|Last revised 2000/01/05|
|Go to assignment #||1||2||3||4||5||6||7||8||9||10||11||12
|Due on Monday||02/14||02/21||02/28||03/06||03/13||03/20||04/03||04/10||04/17||04/24||05/01||05/08|
|Chapter||22||23||23, 24||25||26||27, 28||29||30||31||31||33||33|
I will almost always have hints posted on the web beginning a week before the assignment is due. The day before the problem session I will try to post answers in the same location. I will frequently post solutions after the problem session is over.
In general, I have chosen problems like the more difficult ones in the chapter. It may be helpful to work a few of the simpler problems first, especially if you have trouble getting started on one of the assigned problems. If you have difficulty, see me in Sharp Lab 264, call x2013, or send EMail to firstname.lastname@example.org, preferably before turning the set in. It is as important to know why incorrect attempts do not work as it is to know what the correct solutions are. Please feel free to talk to me at any time either in person, by phone, or by E-MAIL.
The problems will be graded for the existence of diagrams, the use of units, and correct vector notation. Otherwise, a reasonable attempt will get full credit whether it is right or wrong. Since the problems are graded for technique and not for correctness, you can work with each other on the assignments. You should do so. You may also consult others outside the class, but I strongly advise against looking at their solutions before attempting the problems yourself. You must write up your own final copy, unless I have stated otherwise (no Xeroxes!).
Many of these problems are adapted from a book by Tipler, because I already have reliable answers and solutions for those problems. Similar problems appear in our text. I am also modifying some of the problems from the text so that they fit better with the other assigned problems. Whenever the assigned problem is very similar to one in the text, the assignment includes the statement "compare HRW # ...." If an assigned problem is nearly identical to one in the text, the assignment will say "see HRW # ...."
2. Four charges, each of magnitude q, are at the corners of a square of side L. The two charges at one pair of opposite corners are positive, and the other two charges are negative. Find the force exerted on either of the negative charges by the three remaining charges. See figure to the right.
3. (See HRW p. 551 #18 and p. 553 #41). A certain
charge Q is divided into two parts alpha Q and
(1 - alpha) Q, which are then separated by a certain
distance. What must alpha be in order to maximize the
electrostatic repulsion between the two charges? Draw a graph of the
electrostatic force F as a function of alpha and
verify the correctness of your answer.
4. (HRW p. 551 #20). In the figure on the right, two tiny conducting balls of identical mass m and identical charge q hang in equilibrium from nonconducting threads of length L. Assume that theta is so small that tan(theta) can be replaced by its approximate equal sin(theta) [both are in fact essentially equal to theta]. (a) Show that, for equilibrium,
5. (Compare HRW p. 577
# 57) Two charged plates of length L are placed a distance
d apart as shown in the figure to the right. An electron is
injected between the plates, just clearing the lower plate as it
enters. It feels a constant electric field E pointed straight
upward due to the charges on the plates. If the electron's initial
velocity is v0 at an angle theta with respect to the lower
plate, for what values of theta will it not hit the upper plate? If
it clears the upper plate, how far does it get before hitting the
6. (Compare HRW p. 575 #22) Four charges of equal magnitude are arranged at the corners of a square of side L as shown in the figure to the right. Show that the electric field at the midpoint of one of the sides of the square is directed along that side, points toward the negative charge, and has a magnitude E given by
7. A water molecule has its
oxygen atom at the origin, one hydrogen nucleus at
x = 0.077 nanometer, y = 0.058 nm, and the
other hydrogen nucleus at x = - 0.077 nm,
y = 0.058 nm. If the electrons from the hydrogen atoms
are transferred completely to the oxygen atom so that it has a charge
of -2e, what is the dipole moment of the water molecule?
[This characterization of the chemical bonds of water as being totally
ionic overestimates the dipole moment of a water molecule].
8. A positive point charge +Q is at the origin. A dipole of moment p is a distance r away from the charge and points in the radial direction, as shown in the diagram to the right.
