PHY131 HOMEWORK PROBLEMS 2011 JULY 13 - JULY 19
UNIT 2
1. 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.
2. 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
|
|
- +
| +
| +
| + the line of plus marks
L + stands for the line of
| + positive charge
| +
| +
- +
3. 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
|
|
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 ------>
|
|
4. 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.
5. 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?
6. 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? (f) If an electron is released
from rest at point C, what is the electron's kinetic
energy and speed as the electron is passing point B?
7. 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.
8. 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?
GBA NOTES THAT THE ORDERING OF PROBLEMS 9-12 WAS CORRECTED
HERE ON 7/13 at 8:30 PM.
9. Two metal objects, a saw and a wrench, are lying
side-by-side on an non-conducting table; the two
metal objects are not in contact with one another;
they have net charges of +70 pC and -70 pC, and
this results in a 20 V potential difference
between them. (a) What is the capacitance of the
system? (b) If the charges are changed to +200 pC
and -200 pC, what does the capacitance become?
(c) What does the potential difference become?
10. 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, with the touching done
in exactly the same manner each time, 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.
11. Two conducting spheres of radius a are separated
by a distance L >> a; since the distance is large,
neither sphere affects the other's electric field
significantly, and the fields remain spherically
symmetric. (a) If the spheres carry equal but
opposite charges +-q, show that the potential
difference between them is 2kq/a. (b) Write an
expression for the infinitesimal work dW involved
in moving an infinitesimal charge dq (assumed
positive) from the negatively-charged to the
positively-charged sphere (thereby infinitesimally
increasing the amount of separated charge to q+dq).
(c) Integrate your expression to find the work
involved in transferring a total charge Q from one
sphere to the other, assuming both are initially
uncharged.
12. Two square conducting plates measure 5.0 cm on a
side. The plates are parallel, spaced 1.2 mm
apart, and initially uncharged. (a) How much work
is required to transfer 7.2 microC from one plate
to the other? (b) How much work is required to
transfer a second 7.2 microC?
GBA NOTES THAT THE ORDERING OF PROBLEMS 9-12 WAS CORRECTED
HERE ON 7/13 at 8:30 PM.
13. A 4.00 microF capacitor and a 6.00 microF capacitor are
connected in series across a 660-V supply line.
(a) Find the charge on each capacitor and the voltage
across each. (b) The charged capacitors are disconnected
from the line and from each other and then reconnected
to each other, with terminals of like sign together.
Find the final charge on each and the voltage across each.
14. You have three capacitors: capacitors 1 and 3 each
have a capacitance of 0.02 microF and capacitor 2
has a capacitance of 0.01 microF. You first
connect capacitors 2 and 3 in parallel; then you
construct a series capacitor circuit consisting of
a 100 V battery, capacitor 1, and the combination
of capacitors 2 and 3. (a) Draw a circuit diagram
for your capacitor circuit. (b) What is the
equivalent capacitance of your three capacitors
connected in this fashion? (c) What is the charge
on each capacitor in the circuit? (d) What is the
voltage across each capacitor in the circuit?
15. The figure for this problem is Fig. 24-36 (p. 843).
A slab of copper of thickness (0.629)d is thrust
into a parallel-plate capacitor of plate area
115 cm^2 and separation distance d = 1.24 cm; it
is exactly halfway between the plates. Assume that
charges of +-702 pC existed on the two plates
before the slab was introduced, and also that the
battery used to charge the plates was disconnected
before the insertion. (a) What is the electric
field strength in the air gap before and after
insertion of the slab? (b) What is the voltage
between the plates before and after insertion of
the slab? (c) What is the capacitance after the
slab is introduced? (d) What is the stored
electrostatic energy before and after the slab is
inserted? Now suppose that the slab is not made
of copper, but rather of a dieletric material with
kappa = 2.61. Also suppose that an 85.5 V battery
is connected to the capacitor plates while the
dielectric is being inserted. WITHOUT resorting
to a calculation of capacitance, find (e) the
electric field in the gap, and (f) the charge on
the capacitor plates before and after the insertion.
(g) Use your answer to (f) to find the capacitance
after the slab is in place. (h) What is the stored
electrostatic energy before and after the slab is
inserted?
16. You are asked to construct a capacitor having a
capacitance near 1 nF and a breakdown potential in
excess of 10000 V. You think of using the sides of
a tall Pyrex drinking glass as a dielectric, lining
the inside and outside curved surfaces with aluminum
foil to act as the plates. The glass is 15 cm tall
with an inner radius of 3.6 cm and an outer radius
of 3.8 cm. What are the (a) capacitance and (b)
breakdown potential of this capacitor?
17. A certain parallel-plate capacitor is filled with
a dielectric for which Kappa = 5.5. The area of
each plate is 0.034 m^2, and the plates are
separated by 2.0 mm. The capacitor will fail
(short out and burn up) if the electric field
between the plates exceeds 200 kN/C. What is the
maximum energy that can be stored in the capacitor?
18. Two parallel-plate capacitors A and B are connected
in parallel across a 600 V battery. Each plate has
area 80.0 cm^2 and the plate separations are 3.0 mm.
Capacitor A is filled with air; capacitor B is
filled with a dielectric of dielectric constant
Kappa = 2.60. Find the magnitude of the electric
field within (a) the dielectric of capacitor B and
(b) the air of capacitor A. What are the free
charge densities sigma on the higher-potential
plate of (c) capacitor A and (d) capacitor B?
