Chapter 21: ELECTRIC CHARGE AND ELECTRIC FIELD
Exercises 1, 5, 7, 9, 17, 19, 23, 25, 27, 29, 31, 35, 39, 43, 45, 47, 57, 59, 61, 67
Problems: 69(73), 71(75), 73(77), 75(79), 77(80), 81(85), 83(87), 85, 87(90), 95(97), 101(103)
21.69(73). Two positive point charges Q are held fixed on the x-axis at x = a and x = -a. A third positive point charge q,
with mass m, is placed on the x-axis away from the origin at a coordinate x such that |x| << a. The charge q, which is
free to move along the x-axis, is then released. (a) Find the frequency of oscillation of the charge q. (Hint: Review
the definition of simple harmonic motion in Section 13.2. Use the binomial expansion (1 + z)n = 1 + nz + n(n - l)z2/2
+ ..., valid for the case |x| < 1.) (b) Suppose instead that the charge q were placed on the y-axis at a coordinate y
such that |y| << a, and then released. If this charge is free to move anywhere in the xy-plane, what will happen to it?
Explain your answer.
21.71(75). Two small spheres with mass cbarge q, m = 15.0 g are hung by silk threads of length L =
1.20 m from a common point (Fig.). When the spheres are given equal quantities of negative charge, so that q1 = q2 = q, each thread hangs at 6 = 25.0° from the vertical. (a) Draw
a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find
the magnitude of q. (c) Both threads are now shortened to length L = 0.600 m, while the
charges q1 and q2 remain unchanged. What new angle will each thread make with the vertical?
(Hint: This part of the problem can be solved numerically by using trial values for 6 and adjusting the values of 6 until a self-consistent answer is obtained.)
21.73(77). Sodium chloride (NaCl, ordinary table salt) is made up of positive sodium ions (Na+) and negative chloride
ions (Cl-). (a) H a point charge with the same charge and mass as all the Na+ ions in 0.100 mol of NaCl is 2.00 cm
from a point charge with the same charge and mass as all the Cl- ions, what is the magnitude of the attractive force
between these two point charges? (b) If the positive point charge in part (a) is held in place and the negative point
charge is released from rest, what is its initial acceleration? (c) Does it seem reasonable that the ions in NaCl
could be separated in this way? Why or why not? (In fact, when sodium chloride dissolves in water, it breaks up
into Na+ and Cl- ions. However, in this situation there are additioual electric forces exerted by the water molecules
on the ions.) For Cl, atom mass M = 35.45x10-3 kg/mol.
21.75(79). Three identical point charges q are placed at each of three corners of a square of side L. Find the magnitude and direction of the net force on a point charge -3q placed (a) at the center of the square and (b) at the
vacant corner of the square. In each case, draw a free-body diagram showing the forces exerted on the -3q charge
by each of the other three charges.
21.77(80). Three point charges are placed on the y-axis: a charge q at y = a, a charge -2q at the origin, and a charge q at
y = -a. Such an arrangement is called an electric quadrupole. (a) Find the magnitude and direction of the electric
field at pnints on the positive x-axis. (b) Use the binomial expansion to find an approximate expression for the electric field valid for x >> a. Contrast this behavior to that of the electric field of a point charge and that of the electric
field of a dipole.
21.81(85). Two small, copper spheres each have radius 1.00 mm. (a) How many atoms does each sphere contain? (b) Assume that each copper atom contains 29 protons and 29 electrons. We know that electrons and protons have charges
of exactly the same magnitude, but let’s explore the effect of small differences. If the charge of a proton is +e and the
magnitude of the charge of an electron is 0.100% smaller, what is the net charge of each sphere and what force would
one sphere exert on the other if they were separated by 1.00 m?
1
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
21.83(87). A proton is projected into a uniform electric field that points vertically upward and has magnitude E. The
initial velocity of the proton has a magnitude v0 and is directed at an angle α below the horizontal. (a) Find the maximum distance hmax that the proton descends vertically below its initial elevation. You can ignore gravitational forces.
(b) After what horizontal distance d does the proton return to its original elevation? (c) Sketch the trajectory of the
proton. (d) Find the numerical values of hmax and d if E = 500 N/C, v0 = 4.00 x 105 m/s, and α = 30.0°.
21.85. A charge of 12.0 nC is at the origin; a second, unknown charge is at x = 3.00 m, y = 0; and a third charge of -16.0
nC is at x = 5.00 m, y = 0. What are the sign and magnitude of the unknown charge if the net field at x = 8.00 m, y =
0 has magnitude 12.0 N/C and is in the +x-direction?
21.87(90). Positive charge Q is distributed uniformly along the positive y-axis
between y = 0 and y = a. A negative point a charge -q lies on the positive x- axis,
a distance x from the origin (Fig.). (a) Calculate the x- and y-components of the
electric field produced by the charge distribution Q at points on the positive xaxis. (b) Calculate the x- and y-components of the force that the charge
distribution Q exerts on q. (c) Show that if x >> a, Fx ≈ -qQ/4πP0x2 and
Fy ≈ +qQa/8πP0x3. Explain why this result is obtained.
21.95(97). Negative charge -Q is distributed unifonnly around a quarter-circle of radius a that lies in the first quadrant,
with the center of curvature at the origin. Find the x- and y-components of the net electric field at the origin.
21.101(103). An infinite sheet with positive charge per unit area σ lies in the xy-plane. A second infinite sheet with negative charge per unit area -σ lies in the yz-plane. Find the net electric field at all points that do not lie in either of these
planes. Express your answer in terms of the unit vectors i, j, and k.
QUESTIONS
1. How objects become electrically charged? What are building blocks of matter? Describe the structure of atoms.
2. Classify matterials by their electrical properties. Give examples.
3. Compare Newton’s law of universal gravitational attraction and Coulomb’s law for electric charges. What are
similarities and distinctions?
4. What characteristic of gravitational field is a equivalent to electric field intensity E?
5. What are properties of electric field lines? What are those of a uniform electric field?
2
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 22: GAUSS’ LAW
Exercises: 1, 3, 5, 11, 17, 19, 21, 27
Problems: 33(35), 35, 39(45), 43(49), 45(37), 47(39), 51(53), 55(55), 57(57), 61(61), 63(63)
G
22.(33)35. The electric field E1, at one face of a parallelepiped is uniform over the
entire face and is directed out of the face. At the opposite face, the electric field
G
E2 is also uniform over the entire face and is directed into that face (Fig. ).
G
The two faces in question are inclined at 30.00 from the horizontal, while E1,
G
G
G
and E2 are both horizontal; E1 has a magnitude of 2.50x104 N/C, and E2 has a
magnitude of 7.00x104 N/C. (a) Assuming that no other electric field lines cross
the surfaces of the parallelepiped, determine the net charge contained within. (b)
Is the electric field produced ouly by the charges within the parallelepiped, or
is the field also due to charges outside the parallelepiped? How can you tell?
22.35. An insulating sphere with radius 0.120 m has 0.900 nC of charge uniformly distributed throughout its
volume. The center of the sphere is 0.240 m above a large uniform sheet that has charge density -8.00 nC/m2. Find
all points inside the sphere where the electric field is zero. Or, show that there is no such points.
22.39(45). Concentric Spherical Shells. A small conducting spherical shell with inner radius
a and outer radius b is concentric with a larger conducting spherical shell with inner
radius c and outer radius d. The inner shell has total charge +2q, and the outer shell has
charge +4q. (a) 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. Show your results in a graph of the radial component of
G
E as a function of r. (b) What is the total charge on the (i) inner surface of the small shell;
(ii) outer surface of the small shell; (iii) inner surface of the large shell; (iv) outer surface
of the large shell?
22.43(49).Negative charge -Q is distributed uniformly over the surface of a thin spherical insulating shell with radius R.
Calculate the force (magnitude and direction) that the shell exerts on a positive point charge q located (a) a distance r
> R from the center of the shell (outside the shell) and (b) a distance r < R from the center of the shell (inside the
shell).
22.45(37).The Coaxial Cable. A long coaxial cable consists of an inner cylindrical conductor with radius a and an outer
coaxial cylinder 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 . Calculate the electric field
(a) at any point between the cylinders a distance r from the axis and (b) at any point outside the outer cylinder.
(c) Graph 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.
22.47(39). A very long conducting tube (bollow cylinder) has inner radius a and outer radius b. It carries charge per unit
length - , where is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of
charge has charge per unit length + . (a) Calculate the electric field in terms of and the distance r from the axis of
the tube for (i) r < a; (ii) a < r < b; (iii) r > b. Show your results in a graph of E as a function of r. (b) What is the
charge per unitlength on (i) the inner surface of the tube and (ii) the outer surface of the tube?
