2 are repeated until the charge on
tetrahedron B reaches a maximum value; i.e.
the process is repeated until further repetitions
will no longer result in the transfer of additional
charge from tetrahedron A to tetrahedron B.
If the charge on tetrahedron B was q/5 after the first
time it touched tetrahedron A, what is the final charge
on tetrahdedron B? HINT: the way to make progress is
to think carefully about the first and second transfers,
and then about the final attempted transfer (in which
no charge is actually transferred). Be sure to express
your logic clearly and thoroughly when writing up this
question.