(a) Show that the force exerted by the electric field of the point charge on the dipole is attractive and has a magnitude of approximately
(b) Now assume that the dipole is centered at the origin and that the point charge Q is a distance r away along the line of the dipole. From your result in part (a) and Newton's Third Law, show that the magnitude of the electric field of the dipole along the line of the dipole a distance r away is approximately
9. (HRW p. 575 # 26) Electric Quadrupole . The figure shows an electric quadrupole. It consists of two dipoles with dipole moments that are equal in magnitude but opposite in direction. Show that the value of the magnitude of the electric field E on the axis of the quadrupole for points a distance z from its center (assume z >> d) is given by
in which Q (=2 q d2) is known as the quadrupole
moment of the charge distribution. [This is not the only way to
get a quadrupole moment.]
10. (Compare HRW p. 576 #33) (a) A thin, nonconducting rod carries a charge of constant linear charge density lambda and lies on the x-axis from x = 0 to x = a. Show that the y-component of the electric field at any point on the y-axis is given by
Go to solution for part (a)
(b) Show that if the rod extends from x = - b to x = a, the y-component of the electric field at the same point is given by
(c) HRW asks for the special case a = b = L/2. Show that in this case you get
(d) Show that if and only if a = b the x-component of the electric field vanishes.
11. A butterfly net is attached to a ring of radius R, and the ring is held in a plane perpendicular to a constant electric field of magnitude E. What is the electric flux
through the net? The angle theta is the angle between the normal to the net and the electric field.
13. A disk of radius R carries a uniform charge density sigma. (a) Compare the approximation E = sigma / (2 epsilon0 ) with the exact expression for the electric field on the axis of the disk by computing the neglected term as a percentage of sigma / (2 epsilon0 ) for distances of x = R / 300 , x = R / 150 , and x = R / 10 . (b) At what distance is the neglected term 1% of sigma / (2 epsilon0 ) ?
14. (Hint: compare HRW p. 576 #34) In the figure to
the right, two nonconducting rods of length L have charge
q uniformly distributed along their lengths. (a) What is the
linear charge density of each rod? (b) What is the electric field
E due to the left-hand rod at a distance a beyond
its right-hand end? (c) What is the force that one of the rods exerts
on the other? (d) Show that when the distance d between the
rods becomes large compared to the length of the rods, the force tends
toward the expected result q2 / d2.
16. (HRW p. 599 #52). A solid nonconducting sphere of radius R has a nonuniform charge distribution of rho = rhos r / R, where rhos is a constant and r is the distance from the center of the sphere. Show that (a) the total charge on the sphere is Q = pi rhos R 3 and (b) the electric field inside the sphere has a magnitude given by
17. (Compare HRW p. 598 #31) An infinitely long nonconducting cylindrical shell of inner radius a and outer radius b has a uniform volume charge density rho . Find the electric field everywhere.
18. An infinitely long nonconducting solid cylinder of radius R has a charge density of B / r , where r is the distance from the axis of the cylinder. Find the electric field everywhere.
20. Three equal charges lie in the xy plane. Two are on the y-axis at y = - a and y = + a , and the third is on the x axis at x = a . (a) What is the potential V(x) due to these charge at any point on the x-axis? (b) Find Ex along the x-axis from the potential function V(x) . Evaluate your answers to (a) and (b) at the origin and at x = infinity to see if they yield the expected results. (c) If a fourth charge of + q is released at the origin, what is its velocity when it gets very far away from the other three?
21. (HRW p. 624 #41) The figure shows a plastic rod
of length L and uniform positive charge Q lying on
an x axis. With V = 0 at infinity,
find the electric potential at point P1 on the axis, at
distance d from one end of the rod.
22. Four equal charges Q are at the corners of a square of side L. The charges are released one at a time proceeding clockwise around the square. Each charge is allowed to reach its final speed a long distance from the square before the next charge is released. What is the final kinetic energy of (a) the first charge released, (b) the second charge released, (c) the third charge released, and (d) the fourth charge released?