3
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
22.51(53). Thomson’s Model or the Atom. In Thomson’s model, an atom consisted of a sphere
of positively charged material in which were embedded negatively charged electrons, like
chocolate chips in a ball of cookie dough. Consider an atom consisting of two electrons, each
of charge -e, embedded in a sphere of charge + 2e and radius R. In equilibrium, each electron
is a distance d from the center of the atom (Fig.). Find the distance d in terms of the other
properties of the atom.
22.55(55). A Uniformly Charged Slab. A slab of insulating material has thickness 2d and is oriented so that its faces are
parallel to the yz-plane and given by the planes x = d and x = -d. The y- and z-dimensions of the slab are very large
2
compared to d and may be treated as essentially infinite. The charge density of the slab is given by ρ(x) = ρ (x/d)
,
0
where ρ is0 a positive constant (a) Explain why the electric field due to the slab is zero at the center of the slab (x =
0). (b) Using Gauss’s law, find the electric field due to the slab (magnitude and direction) at all points in space.
22.57(57). A nonuniform, but spherically symmetric, distribution of charge has a charge density ρ(r) given as follows:
where ρ 0 = 3Q/πR3 is a positive constant. (a) Show that the total charge contained in the charge distribution is Q.
(b) Show that the electric field in the region r ≥ R is identical to that produced by a point charge Q at r = 0.
(c) Obtain an expression for the electric field in the region r ≤ R. (d) Graph the electric field magnitude E as a function of r. (e) Find the value of r at which the electric field is maximum, and find the value of that maximum field.
22.61(61). (a) An insulating sphere with radius a has a uniform charge density ρ. The sphere
G G
is not centered at the origin but at r = b. Show that the electric field inside the sphere is
G
G G
given by E = ρ(r - b)/3P0 . (b) An insulating sphere of radius R has a spherical hole of
radius a located within its volume and centered a distance b from the center of the sphere,
where a < b < R (a cross section of the sphere is shown in Fig.). The solid part of the
sphere has a uniform
volume charge density ρ. Find the magnitude and direction of the
G
G
electric field E inside the hole, and show that E is uniform over the entire hole. [Hint: Use
the principle of superposition and the result of part (a).]
22.63(63). Positive charge Q is distributed uniformly over each of two spherical volumes
with radius R. One sphere of charge is centered at the origin and the other at x = 2R
(Fig.). Find the magnitude and direction of the net electric field due to these two distributions of charge at the following points on the x-axis: (a) x = 0; (b) x = R/2; (c) x =
R; (d) x = 3R.
QUESTIONS
1. A point charge is located at the center of a spherical Gauss’ surface. How the flux will be changed if we: (a) replace
the spherical surface by a cube of the same volume with the sphere; (b) shift the point charge from the center of the
sphere; (c) take the charge out of the sphere; (d) place another charge outside the sphere; (e) place another charge
inside the sphere
2. Would Gauss’ law still be valid if Coulomb force between two point charges were not proporttional to square of the
distance between them?
4
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 23: ELECTRIC POTENTIAL
Exercises: 1, 3, 5, 9, 11, 13, 17, 21, 29, 31, 33, 35, 37, 39, 41, 43
Problems: 49(53), 51(55), 55(59), 57(61), 59(63), 61(65), 63(66), 79(79), 81(81), 83(83)
23.49(53). A particle with charge +7.60 nC is in a unifonn electric field directed to the left. Another force, in addition to
the electric force, acts on the particle so that when it is released from rest, it moves to the right. After it has moved
8.00 cm, the additional force has done 6.50 x 10-5 J of work and the particle has 4.35 x 10-5 J of kinetic energy.
(a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end
point? (c) What is the magnitude of the electric field?
23.51(55). A vacuum tube diode consists of concentric cylindrical electrodes, the negative cathode and the positive
anode. Because of the accumulation of charge near the cathode, the electric potential between the electrodes is not a
linear function of the position, even with planar geometry, but is given by V(x) = Cx4/3, where x is the distance from
the cathode and C is a constant, characteristic of a particular diode and operating conditions. Assume that the distance
between the cathode and anode is 13.0 mm and the potential difference between electrodes is 240 V. (a) Determine
the value of C. (b) Obtain a fonnula for the electric field between the electrodes as a function of x. (c) Determine the
force on an electron when the electron is halfway between the electrodes.
23.55(59). The H2+ Ion. The H2+ ion is composed of two protons, each of charge +e = 1.60 x 10-19 C, and an electron of
charge -e and mass 9.11 x 10-31 kg. The separation between the protons is 1.07 x 10-10 m. The protons and the electron may be treated as point charges. (a) Suppose the electron is located at the point midway between the two protons. What is the potential energy of the interaction between the electron and the two protons? (Do not include the
potential energy due to the interaction between the two protons.) (b) Suppose the electron in part (a) has a velocity
of magnitude 1.50 x 106 m/s in a direction along the perpendicular bisector of the line connecting the two protons.
How far from the point midway between the two protons can the electron move? Because the masses of the protons
are much greater than the electron mass, the motions of the protons are very slow and can be ignored. (Note: A
realistic description of the electron motion requires the use of quantum mechanics, not Newtonian mechanics.)
23.57(61).Coaxial Cylinders. A long metal cylinder with radius a is supported on an insulating stand on the axis of a
long, hollow, metal tube with radius b. The positive charge per unit length on the inner cylinder is λ. and there is an
equal negative charge per unit length on the outer cylinder. (a) Calculate the potential V(r) for (i) r < a; (ii) a < r
< b; (iii) r > b. (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take V = 0 at r
= b. (b) Show that the potential of the inner cylinder with respect to the outer is Vab = Va − Vb = (λ /2π P0 )ln(b /a).
(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has
Vab 1
magnitude E =
ln( b a ) r
(d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?
21.59(63).Deflection in a CRT. Cathode-ray tubes (CRTs) are often found in oscilloscopes and computer monitors. In
Fig. an electron with an initial speed of 6.50 x 106 m/s is projected along the axis midway between the deflection
plates of a cathode-ray tube. The wriform electric field between the plates has a magnitude of 1.10 x 103 V/m and
is upward. (a) What is the force (magnitude and direction) on the electron when it is between the plates? (b) What
is the acceleration of the electron (magnitude and direction) when acted on by the force in part (a)? (c) How far
below the axis has the electron moved when it reaches the end of the plates? (d) At what angle with the axis is it
moving as it leaves the plates? (e) How far below the axis will it strike the fluorescent screen S?
5
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
23.61(65). Electrostatic precipitators use electric forces to remove pollutant particles from smoke, in particular in the smokestacks of coal-burning power plants.
One form of precipitator consists of a vertical, hollow, metal cylinder with a thin
wire, insulated from the cylinder, running along its axis (Fig.). A large potential
difference is established between the wire and the outer cylinder, with the wire
at lower potential. This sets up a strong radial electric field directed inward. The
field produces a region of ionized air near the wire. Smoke enters the precipitator at the bottom, ash and dust in it pick up electrons, and the charged pollutants
are accelerated toward the outer cylinder wall by the electric field. Suppose the
radius of the central wire is 90.0 m, the radius of the cylinder is 14.0 cm, and a
potential difference of 50.0 kV is estabished betWeen the wire and the cylinder.
Also assume that the wire and cylinder are both very long in comparison to the
cylinder radius. so the results of Problem 23.61 apply. (a) What is the magnitude
of the electric field midway between the wire and the cylinder wall? (b) What
magnitude of charge must a 30.0-/Lg ash particle have if the electric field computed in part (a) is to exert a force ten times the weight of the particle?
23.63(66). A disk with radius R has uniform surface charge density σ. (a) By regarding the disk as a series of thin concentric rings, calculate the electric potential V at a point on the disk’s axis a distance x from the center of the disk.
Assume that the potential is zero at infinity. (Hint: Use the result of Example 23.11 in Section 23.3.) (b) Calculate
-∂V/∂x. Show that the result agrees with the expression for Ex calculated in Example 21.12 (Section 21.5).
23.79(79). Electric charge is distributed uniformly along a thin rod of length
a, with total charge Q. Take the potential to be zero at infinity. Find the
potential at the following points (Fig.): (a) point P, a distance x to the
right of the rod, and (b) point R, a distance y above the right-hand end of
the rod. (c) In parts (a) and (b), what does your result reduce to as x or y
becomes much larger than a?
23.81(81). Two metal spheres of different sizes are charged such that the electric potential is the same at the surface of
each. Sphere A has a radius three times that of sphere B. Let QA and QB be the charges on the two spheres, and let
EA and EB be the electric-field magnitudes at the surfaces of the two spheres. What are (a) the ratio QB /QA and
(b) the ratio EB/EA ?