(e) What is the induced charge density sigma' on
the surface of the dielectric which is nearest to
the higher-potential plate of capacitor B.
19. Two parallel plates of area 100 cm^2 are given charges
of equal magnitudes 0.89 microC but opposite signs.
The electric field within the dielectric material
filling the space between the plates is 1.4 MV/m.
(a) Calculate the dielectric constant of the material.
(b) Determine the magnitude of the charge induced on
each dielectric surface. (c) The capacitor will fail
(short out and burn up) if the electric field
between the plates exceeds 20 MV/m; as a result the
maximum energy that can be stored in this capacitor
is 0.292 J. Find the separation distance between the
plates.
20. The electron beam that "paints" the image on a
certain oscilloscope screen contains 5 million
electrons per cm of its length. If the electrons
move toward the screen at 60 million m/s, how much
current does the beam carry? What is the direction
of this current?
21. A 3.90 A current runs through a 12-gauge copper wire
(diameter 2.05 mm) and through the filament of a light
bulb to which the wire is attached. Copper has
8.5 X 10^{28} free electrons per cubic meter.
(a) How many electrons pass any cross-section of the
wire, or any cross-section of the filament, each second?
(b) What is the current density in the wire?
(c) At what speed does a typical electron pass by any
given point in the wire?
(d) If you were to use wire of twice the diameter (with
the current unchanged), which of the above answers would
change? Would those answers increase or decrease?
(e) Assume that the filament is tungsten, with a free
electron density of 4.22x10^{28} free electrons per cubic
meter. If the diameter of the filament is 0.62 mm, then
what is the current density in the filament?
(f) At what speed does a typical electron pass by any
given point in the filament?
(g) If the filament is 38 mm long, how long does it take
for an electron to pass through the filament?
22. A power plant produces 1000 MW to supply a city
40 km away. Current flows from the power plant
on a single wire of resistance 0.050 Ohms/km,
through the city, and returns via the ground,
assumed to have negligible resistance. At the
power plant the voltage between the wire and the
ground is 115 kV. (a) What is the current in
the wire? (b) What fraction of the power is
lost in transmission? (c) What should be the
power line voltage if the transmission loss is
not to exceed 2.0%.
23. The available kinetic energy per unit volume, due to the
drift speed of the conduction electrons in a current-
carrying conductor, can be defined as
K/volume=n((1/2)m(v_d)^2). Evaluate K/volume for the
copper wire and current of Example 25.1 (page 850).
(b) Calculate the total change in electric potential
energy for the conduction electrons in 1.0 cm^3 of
copper if they fall through a potential drop of 1.0 V.
How does your answer compare to the available kinetic
energy in 1.0 cm^3 due the the drift speed?
24. A defective starting motor in a car draws 300 A from
the car's 12 V battery, dropping the battery terminal
voltage to only 6 V. A good starter motor should draw
only 100 A. What will the battery terminal voltage
be with a good starter?
25. You have three resistors: resistors 2 and 3 each
have a resistance of 40 kOhms and resistor 1
has a resistance of 30 kOhms. You first
connect resistors 2 and 3 in parallel; then you
construct a series circuit consisting of a 100 V
battery, resistor 1, and the combination
of resistors 2 and 3. (a) Draw a circuit diagram
for your circuit. You have three voltmeters, each
with a different internal resistance. You will
use each voltmeter, in turn, to measure the voltage
across resistor 1 while the circuit is in operation.
What will be the reading when the voltage is
measured with a (b) 50-kOhm voltmeter,
(c) a 250-kOhm voltmeter, and (d) a digital meter
with a 10-MOhm resistance?
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 and Freedman - 12th Ed - CHAPTER 24
9. In showing your work for this problem, you should first
work out, from the definition of capacitance (i.e. use
the method for finding capacitance as described in
lecture), the capacitance per unit length for infinitely
long coaxial cylinders. Then apply that result to these
coaxial cylinders of finite length.
64. Add part (c) What is the energy stored in each
combination before the switch S is thrown?
Young and Freedman - 12th Ed - CHAPTER 25
48. In part (b), Y&F want the NET power output of the
battery, i.e. the power output of the EMF minus the
power dissipated in the internal resistance of the
battery. In part (c), you are to assume that the 8 volt
battery is rechargeable; i.e., running current "backward"
through the battery will result in the conversion of
electric potential energy into chemical energy of the
battery. Also in part (d), Y&F are again asking for the
net rate of energy conversion, i.e. the rate of
production of thermal energy in the internal resistance
of the battery plus the rate of energy storage in the
battery's chemicals. (Running current "backward" through
a non-rechargeable battery would result in a dramatic
increase in the internal resistance of the battery; i.e.,
all of the energy would then be converted into thermal
energy.)
60. In my copy of Y&F 12e, there is a misprint in this problem.
The total length of the composite wire is 2.0 m (not 2.0 mm).
64. It is very important to do this problem by the general
method for finding resistance: (1) imagine a selected
current I flowing between the two relevant locations on
the conducting material; (2) find J in the material in
terms of I and in terms of location within the material,
then E in the material (in terms of location and I), and
then the voltage V between the two relevant locations;
and (3) use the DEF of resistance for an Ohmic material,
namely R=V/I -- your selected I will drop out, leaving
only geometry and resistivity.
Young and Freedman - 12th Ed - CHAPTER 26
62. Add the following parts:
(b) What is the amount of current through the 24.0-V battery?
(c) What is the amount and direction of current through the
2.0-Ohm resistor?
HINT: It is easier to answer parts (b) and (c) before
answering part (a).