23. (Compare HRW p. 624 # 46) The electric potential in a region of space is given by
Find the electric field at all points (x, y, z)
24. Consider two concentric spherical metal shells of radii a and b where b > a . The outer shell has a charge Q , but the inner shell is grounded [it is at the same potential as a point at infinity]. This means that the inner shell is at zero potential and implies that electric-field lines leave the outer shell and go to infinity but that other electric-field lines leave the outer shell and end on the inner shell. Find the charge on the inner shell.
25. A nonconducting sphere of radius R has a volume charge density rho = rho0 r / R , where rho0 is a constant. (a) Show that the total charge is Q = pi R3 rho0 . (b) Use Gauss's Law to find the electric field Er everywhere. (c) From the electric field, find the electric potential everywhere, assuming that V = 0 at infinity. (Remember that V is continuous at r = R .
27. Consider two parallel-plate capacitors, C1 and C2, that are connected in parallel. The capacitors are identical except that C2 has a dielectric inserted between its plates. A voltage source of V volts is connected across the capacitors to charge them and is then disconnected. (a) What is the charge on each capacitor? (b) What is the total stored energy of the capacitors? (c) The dielectric is removed from C2. What is the final stored energy of the capacitors? (d) What is the final voltage across the two capacitors?
28. (Compare HRW p. 650 # 63) A parallel-plate capacitor of area A and separation d is charged to a potential difference V and is then removed from the charging source. A dielectric slab of constant kappa, thickness d, and area A/2 is inserted along one edge of the capacitor as shown in the figure. Let sigma1 be the free charge density at the conductor-dielectric surface and sigma2 be the free charge density at the conductor-air surface. (a) Why must the electric field have the same value inside the dielectric as in the free space between the plates? (b) Show that sigma1 = kappa sigma2. (c) Show that the new capacitance is
and that the new potential difference is
2 V / ( kappa + 1 )
29. HRW p. 650 # 68: The space between two concentric conducting spherical shells of radii b and a (where b > a) is filled with a substance of dielectric constant kappa. A potential difference V exists between the inner and outer shells. Determine (a) the capacitance of the device, (b) the free charge q on the inner shell, and (c) the charge q' induced along the surface of the inner shell.
30. The wires in a house must be large enough in diameter so that they do not get hot enough to start a fire. Suppose a certain wire is to carry a current of 20 A, and it is determined that the joule heating of the wire should not exceed 2 W/m. What diameter must a copper wire have to be "safe" for this current?
31. HRW p. 672 #59: A linear accelerator produces a pulsed beam of electrons. The pulse current is 0.50 A, and the pulse duration is 0.10 microsecond. (a) How many electrons are accelerated per pulse? (b) What is the average current for a machine operating at 500 pulses/s? (c) If the electrons are accelerated to an energy of 50 MeV, what are the average and peak powers of the accelerator?
32. HRW p. 671 #41: A resistor has the shape of a truncated right-circular cone (see figure 27-25 in text). The end radii are a and b, and the altitude is L. If the taper is small, we may assume that the current density is uniform across any cross section. (a) Calculate the resistance of this object. (b) Show that your answer reduces to rho L / A for the special case of zero taper (that is, for a = b).
33. A resistor of size R1 is
placed in series with a parallel combination of resistors
R2 and R3. A battery of
voltage V is connected across the entire system, as shown in
the diagram. Find the effective resistance of the system, the voltage
across each resistor, the current through each resistor, and the
current through the battery. [When you get the correct answers, keep
them for use in the assignment on AC circuits.]
34. For the circuit shown
in the figure, find (a) the current in each resistor, (b) the power
supplied by each EMF, and (c) the power dissipated in each
resistor. (d) Is energy conservation obeyed?
35. Nine resistors of
resistance R are connected as shown in the figure, and a
potential difference V is applied between points a
and b. What is the equivalent resistance of this network? (b)
Find the current in each of the nine resistors.
36. Find the current through each of the elements in
the diagram to the right. The use of a symbolic-algebra computer
program is recommended. [When you have correct answers, keep them for
use in the chapter on AC circuits.]