23.83(83). A metal sphere with radius R1 has a charge Q1. Take the electric potemial to be zero at an infinite distance
from the sphere. (a) What are the electric field and electric potential at the surface of the sphere? This sphere is now
connected by a long, thin conducting wire to another sphere of radius R2 that is several meters from the first sphere.
Before the connection is made. this second sphere is uncharged. After electrostatic equilibrium has been reached,
what are (b) the total charge on each sphere; (c) the electric potential at the surface of each sphere (d) the electric
field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on
each sphere.
QUESTIONS
1. What are equipotential (isopotential) surfaces of: (a) a uniform electric field (b) spherical charge distribution (c) an
infinite charged rod?
2. What is electric potential inside a conductor?
3. How does electric field look like (its intensity vector or lines of flux) in a vinicity of an isopotential surface?
6
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 24: CAPACITANCE AND DIELECTRICS
Exercises: 3, 11, 13, 15, 17, 23, 25, 29, 35, 37, 39, 41, 45, 49,
Problems: 51(51), 57, 59(59), 61(61), 63(63), 65(65), 67(67), 69, 71(71)
24.51(51). A parallel-plate air capacitor is made by using two plates 16 cm square, spaced 9.4 mm apart. It is connected
to a l2-V battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field
between the plates? (d) What is the energy stored in the capacitor?
24.57. A 4.00-μF and a 6.00-μF capacitors are connected in parallel across a 660-Vsupply 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 with terminals of unlike sign together. Find the final charge on each and
the voltage across each.
24.59(59). In Fig. below C1 = C5 =8.4 μF and C2 = C3 = C4 = 4.2 μF. The
applied potential is Vab = 220 V. (a) What is the equivalent capacitance
of the network between points a and b? (b) Calculate the charge on
each capacitor and the potential difference across each capacitor.
24.61(61). Three capacitors having capacitances of 8.4, 8.4, and 4.2 μF are connected in series across a 36-V potential
difference. (a) What is the charge on the 4.2-μF capacitor? (b) What is the total energy stored in all three capacitors?
(c) The capacitors are disconnected from the potential difference without allowing them to discharge. They are then
reconnected in parallel with each other. with the positively charged plates connected together. What is the voltage
across each capacitor in the parallel combination? (d) What is the total energy now stored in the capacitors?
24.63(63). In Fig. below each capacitance C1 is 6.9 μF, and each
capacitance C2 is 4.6 μF. (a) Compute the equivalent capacitance
of the network between points a and b. (b) Compute the charge
on each of the three capacitors nearest a and b when Vab = 420V.
(c) With 420V across a and b, compute Vcd.
24.65(65). A parallel-plate capacitor with only air between the plates is charged by connecting it to a battery. The
capacitor is then disconnected from the battery, without any of the charge leaving the plates. (a) A voltmeter reads
45.0 V when placed across the capacitor. When a dielectric is inserted between the plates, completely filling the
space, the voltmeter reads 11.5 V. What is the dielectric constant of this material? (b) What will the voltmeter read
if the dielectric is now pulled partway out so it fills only one-third of the space between the plates?
24.47(47). Capacitance of the Earth. (a) Discuss how the concept of capacitance can also be applied to a single conductor. (Hint: In the relationship C = Q/Vab, think of the second conductor as being located at infinity.) (b) Use Eq.
(24.1) to show that C = 4πP0R for a solid conducting sphere of radius R. (c) Use your result in part (b) to calculate
the capacitance of the earth, which is a good conductor of radius 6380 km. Compare to typical capacitors used in
electronic circuits that have capacitances ranging from 10 pF to 100 μF.
24.69. A uniformly charged sphere of radius R has total charge Q. Calculate the electric-field energy density at a point a
distance r from the center of the sphere for (a) r < R (b) r > R (c) Calculate the total electric-field energy.
24.71(71). A parallel-plate capacitor has the space between the plates filled with two slabs of dielectric, one with constant K1, and one with constant K2 (Fig.). Each slab has thickness d/2, where d is the plate separation. Show that the
2P0 A ⎛ K1K 2 ⎞
capacitance is
C=
⎜
⎟.
d ⎝ K1 + K 2 ⎠
7
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 25: CURRENT, RESISTANCE, AND ELECTROMOTIVE FORCE
Exercises: 3, 5, 9, 15, 21, 23, 25, 27, 35, 37, 41, 45, 47, 49
Problems: 53 (57), 55(59), 59(63), 61(65), 65(69), 69(73), 71(75), 75(79), 77(83)
25.53(57). An electrical conductor designed to carry large currents has a circular cross section 2.50 mm in diameter
and is 14.0 m long. The resistance between its ends is 0.104 Ω . (a) What is the resistivity of the material? (b) If
the electric-field magnitude in the conductoris 1.28 V/m. what is the total current? (c) If the material has 8.5 x 1028
free electrons per cubic meter, find the average drift speed under the conditions of part (b).
25.55(59). On your first day at work as an elec1rical technician, you are asked to determine the resistance per meter
of a long piece of wire. The company you work for is poorly equipped. You find a battery, a voltmeter, and an
ammeter, but no meter for directly measuring resistance (an ohmmeter). You put the leads from the voltmeter
across the terminals of the battery, and the meter reads 12.6 V. You cut off a 20.0-m length of wire and connect
it to the battery, with an ammeter in series with it to measure the current in the wire. The ammeter reads 7.00 A.
You then cut off a 40.0-m length of wire and connect it to the battery, again with the ammeter in series to
measure the current. The ammeter reads 4.20 A. Even though the equipment you have available to you is limited,
your boss assures you of its high quality: The ammeter has very small resistance, and the voltmeter has very
large resistance. What is the resistance of one meter of wire?
25.59(63). A material of resistivity ρ is formed into a solid, truncated cone of height h and radii
r1 and r2 at either end (Fig.). (a) Calculate the resistance of the cone between the two flat
end faces. (Hint: Imagine slicing the cone into very many thin disks, and calculate the resistance of one such disk.) (b) Show that your result agrees with Eq. (25.10) when r1 = r2.
25.61(65). Leakage in a Dielectric. Two parallel plates of a capacitor have equal and opposite charges Q. The
dielectric has a dielectric constant K and a resistivity ρ. Show that the “leakage” current I carried by the
dielectric is given by I = Q/KP0ρ.
25.65(69). The potential difference across the tenninals of a battery is 8.4 V when there is a current of 1.50 A in the
battery from the negative to the positive tenninal. When the current is 3.50 A in the reverse direction, the
potential difference becomes 9.4 V. (a) What is the internal resistance of the battery? (b) What is the emf of the
battery?
25.69(73). A 12.6-V car battery with negligible internal resistance is connected to a series combination of a 3.2-Ω
resistor that obeys Ohm’s law and a thermistor that does not obey Ohm’s law but instead has a current-voltage
relationship V = αI + βI2, with α = 3.8 Ω and β = 1.3 Ω / A. What is the current through the 3.2-Ω resistor?
25.71(75). A 12.6-V car battery with negligible internal resistance is connected to a series combination of a 3.2-Ω
resistor that obeys Ohm’s law and a thermistor that does not obey Ohm’s law but instead has a current-voltage
relationship V = αI + βI2, with α = 3.8 Ω and β = 1.3 Ω / A. What is the current through the 3.2-Ω resistor?
8
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
25.75(79). In the circuit of Fig. below, find (a) the current through the 8.0-Ω resistor and (b) the total rate of
dissipation of electrical energy in the 8.0-Ω resistor and in the internal resistance of the batteries. (c) In one of
the batteries, chemical energy is being converted into electrical energy. In which one is this happening, and at
what rate? (d) In one of the batteries, electrical energy is being converted into chemical energy . In which one
is this happening, and at what rate? (e) Show that the overall rate of production of electrical energy equals the
overall rate of consumption of electrical energy in the circuit.
25.77(83). The Tolman-Stewart experiment in 1916 demonstrated that the free charges in a metal have negative
charge and provided a quantitative measurement of their charge-to-mass ratio,|q|/m. The experiment consisted
of abruptly stopping a rapidly rotating spool of wire and measuring the potential di fference that this produced
between the ends of the wire. In a simplified model of this experiment, consider a metal rod of length L that is
given a uniform acceleration a to the right. Initially the free charges in the metal lag behind the rod’s motion, thus
setting up an electric field E in the rod. In the steady state this field exerts a force on the free charges that makes
them accelerate along with the rod. (a) Apply ΣF = ma to the free charges to obtain an expression for |q|/m in terms
of the magnitudes of the induced electric field E and the accelemtion a. (b) If all the free charges in the metal rod
have the same acceleration, the electric field E is the same at all points in the rod. Use this fact to rewrite the
expression for |q|/m in terms of the potential V be between the ends of the rod (Fig. below). (c) If the free charges
have negative charge, which end of the rod, b or c, is at higher potential? (d) If the rod is 0.50 m long and the free
charges are electrons (charge q = -1.60x10-19 C, mass 9.11x10-31 kg), what magnitude of acceleration is required
to produce a potential difference of 1.0 mV between the ends of the rod? (e) Discuss why the actual
experiment used a rotating spool of thin wire rather than a moving bar as in our simplified analysis.
QUESTIONS
1. Sketch Current-Voltage Relationships for: (a) a semiconductor diode; (b) a dielectric; (c) a superconductor; (d) a
phototube (photocell)? Qualitatively describe your sketches.
2. What’s the origin of resistance? Explain why resistance of almost conductors increases when they’re heated up?
3. The expression for power P = I2R implies that the heat radiated from a conductor varies proportionally to its
resistance. On the other side, the other expression P = U2/R shows an inverted relationship between power and
resistance. How could you resolve this paradox?
4. What force drives charge inside: (a) a battery; (b) a generator; (c) a photocell?
9
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 27: MAGNETIC FIELD AND MAGNETIC FORCES
Exercises: 1, 3, 11, 13, 15, 19, 21, 25, 31, 33, 37, 39, 47, 49, 51
Problems: 53(53), 57(57), 61(61), 65, 67(67), 71(77), 75(75), 79(79), 83(83), 85(85)
!
27.53(53). When a particle of charge q > 0 moves with a velocity of v1 at 45.0° the +x-axis in the xy-plane, a uniform
!
!
magnetic field exerts a force F1 along the -z-axis (Fig. below). When the same particle moves with a velocity v2
!
!
with the same magnitude as v1 but along the +z-axis. a force F2 of magnitude F2 is exerted on it along the +x-axis.
(a) What are the magnitude (in terms of q, v1 and F2) and direction of the magnetic field? (b) What is the magni!
tude of F1 in terms of F2?
27.57(57). The magnetic poles of a small cyclotron produce a magnetic field with magnitude 0.85 T. The poles
have a radius of 0.40 m, which is the maximum radius of the orbits of the accelerated particles. (a) What is the
maximum energy to which protons (q = 1.60x10-19 C, m = 1.67x10-27 kg) can be accelerated by this cyclotron?
Give your answer in electron volts and in joules. (b) What is the time for one revolution of a proton orbiting
at this maximwn radius? (c) What would the magnetic-field magnitude have to be for the maximum energy
to which a proton can be accelerated to be twice that calculated in part (a)? (d) For B = 0.85 T. what is the maximum energy to which alpha particles (q = 3.20x10-19 C, m = 6.65x10-27 kg) can be accelerated by this cyclotron?
How does this compare to the maximwn energy for protons?
27.61(61). A particle with negative charge q and mass m = 2.58x10-15 kg is traveling through a region containing a
!
!
!
uniform magnetic field
B
=
(0.120
T)k.
At a particular instant of time the velocity of the particle is v = (1.05x106
!
!
!
!
m/s)( -3i + 4j + 12k) and the force F on the particle has a magnitude of 1.25 N. (a) Determine the charge q. (b)
!
Determine the acceleration a of the particle. (c) Explain why the path of the particle is a helix, and determine
the radius of curvature R of the circular component of the helical path. (d) Determine the cyclotron frequency of the
particle. (e) Although helical motion is not periodic in the full sense of the word, the x- and y-coordinates do
vary in a periodic way. If the coordinates of the particle at t = 0 are (x, y, z) = (R, 0, 0), determine its coordinates
at a time t = 2T, where T is the period of the motion in the xy-plane.
10
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
27.65. The cube in the fig., 75.0 cm on a side, is in a uniform magnetic field of 0.860 T parallel to the x-axis.
The wire abcdef carries a current of 6.58 A in the direction indicated. (a) Determine the magnitude and
direction of the force acting on each of the segments ab, bc, cd, de and ef. (b) What are the magnitude nd
direction of the total force on the wire?
y
c
b
I
f
B
e
a
z
x
d
27.67 (67). A straight piece of conducting wire with mass M and length L is
placed on a frictionless incline tilted at an angle θ from the horizontal (Fig.).
There is a uniform, vertical magnetic field B at all points (produced by an
arrangement of magnets not shown in the figure). To keep the wire from
sliding down the incline, a voltage source is attached to the ends of the
wire. When just the right amount of current flows through the wire, the wire
remains at rest. Determine the magnitude and direction of the current in the
wire that will cause the wire to remain at rest. Copy the figure and draw the
direction of the current on your copy. In addition, show in a free-body diagram all the forces that act on the wire.
27.71(77). A thin, uniform rod with negligible mass and length 0.200 m is attached to the floor by a frictionless
hinge at point P (Fig.). A horizontal spring with force constant k = 4.80 N/m connects the other end of
the rod to a vertical wall. The rod is in a uniform magnetic field B = 0.340 T directed into the plane
of the figure. There is current I = 6.50 A in the rod, in the direction shown. (a) Calculate the torque
due to the force on the rod, for an axis at P. Is it correct to take the total magnetic force to act at the
center of gravity of the rod when calculating the torque? Explain. (b) When the rod is in equilibrium
and makes an angle of 53.00 with the floor, is the spring stretched or compressed? (c) How much energy is
stored in the spring when the rod is in equilibrium?
11
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
27.75(75). The rectangular loop of wire shown in Fig. below has a
mass of 0.15 g per centimeter of length and is pivoted about side
ab on a frictionless axis. The current in the wire is 8.2 A in the
direction shown. Find the magnitude and direction of the magnetic
field parallel to the y-axis that will cause the loop to swing up
until its plane makes an angle of 30.00 with the yz-plane.
27.79(79). A Voice Coil. It was shown in Section 27.7 that the net force on a current loop in a uniform magnetic
field is zero. The magnetic force on the voice coil of a loudspeaker is nonzero because the magnetic field at
the coil is not uniform. A voice coil in a loudspeaker has 50 turns of wire and a diameter of 1.56 cm,
and the current in the coil is 0.950 A. Assume that the magnetic field at each point of the coil has a
constant magnitude of 0.220 T and is directed at an angle of 60.00 outward from the normal to the plane
of the coil (Fig. below). Let the axis of the coil be in the y-direction. The current in the coil is in the
direction shown (counterclockwise as viewed from a point above the coil on the y-axis). Calculate the
magnitude and direction of the net magnetic force on the coil
27.83(83). An insulated wire with mass m = 5.40x10-5 kg is bent into the shape of an inverted U such that
the horizontal part has a length l = 15.0 cm. The bent ends of the wire are partially immersed in two
pools of mercury, with 2.5 cm of each end below the mercury’s surface. The entire structure is in a
region containing a uniform 0.00650-T magnetic field directed into the page (Fig. below). An electrical
connection from the mercury pools is made through the ends of the wires. The mercury pools are connected to a 1.50-V battery and a switch S. When switch S is closed, the wire jumps 35.0 cm into the
air, measured from its initial position. (a) Determine the speed v of the wire as it leaves the mercury.
(b) Assuming that the current I through the wire was constant from the time the switch was closed until
the wire left the mercury, determine I. (c) Ignoring the resistance of the mercury and the circuit wires,
determine the resistance of the moving wire.
12
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
27.85(85). A circular loop of wire with area A lies in the xy-plane. As viewed along the z-axis looking in
the -z-direction toward the origin, a current I is circulating clockwise around the loop. The torque
!
!
!
produced by an external magnetic field B is given by τ = D(4i - 3j). where D is a positive constant,
!!
and for this orientation of the loop the magnetic potential energy U = -μ.B is negative. The magnitude
of the magnetic field is B0 = 13D/IA. (a) Determine the vector magnetic moment of the current loop.
!
(b) Determine the components Bx, By. and Bz of B.
QUESTIONS
1. What common features do magnetic and electric fields share? What are distinctions between them?
2. State a Gauss’ law for magnetic field.
3. You’re handling a magnet with its poles coverd by Red and Blue paints. You decide to divine the magnet into
two pieces of single colors and place them far from each other. Draw magnetic field lines before and after
division.
4. A charged particle can be deflected either by an electric either by a magnetic field. Which way is better? Why?
13
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 28: SOURCES OF MAGNETIC FIELD
Exercises: 1, 5, 9, 11, 15, 17, 19, 21, 25, 29, 33, 35, 41
Problems: 47(52), 49(54), 53(58), 55( 59), 61(64), 69(71), 73(75), 75(77), 79(81), 81(83)
28.47(52). A long, straight wire carries a current of 2.50 A. An electron is traveling in the vicinity of the wire. At the
instant when the electron is 4.50 cm from the wire and traveling with a speed of 6.00x104 m/s directly toward
the wire, what are the magnitude and direction (relative to the direction of the current) of the force that the
magnetic field of the current exerts on the electron?