38. A particle of charge q and mass M moves in a circle of radius r and with angular velocity omega. (a) Show that the average current is I = q omega / (2 pi ) and that the magnetic moment has the magnitude mu = (1/2) q omega r2. (b) Show that the angular momentum of this particle has the magnitude L = M r2 omega and that the magnetic moment and angular momentum vectors are related by mu = [q/(2M)]L.
39. A current-carrying wire is bent into a semicircular loop of radius R, which lies in the xy plane. There is a uniform magnetic field B = B 1z perpendicular to the plane of the loop. Show that the force acting on the loop is F = 2 I R B 1y.
40. A nonconducting rod
of mass M and length l has a uniform charge per unit
length lambda and rotates with angular velocity
omega about an axis through one end and perpendicular to the
rod. (a) Consider a small segment of the rod of length dx and
charge dq = lambda dx at a distance x
from the pivot. Show that the magnetic moment of this segment is
(1/2) lambda omega x2 dx. (b)
Integrate your result to show that the total magnetic moment of the
mu = (1/6) lambda omega l3
(c) Show that the magnetic moment mu and angular
momentum L are related by
mu = [(Q/(2M)] L,
where Q is the total charge on the rod.
42: HRW p. 747 # 38: In the figure to the right (HRW figure 30-55), the long straight wire carries a current of 30 A and the rectangular loop carries a current of 20 A. Calculate the resultant force acting on the loop. Assume that the distance a = 1.0 cm, b = 8.0 cm, and L = 30 cm.
43: HRW p. 749 #50. Figure 30-64 of the text shows a cross section of a long cylindrical conductor of radius a containing a long cylindrical hole of radius b. The axes of the cylinder and hole are parallel and are a distance d apart; a current i is uniformly distributed in the conductor outside the hole. (a) Use superposition to show that the magnetic field at the center of the hole is
mu0 i d B = ------------------------ 2 pi (a2 - b2)
(b) Discuss the two special cases b = 0 and
d = 0. (c) Use Ampere's Law to show that the
magnetic field in the hole is uniform.
(Hint: Regard the cylindrical hole as filled with two equal currents moving in opposite directions, thus canceling each other. Assume that each of these currents has the same current density as that in the actual conductor. Then superimpose the fields due to two complete cylinders of current, of radii a and b, each cylinder having the same current density.)
44. A solenoid has n turns per unit length and radius R and carries a current I. Its axis is along the x axis with one end at x = -L/2 and the other end at x = +L/2, where L is the total length of the solenoid. Show that the magnetic field B at a point on the axis outside the solenoid is given by
B = (1/2) mu0 n l (cos theta1 - cos theta2)
x + (1/2) L cos theta1 = ------------------------- [ R2 + (x + L/2)2 ]1/2
x - (1/2) L cos theta2 = ------------------------- [ R2 + (x - L/2)2 ]1/2
45. In Problem 44, a formula for the magnetic field along the axis of a solenoid is given. For x >> L and L > R, the angles theta1 and theta2 in that problem are very small, so that the small angle approximation cos theta = 1 - theta2 / 2 is valid. (a) Draw a diagram and show that
R tan (theta1) = ------ x + L/2
R tan (theta2) = ------- x - L/2
(b) Show that the magnetic field at a point far from either end of the solenoid can be written
mu0 qm qm B = ---- [ -- - -- ] 2 pi r12 r22
where r1 = x - L/2 is the distance to the near end of the solenoid, r2 = x + L/2 is the distance to the far end, and qm = n I pi R2 is the magnetic moment of the solenoid divided by its length.
This is the priciple of the commercial alternating-current generator. (b) Design a loop that will produce an emf of 150 V when rotated at 60.0 rev/s in a magnetic field of 0.500 T.