28.49(54). In Fig. the battery branch of the circnit is very far from the two
horizontal segments containing two resistors. These horizontal
segments are separated by 5.00 cm, and they are much longer than
5.00 cm. A proton (charge +e) is fired at 650 km/s from a point
midway between the upper two horizontal segments of the circuit.
The initial velocity of the proton is in the plane of the circuit and
is directed toward the upper wire. Find the magnitude and direction
of the initial magnetic force on the proton.
!
28.53(58). A neophyte magnet designer tells you that he can produce a magnetic field B in vacuum that points
!
!
everywhere in the x-direction and that increases in magnitude with increasing x. That is, B = B0(x/a)i, where
B0 and a are constants with units of teslas and meters, respectively. Use Gauss’s law for magnetic fields to
show that this claim is impossible. (Hint: Use a Gaussian surface in the shape of a rectangular box, with edges
parallel to the x-, y-, and z-axes.)
28.55(59). Two long, straight, parallel wires are 1.00 m apart (Fig. below). The
wire on the left carries a current I1 of 6.00 A into the plane of the paper.
(a) What must the magnitude and direction of the current I2 be for the net
field at point P to be zero? (b) Then what are the magnitude and direction of
the net field at Q? (c) Then what is the magnitude of the net field at S?
28.61(64). The long, straight wire AB shown in Fig. carries a
current of 14.0 A. The rectangular loop whose long edges are
parallel to the wire carries a current of 5.00 A. Find the
magnitude and direction of the net force exerted on the loop by
the magnetic field of the wire.
28.69(71). A long, straight wire with a circular cross section of radius R carries a current I. Assume that the
current density is not constant across the cross section of the wire, but rather varies as J = α.r, where α is
a constant. (a) By the requirement that J integrated over the cross section of the wire gives the total current
I, calculate the constant α in terms of I and R. (b) Use Ampere’s law to calculate the magnetic field B(r)
for (i) r R and (ii) r R. Express your answers in terms of I.
14
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
28.73(75). Knowing Magnetic Fields Inside and Out. You are given a hollow copper cylinder with inner radius
a and outer radius 3a. The cylinder’s length is 200a and its electrical resistance to current flowing down
its length is R. To test its suitability for use in a circuit, you connect the ends of the cylinder to a
voltage source, causing a current I to flow down the length of the cylinder. The current is spread
uniformly over the cylinder’s cross section. You are interested in knowing the strength of the magnetic field
that the current produces within the solid part of the cylinder, at a radius 2a from the cylinder axis. But
since it’s not easy to insert a magnetic-field probe into the solid metal, you decide instead to measure
the field at a point outside the cylinder where the field should be as strong as at radius 2a. At what
distance from the axis of the cylinder should you place the probe?
28.75(77). A long, straight,
solid cylinder, oriented with its axis in the z-direction, carries a current whose
!
current density is J. The current density, although symmetrical about the cylinder axis, is not constant
but varies according to the relationship
where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and I0 is a constant having
units of amperes. (a) Show that I0 is the total current passing through the entire cross section of the wire. (b) Using
!
Ampere’s law, derive an expression for the magnitude of the magnetic field B in the region r a. (c) Obtain an
expression for the current I contained in a circular cross section of radius r a and centered at the cylinder axis.
!
(d) Using Ampere’s law, derive an expression for the magnitude of the magnetic field B in the region r a. How
do your results in parts (b) and (d) compare for r = a?
28.79(81). An Infinite Current Sheet. Long, straight conductors with square
cross sections and each carrying current I are laid side by side to
form an infinite current sheet (Fig.). The conductors lie in the xy-plane,
are parallel to the y-axis, and carry current in the +y-direction There
are n conductors per unit length measured along the x-axis. (a) What
are the magnitude and direction of the magnetic field a distance a
below the current sheet? (b) What are the magnitude and direction of
the magnetic field a distance a above the current sheet?
*28.81(83). A piece of iron has magnetization M = 6.50x104 A/m. Find the average magnetic dipole moment
per atom in this piece of iron. Express your answer both in A.m2 and in Bohr magnetons. The density
of iron is given in Table 14.1. and the atomic mass of iron (in grams per mole) is given in Appendix
D. The chemical symbol for iron is Fe.
QUESTIONS
1. The expression B = μ 0I/2πr shows that the magnetic field in a vicinity of a wire is extremely strong. Applying
Laplace’s law one might come to a conclusion that the magnetic field would exert a very strong force on the wire.
Then, why the wire is still at equilibrium?
2. Will the formula B = μ 0In still valid for a solenoid with SQUARE cross section instead of a circular one?
3. Design an experimental setup to measure the magnetic moment of a compass needle, using as less as possible
equipments.
15
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 29: ELECTROMAGNETIC INDUCTION
Exercises: 1, 3, 5, 9, 17, 19, 23, 25, 27, 29, 35, 37, 39
Problems: 43(44), 45, 49(49), 55(55), 61(61), 63(63), 65(65), 67(67), 69(70), 71(72)
29.43(44). A Changing Magnetic Field. You are testing a new data-acquisition system. This system allows you to
record a graph of the current in a circuit as a function of time. As part of the test, you are using a circuit made up
of a 4.00-cm-radius, 500-turn coil of copper wire connected in series to a 600-Ω resistor. Copper has resistivity
1.72xl0-8 Ω.m, and the wire used for the coil has diameter 0.0300 mm. You place the coil on a table that is
tilted 30.0° from the horizontal and that lies between the poles of an electromagnet. The electromagnet generates
a vertically upward magnetic field that is zero for t < 0, equal to (0.120 T)(l - cosπt) for 0 ≤ t ≤ 1.00 s, and
equal to 0.240 T for t > 1.00 s. (a) Draw the graph that should be produced by your data-acquisition system.
(This is a full-featured system, so the graph will include labels and numerical values on its axes.) (b) If you
were looking vertically downward at the coil, would the current be flowing clockwise or counterclockwise?
29.45. A cicular coil of wire had a radius of 0.500 m, 20 turns, and a total resistance of 1.57 Ω. The coil lies in the xy!
plane. The coil is in a uniform megnetic field B that is in the -z-direction, which is directed away from you as you
view the coil. The magnitude B of the field depends on timeasfollows: it increases at a constant rate form 0 at t =
0 to 0.800 T at t = 0.500 s; is constant at 0.800 T from t = 0.500 s to t = 1.00 s; decreases at a constant rate from
0.800 T at t = 1.00 s to 0 at t = 2.00 s. (a) Graph B versus t for t from 0 to 2.00 s. (b) Graph the current I induced
in the coil versus t for t from 0 to 2.00 s. Let counterclockwise currents be positive and clockwise currents be
negative. (c) What is the maximum induced electric field magnitude incoil during 0- to 2.00-s time interval?
29.49(49). In Fig. the loop is being pulled to the right at constant speed v. A
constant current I flows in the long wire, in the direction shown. (a) Calculate
the magnitude of the net emf E induced in the loop. Do this two ways: (i) by
using Faraday’s law of induction and (ii) by looking at the emf induced in
each segment of the loop due to its motion. (b) Find the direction (clockwise
or counterclockwise) of the current induced in the loop. Do this two ways:
(i) using Lenz’s law and (ii) using the magnetic force on charges in the loop.
(c) Check your answer for the emf in part (a) in the following special cases to
see whether it is physically reasonable: (i) The loop is stationary; (ii) the loop
is very thin, so a → 0; (iii) the loop gets very far from the wire.
29.55(55). Terminal Speed. A conducting rod with length L, mass m, and resistance R moves without friction on
!
metal rails as shown in Fig. below. A uniform magnetic field B is directed into the plane of the figure.
!
The rod starts from rest and is acted on by a constant force F directed to the right. The rails are infinitely long
and have negligible resistance. (a) Graph the speed of the rod as a function of time. (b) Find an expression for
the terminal speed (the speed when the acceleration of the rod is zero).
16
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
29.61(61). The long, straight wire shown in Fig. below carries constant current I. A metal bar with length L is moving
!
at constant velocity v, as shown in the figure. Point a is a distance d from the wire. (a) Calculate the emf induced
in the bar. (b) Which point, a or b, is at higher potential? (c) If the bar is replaced by a rectangular wire loop of
resistance R, what is the magnitude of the current induced in the loop?
29.63(63). A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad/s about an axis through one end
and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a
magnitude of 0.650 T. (a) What is the induced emf in the rod? (b) What is the potential difference between its
ends? (c) Suppose instead the rod rotates at 8.80 rad/s about an axis through its center and perpendicular to the
rod. In this case, what is the potential difference between the ends of the rod? Between the center of the rod and
one end?
29.65(65). A rectangular loop with width L and a slide wire with mass m are as shown in Fig. below. A uniform
!
magnetic field B is directed perpendicular to the plane of the loop into the plane of the figure. The slide wire is
given an initial speed of v0 and then released. There is no friction between the slide wire and the loop, and the
resistance of the loop is negligible in comparison to the resistance R of the slide wire. (a) Obtain an expression
for F, the magnitude of the force exerted on the wire while it is moving at speed v. (b) Show that the distance
x that the wire moves before coming to rest is x = mv0R/a2B2.
!
29.67(67). The magnetic field B, at all points within a circular region of radius
R, is uniform in space and directed into the plane of the page as shown in
Fig. (The region could be a cross section inside the windings of a long,
straight solenoid.) If the magnetic field is increasing at a rate dB/dt, what
are the magnitude and direction of the force on a stationary positive point
charge q located at points a, b, and c? (Point a is a distance r above the
center of the region, point b is a distance r to the right of the center, and
point c is at the center of the region.
17
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
29.69(70). Falling Square Loop. A vertically oriented, square loop of copper wire falls from a region where the
!
field B is horizontal, uniform, and perpendicular to the plane of the loop, into a region where the field is zero.
The loop is released from rest and initially is entirely within the magnetic-field region. Let the side length of
the loop be s and let the diameter of the wire be d. The resistivity of copper is ρ R and the density of copper
is ρ m. If the loop reaches its terminal speed while its upper segment is still in the magnetic-field region, find
an expression for the terminal speed.
29.71(72). A capacitor has two parallel plates with area A separated by a distance d. The space between plates is
filled with a material having dielectric constant K. The material is not a perfect insulator but has resistivity
ρ. The capacitor is initially charged with charge of magnitude Q0 on each plate that gradually discharges
by conduction through the dielectric. (a) Calculate the conduction current density jC(t) in the dielectric.
(b) Show that at any instant the displacement current density in the dielectric is equal in magnitude to
the conduction current density but opposite in direction, so the total current density is zero at every instant.
QUESTION:
1. A conducting sheet is at rest in a magnetic field. If one tried to pull the sheet out, or, tried to push it deeper into the
magnetic field, then there would arise a force against the motion of sheet. What’s the origin og that force?
2. What principe in chemistry is equivalent to Lenz’ rule ins physics?
3. When a magnetic field varies in time, an electric field is generated. What are distinctions between this electric field
and the one, generated by a system of fixed charges.
4. Describe the levitating-frog experiment, proposed by the Nobel prize laureat A. Geim, that was awarded “craziest
invention” in early 2000s.
18
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 30: INDUCTANCE
Exercises: 1, 3, 9, 13, 17, 19, 23, 29, 33, 41
Problems: 47, 51(49), 55(54), 57(55), 59(57), 63(61), 65(63), 69(67), 71(69), 73(71), 75(75)
30.47. The current in a coil of wire is initially zero but increases at a constant
rate; after 12.0 s it is 48.0 A. The changing current induces an emf of magnitude 30.0 V. (a) Determine the
inductance of the coil. (b) Determine the total magnetic flux through the coil when the current is 48.0 A.
(c) If the resistance of the coil is 60.0 Ω, determine the ratio of the rate at which energy is being stored in
the magnetic field to the rate at which electrical energy is being dissipated by the resistance at the instant
when the current is 48.0 A.
30.51(49). A Coaxial Cable. A small solid conductor with radius a is supported by insulating, nonmagnetic
disks on the axis of a thin-walled tube with inner radius b. The inner and outer conductors carry
equal currents i in opposite directions. (a) Use Ampere’s law to find the magnetic field at any point
in the volume between the conductors. (b) Use the energy density for a magnetic field to calculate
the energy stored in a thin, cylindrical shell between the two conductors. Let the cylindrical shell
have inner radius r, outer radius r + dr, and length l. (c) integrate your result in part (b) over the
volume between the two conductors to find the total energy stored in the magnetic field for a length
l of the cable. (d) Use your result in part (c) to calculate the inductance L of a length l of the cable.
Compare your result to L calculated in part (d) of Problem 30.50.
30.55(54). In Fig., suppose that E = 60.0 V, R = 240 Ω , and L =
0.160 H. With switch S 2 open, switch S1 is left closed until a
constant current is established. Then S 2 is closed and S 1
opened, taking the battery out of the circuit. (a) What is the
total energy initially stored in the inductor? (b) At t =
4.00x10 -4 s, at what rate is the energy stored in the
inductor decreasing? (c) At t = 4.00x10 -4 s, at what rate is
electrical energy being converted into thermal energy in the
resistor? (d) Obtain an expression for the rate at which
electrical energy is being converted into thermal energy in
the resistor as a function of time. Integrate this expression
from t = 0 to t =
to obtain the total electrical energy
dissipated in the resistor. Compare your result to that of part (a).
30.57(55). The equation -iR - Ldi/dt - q/C = 0 may be converted into an energy relationship. Multiply
both sides of this equation by -i = -dq/dt. The first term then becomes i2 R. Show that the second
term can be written as d(Li 2 / 2 )/dt. and that the third term can be written as d(q 2 /2C)/dt. What does
the resulting equation say about energy conservation in the circuit?
30.59(57). An Electromagnetic Car Alarm. Your latest invention is a car alarm that produces sound
at a particularly annoying frequency of 3500 Hz. To do this. the car-alarm circuitry must produce
an alternating electric current of the same frequency. That’s why your design includes an
inductor and a capacitor in series. The maximum voltage across the capacitor is to be 12.0 V
(the same voltage as the car battery). To produce a sufficiently loud sound, the capacitor must
store 0.0160 J of energy. What values of capacitance and inductance should you choose for your caralarm circuit?
19
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
30.63(61). In the lab, you are trying to find the inductance and internal resistance of a solenoid. You
place it in series with a battery of negligible internal resistance, a 10.0- Ω resistor, and a switch.
You then put an oscilloscope across one of these circuit elements to measure the voltage across
that circuit element as a function of time. You close the switch, and the oscilloscope shows
voltage versus time as shown in Fig. below. (a) Across which circuit element (solenoid or
resistor) is the oscilloscope connected? How do you know this? (b) Why doesn’t the graph
approach zero as t → ∞ ? (c) What is the emf of the battery? (d) Find the maximum current in
the citcuit. (e) What are the internal resistance and self-inductance of the solenoid?
30.65(63). In the circuit shown in Fig. below, switch S is closed at time t = 0 with no charge
initially on the capacitor. (a) Find the reading of each ammeter and each voltmeter just after
S is closed. (b) Find the reading of each meter after a long time has elapsed. (c) Find the
maximum charge on the capacitor. (d) Draw a qualitative graph of the reading of voltmeter
V2 as a function of time.
30.69(67). In the circuit shown in Fig. E = 60.0 V, R 1 = 40.0 Ω,
R 2 = 25.0 Ω, and L = 0.300 H. Switch S is closed at t =
0. Just after the switch is closed, (a) what is the potential
difference V ab across the resistor R 1; (b) which point, a or b,
is at a higher potential; (c) what is the potential difference
V cd across the inductor L; (d) which point, c or d, is at a
higher potential? The switch is left closed a long time and
then opened. Just after the switch is opened, (e) what is the
potential difference V ab . across the resistor R 1; (f) which point, a or b, is at a higher potential;
(g) what is the potential difference V cd across the inductor L? (h) which point, c or d, is at
a higher potential?
20
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
30.71(69). Consider the circuit shown in Fig. Let E = 36.0 V, R0 = 50.0 Ω, and
L = 4.00 H. (a) Switch S1 is closed and switch S2 is left open. Just after
S1 is closed, what are the current i0 through R0 and the potential
differences Vac and Vcb? (b) After S1 has been closed a long time (S2 is
still open) so that the current has reached its final, steady value, what
are i0,Vac and Vcb? (c) Find the expressions for i0,Vac and Vcb as functions of the
time t since S1 was closed. Your results should agree with part (a) when
t = 0 and with part (b) when t → ∞ . Graph i0,Vac and Vcb versus time.
30.73(71). In the circuit shown in Fig., the switch has been open for a long
time and is suddenly closed. Neither the battery nor the inductors have
any appreciable resistance. Review the results of Problem 30.49. (a)
What do the ammeter and voltmeter read just after S is closed? (b)
What do the ammeter and the voltmeter read after S has been closed
a very long time? (c) What do the S ammeter and the voltmeter read
0.115 ms after S is closed?