47. HRW p. 780 # 29: A rectangular loop of wire with length a, width b, and resistance R is placed near an infinitely long wire carrying current i, as shown in Fig. 31-53 of the text. The distance from the long wire to the center of the loop is r. Find (a) the magnitude of the magnetic flux through the loop and (b) the current in the loop as it moves away from the long wire with speed v. [Neglect the magnetic field generated by the current in the loop itself. - MVB]
48. HRW p. 781 # 38: In Fig. 31-57 of the text a conducting rod of mass m, resistance R, and length L slides without friction on two long, horizontal, resistanceless rails. A uniform vertical magnetic field B fills the region in which the rod is free to move. The generator G supplies a constant current i directed as shown in the figure. (a) Find the velocity of the rod as a function of time, assuming it to be at rest at t = 0. The generator is now replaced by a battery that supplies a constant EMF. (b) Show that the velocity of the rod now approaches a constant terminal value v and give its magnitude and direction. (c) What is the current in the rod when this terminal velocity is reached? (d) Analyze this situation and that with the generator from the point of view of energy transfers.
49. A long solenoid has n turns per unit length and carries a current given by I = I0 sin (omega t). The solenoid has a circular cross section of radius R. Find the induced electric field at a radius r from the axis of the solenoid for (a) r < R and (b) r > R.
51. Show that the inductance of a toroid of rectangular cross section, half of which is shown in the figure, is given by
where N is the total number of turns, a is the
inside radius of the toroid, b is its outside radius, and
h is its height.
52. HRW p. 783 # 69. In figure 31-65 of the text, E = 100 V, R1 = 10.0 Ohms, R2 = 20 Ohms, R3 = 30 Ohms, and L = 2.00 H. Find the values of i1 and i2 (a) immediately after the closing of switch S, (b) a long time later, (c) immediately after the reopening of switch S, and (d) a long time after the reopening.
For the circuit in the figure, (a) find the rate of change of current
in each inductor and in the resistor just after the switch is
closed. (b) What is the current through the resistor after a long time
54. In the circuit in the
figure, the switch is closed at t = 0. From time
t to time L/R, find (a) the total energy that has
been supplied by the battery, (b) the total energy that has been
dissipated in the resistor, and (c) the energy that has been stored
in the inductor. Hint: Find the rates as functions of time
and integrate from t = 0 to
t = L/R
56. A resistance R and a 1.4-H inductance are in series across a 60-Hz AC voltage. The voltage across the resistor is 30 V and the voltage across the inductor is 40 V. (a) What is the value of the resistance? (b) What is the AC input voltage?
57. A coil draws 15 A when connected to a 220-V AC 60-Hz line. When it is in series with a 4-ohm resistor and the combination is connected to a 100-V battery, the battery current after a long time is observed to be 10 A. (a) What is the resistance in the coil? (b) What is the inductance of the coil?
59. The circuit shown in the figure is called an RC high-pass filter because high input frequencies are transmitted with greater amplitude than low input frequencies. (a) If the input voltage is
Vin = V0 cos(omega t),
show that the output voltage [the potential difference between the two black dots in the figure] is
V0 cos (omega t + phi) Vout = ------------------------- [ 1 + (omega R C)-2 ]1/2and determine the value of phi. (b) At what angular frequency is the output voltage half the input voltage? (c) Sketch a graph of the amplitude of Vout/V0 as a function of omega.
60. A resistor and an
ideal inductor are connected in parallel across an emf
V = V0 cos(omega t). Show that
(a) the current in the resistor is IR = (V0/R) cos(omega t),
(b) the current in the inductor is IL = (V0/XL)cos(omega t - pi / 2),
(c) I = IR + IL = Imaxcos(omega t - delta) where tan(delta) = R/XL and Imax = V0/Z with Z -2 = R -2 + XL -2.
61. Solve for the amplitudes of all the currents in a circuit which has a capacitor in series with a parallel combination of a resistor and an ideal inductor. Use the values R=10 ohms, |XL|=30 ohms, |XC|=10 ohms, and |V0|=20 volts.
62. Find the amplitudes of all the currents in the
3-loop circuit in the figure. Use
R1 = 10 ohms,
R2 = 40 ohms,
R3 = 20 ohms,
XL = 30 ohms,
XC = 10 ohms, and
V = 15 Volts. Do the calculation numerically or use a
computer-algebra program to work the problem analytically.