30.75(75). Demonstrating Inductance. A common demonstration of inductance employs a circuit such
as the one shown in Fig. from Problem 30.69(67). Switch S is closed, and the light bulb (represented
by resistance R 1) just barely glows. After a period of time, switch S is opened, and the bulb lights
up brightly for a short period of time. To understand this effect, think of an inductor as a
device that imparts an “inertia” to the current, preventing a discontinuous change in the current
through it. (a) Derive, as explicit functions of time, expressions for i 1 (the current through the light
bulb) and i 2 (the current through the inductor) after switch S is closed. (b) After a long period of
time, the currents i 1 and i 2 reach their steady-state values. Obtain expressions for these steadystate currents. (c) Switch S is now opened. Obtain an expression for the current through the
inductor and light bulb as an explicit function of time. (d) You have been asked to design
a demonstration apparatus using the circuit shown in the Fig. with a 22.0-H inductor and a 40.0W light bulb. You are to connect a resistor in series with the inductor, and R 2 represents the sum
of that resistance plus the internal resistance of the inductor. When switch S is opened, a transient
current is to be set up that starts at 0.600 A and is not to fall below 0.150 A until after 0.0800
s. For simplicity, assume that the resistance of the light bulb is constant and equals the resistance
the bulb must have to dissipate 40.0 W at 120 V. Determine R 2 and E for the given design considerations. (e) With the numerical values determined in part (d), what is the current through the light
bulb just before the switch is opened? Does this result confirm the qualitative description of what
is observed in the demonstration?
QUESTIONS
1. If a solenoid were virtually divided into three equal solenoids, which one of them, the middle, the left or the
right had the greatest (smallest) value of self-inductance?
2. N conducting rings are connectect in series to form an inductor (note that a solenoid is just one of
many possibilities). Find the configuration that has the greatest value of self-inductance.
3. A LR-circuit is connected to a battery. Can the induced emf of the inductor be greater than the emf of the
battery at some instant?
4. Estimate the highest frequency generated by a LC-circuit.
21
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 31 & 32: ELECTROMAGNETIC OSCILATIONS & WAVES
Chapter 31: 45 (49), 57(59), 59(61), 61 (63)
Chapter 32: 39(41), 45(47), 47(49), 49(51), 51(55)
31.45(49). A High-Pass Filter. One application of L-R-C series circuits is to
high-pass or low-pass filters, which filter out either the low- or highfrequency components of a signal. A high-pass filter is shown in Fig.,
where the output voltage is taken across the L-R combination. (The L-R
combination represents an inductive coil that also has resistance due to the
large length of wire in the coil.) Derive an expression for Vout/Vs, the ratio
of the output and source voltage amplitudes, as a function of the angular
frequency ω of the source. Show that when ω is small, this ratio is
proportional to ω and thus is small, and show that the ratio approaches
unity in the limit of large frequency.
31.57(59). In an L-R-C series circuit the magnitude of the phase angle is 54.00, with the source voltage lagging the
current. The reactance of the capacitor is 350 Ω , and the resistor resistance is 180 Ω . The average power delivered by the source is 140 W. Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of
the source.
31.59(61). In an L-R-C series circuit, the source has a voltage amplitude of 120 V, R = 80.0 Ω , and the reactance of the capacitor is 480 Ω . The voltage amplitude across the capacitor is 360 V. (a) What is the current
amplitude in the circuit? (b) What is the impedance? (c) What two values can the reactance of the inductor
have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance
angular frequency? Explain.
31.61(63). The current in a certain circuit varies with time as shown in Fig. below. Find the average current and the
rms current in terms of I0.
32.39(41). A small helium-neon laser emits red visible light with a power of 3.20 mW in a beam that has a
diameter of 2.50 mm. (a) What are the amplitudes of the electric and magnetic fields of the light? (b) What are
the average energy densities associated with the electric field and with the magnetic field? (c) What is the total
energy contained in a 1.00-m length of the beam?
22
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
32.45(47). A cylindrical conductor with a circular cross section has a radius a and a resistivity ρ and carries a constant
!
current I. (a) What are the magnitude and direction of the electric-field vector E at a point just inside the wire at a
!
distance a from the axis? (b) What are the magnitude and direction of!the magnetic-field vector B at the same
!
point? (c) What are the magnitude and direction of the Poynting vector S at the same point? (The direction of S is
the direction in which electromagnetic energy flows into or out of the conductor.) (d) Use the result in part (c)
!
to find the rate of flow of energy into the volume occupied by a length l of the conductor. (Hint: Integrate S over
the surface of this volume.) Compare your result to the rate of generation of thermal energy in the same volume.
Discuss why the energy dissipated in a current-carrying conductor, due to its resistance, can be thought of as
entering through the cylindrical sides of the conductor.
32.47(49). A circular loop of wire can be used as a radio antenna. If a l8.0-cm-diameter antenna is located 2.50
km from a 95.0-MHz source with a total power of 55.0 kW, what is the maximum emf induced in the loop?
(Assume that the plane of the antenna loop is perpendicular to the direction of the radiation’s magnetic field
and that the source radiates uniformly in all directions.)
32.49(51). Flashlight to the Rescue. You are the sole crew member of the interplanetary spaceship T: 1339
Vorga, which makes regular cargo runs between the earth and the mining colonies in the asteroid belt. You
are working outside the ship one day while at a distance of 2.0 AU from the sun [1 AU (astronomical unit)
is the average distance from the earth to the sun, 149,600,000 km]. Unfortunately, you lose contact with the
ship’s hull and begin to drift away into space. You use your spacesuit’s rockets to try to push yourself back
toward the ship, but they run out of fuel and stop working before you can return to the ship. You find
yourself in an awkward position, floating 16.0 m from the spaceship with zero velocity relative to it.
Fonunately, you are carrying a 200-W flashlight. You turn on the flashlight and use its beam as a “light
rocket” to push yourself back toward the ship. (a) If you, your spacesuit, and the flashlight have a combined
mass of 150 kg, how long will it take you to get back to the ship? (b) Is there another way you could use
the flashlight to accomplish the same job of returning you to the ship?
32.51(55). Interplanetary space contains many small particles referred to as interplanetary dust. Radiation pressure from
the sun sets a lower limit on the size of such dust particles. To see the origin of this limit, consider a spherical dust
particle of radius R and mass density ρ. (a) Write an expression for the gravitational force exerted on this particle
by the sun (mass M) when the particle is a distance r from the sun. (b) Let L represent the luminosity of the sun, equal
to the rate at which it emits energy in electromagnetic radiation. Find the force exerted on the (totally absorbing)
particle due to solar radiation pressure, remembering that the intensity of the sun’s radiation also depends on the
distance r. The relevant area is the cross-sectional area of the particle, not the tota1 surface area of the particle. As
part of your answer, explain why this is so. (c) The mass density of a typical interplanetary dust particle is about
3000 kg/m3. Find the particle radius R such that the gravitational and radiation forces acting on the particle are
equal in magnitude. The luminosity of the sun is 3.9x106 W. Does your answer depend on the distance of the
particle from the sun? Why or why not? (d) Explain why dust particles with a radius less than that found in part
(c) are unlikely to be found in the solar system. [Hint: Construct the ratio of the two force expressions found in
parts (a) and (b).]
QUESTIONS
1. What engineering difficulities one would face when he wants to construct a LC circuit with frequency of 0,01 Hz (or
1010 Hz)?
2. What is the ratio of wave length in vacuum to the wave length in the coaxial cabel, knowing dielectric and magnetic
permittivities of the cabel?
23
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 18: THERMAL PROPERTIES OF MATTER
Problems: 55(55), 61(61), 65(65), 69(73), 71(75), 73(77), 77( 81), 85(89)
18.55(55). A cylinder 1.00 m tall with inside diameter 0.120 m is used to hold propane gas (molar mass 44.1 g/mol)
for use in a barbecue. It is initially filled with gas until the gauge pressure is 1.30x106 Pa and the temperature
is 22.00 C. The temperature of the gas remains constant as it is partially emptied out of the tank, until the gauge
pressure is 2.50x105 Pa. Calculate the mass of propane that has been used.
18.61(61). A balloon whose volume is 750 m3 is to be filled with hydrogen at atmospheric pressure (1.01x105 Pa).
(a) If the hydrogen is stored in cylinders with volumes of 1.90 m3 at a gauge pressure of 1.20x106 Pa, how
many cylinders are required? Assume that the temperature of the hydrogen remains constant. (b) What is the
total weight (in addition to the weight of the gas) that can be supported by the balloon if the gas in the balloon
and the surrounding air are both at 15.0° C? The molar mass of hydrogen (H2 ) is 2.02 g/mol. The density of
air at 15.00 C and atmospheric pressure is 1.23 kg/m3 . See Chapter 14 for a discussion of buoyancy. (c) What
weight could be supported if the balloon were filled with helium (molar mass 4.00 g/mol) instead of hydrogen,
again at 15.0° C?
18.65(65). How Many Atoms Are You? Estimate the number of atoms in the body of a 50-kg physics student.
Note that the human body is mostly water, which has molar mass 18.0 g/mol, and that each water molecule
contains three atoms.
18.69(73). The Lennard-Jones Potential. A commouly used potential-energy function for the interaction of two
molecules (see Fig. 18.8) is the Lennard-Jones 6-12 potential U (r ) = U 0 ⎡⎣( R0 / r )12 − 2( R0 / r )6 ⎤⎦
where r is the distance between the centers of the molecules and U0 and R0 are positive constants. The
corresponding force F(r) is given in Eq. (13.26). (a) Graph U(r) and F(r) versus r. (b) Let r1 be the value of r
at which U(r) = 0, and let r2 be the value of r at which F(r) = 0. Show the locations of r1 and r2 on your graphs
of U(r) and F(r). Which of these values represents the equilibrium separation between the molecules? (c) Find
the values of r1 and r2 in terms of R0, and find the ratio r1/ r2 . (d) If the molecules are located a distance r2
apart [as calculated in part (c)], how much work must be done to pull them apart so that r → ∞ ?
18.71(75). The speed of propagation of a sound wave in air at 27° C is about 350 m/s. Calculate, for comparison, (a)
vrms for nitrogen molecules and (b) the rms value of vx at this temperature. The molar mass of nitrogen (N2 ) is
28.0 g/mol.
18.73(77). (a) Show that a projectile with mass m can “escape” from the surface of a planet if it is launched
vertically upward with a kinetic energy greater than mgRp, where g is the acceleration due to gravity at the
planet’s surface and Rp is the planet’s radius. Ignore air resistance. (b) If the planet in question is the earth, at
what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass 28.0 g/mol)
equal that required to escape? What about a hydrogen molecule (molar mass2.02 g/mol)? (c) Repeat part (b)
for the moon, for which g = 1.63 m/s2 and Rp = 1740 km. (d) While the earth and the moon have similar
average surface temperatures, the moon has essentially no atmosphere. Use your results from parts (b) and (c)
to explain why.
24
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
18.77(81). It is possible to make crystalline solids that are only one layer of atoms thick. Such “two-dimensional’’
crystals can be created by depositing atoms on a very flat surface. (a) If the atoms in such a two-dimensional
crystal can move only within the plane of the crystal, what will be its molar heat capacity near room temperature?
Give your answer as a multiple of R and in J/mol.K. (b) At very low temperatures, will the molar heat capacity
of a two-dimensional crystal be greater than, less than, or equal to the result you found in part (a)? Explain why.
18.77(81). The Dew Point. The vapor pressure of water decreases as the temperature decreases. If the amount of
water vapor in the air is kept constant as the air is cooled, a temperature is reached, called the dew point, at which
the partial pressure and vapor pressure coincide and the vapor is saturated. If the air is cooled further, vapor
condenses to liquid until the partial pressure again equals the vapor pressure at that temperature. The temperature
in a room is 30.0° C. A meteorologist cools a metal can by gradually adding cold water. When the can temperature
reaches I6.0°C, water droplets form on its outside surface. What is the relative humidity of the 30.00 C air in the
room? The table lists the vapor pressure of water at various temperatures:
25
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 19: THE FIRST LAW OF THERMODYNAMICS
Problems: 43(41), 44(42), 48(46), 55(57), 61(63), 63(65), 64(66), 65(67), 69 (69)
19.43(41). When a system is taken from state a to state b in Fig. along the path
acb, 90.0 J of heat flows into the system and 60.0 J of work is done by the
system. (a) How much heat flows into the system along path adb if the work done
by the system is 15.0 J? (b) When the system is returned from b to a along the
curved path, the absolute value of the work done by the system is 35.0 J. Does
the system absorb or liberate heat? How much heat? (c) If Ua = 0 and Ud = 330 J
find the heat absorbed in the processes ad and db.
19.44(42). A thermodynamic system is taken from state a to state c in Fig. along either path
abc or path adc. Along path abc, the work W done by the system is 450 J. Along
path adc, W is 120 J. The internal energies of each of the four states shown in the
gure are Ua = 150 J, Ub = 240 J, Uc = 680 J, and Ud = 330 J. Calculate the heat ow
Q for each of the four processes ab, bc, ad, and dc. In each process, does the system
absorb or liberate heat?
p
c
b
a
d
V
O
p
c
b
d
a
V
O
19.48(46). Three moles of an ideal gas are taken around the cycle acb shown in. For
this gas, Cp = 29.1 J/mol.K. Process ac is at constant pressure, process ba is at
constant volume, and process cb is adiabatic. The temperatures of the gas in states a,
c, and b are Ta = 300 K, Tc = 492 K, and Tb = 600 K. Calculatethe total work W for
the cycle.
p
b
a
O
c
V
19.55(57). A Thermodynamic Process in an Insect. The African bombardier
beetle (Stenaptinus insignis) can emit a jet of defensive spray from the movable
tip of its abdomen (Fig.). The beetle’s body has reservoirs of two different
chemicals; when the beetle is disturbed, these chemicals are combined
in a reaction chamber, producing a compound that is warmed from 20°C to
100°C by the heat of reaction. The high pressure produced allows the compound
to be sprayed out at speeds up to 19 m/s (68 km/h), scaring away predators of all kinds. (The beetle shown in
the gure is 2 cm long). Calculate the heat of reaction of the two chemicals (in J/kg). Assume that the
speci c heat of the two chemicals and the spray is the same as that of water, 4.19 x 103 J/kg.K and that the
initial temperature of the chemicals is 20°C.
19.61(63). A monatomic ideal gas expands slowly to twice its original volume, doing 300 J of work in the process.
Find the heat added to the gas and the change in internal energy of the gas if the process is (a) isothermal;
(b) adiabatic; (c) isobaric.
19.63(65). A cylinder with a piston contains 0.250 mol of oxygen at 2.40 x 10 5 Pa and 355 K. The oxygen may be
treated as an ideal gas. The gas rst expands isobarically to twice its original volume. It is then compressed
isothermally back to its original volume, and nally it is cooled isochorically to its original pressure. Compute
(a) the work done by the gas, the heat added to it, and its internal energy change during the initial expansion; (b)
the work done, the heat added, and the internal energy change during the nal cooling; (c) the internal energy
change during the isothermal compression.
26
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
19.64(66). A cylinder with a piston contains 0.150 mol of nitrogen at 1.80 x 10 5 Pa and 300 K. The nitrogen may
be treated as an ideal gas. The gas is rst compressed isobarically to half its original volume. It then expands
adiabatically back to its original volume, and nally it is heated isochorically to its original pressure. (a)
Show the series of processes in a pV-diagram. (b) Compute the temperatures at the beginning and end of
the adiabatic expansion. (c) Compute the minimum pressure.
19.65(67). Use the conditions and processes of Problem 19.64(66) to compute (a) the work done by the gas, the
heat added to it, and its internal energy change during the initial compression; (b) the work done by the gas, the
heat added to it, and its internal energy change during the adiabatic expansion; (c) the work done, the heat added,
and the internal energy change during the nal heating.
19.69(69). The van der Waals equation of state, an approximate representation of the behavior of gases at high
presure, is given by Eq.: (p + an2/V2).(V - nb) = nRT, where a and b are constants having different values for
different gases. (In the special case of a = b = 0, this is the ideal-gas equation). (a) Calculate the work done by
a gas with this equation of state in an isothermal expansion from V1 to V2 . Show that your answer agrees with
the ideal-gas result found in Example 19.1 (Section 19.2) when you set a = b = 0. (b) For ethane gas (C2H6), a =
0.554 J.m3/mol2 and b = 6.38 x l0-5 m3/mol. Calculate the work W done by 1.80 mol of ethane when it expands
from 2.00 x 10-3 m3 to 4.00 x 10-3 m3 at a constant temperature of 300 K. Do the calculation using (i) the van
der Waals equation of state and (ii) the ideal-gas equation of state. (c) How large is the difference between the
two results for W in part (b)? For which equation of state is W larger? Use the interpretation of the terms a
and b given in Section 18.1 to explain why this should be so. Are the diffferences between the two equations
of state important in this case?
27
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
Chapter 20: THE SECOND LAW OF THERMODYNAMICS
Problems: 41 (42), 43(44), 45(46), 47(48), 49(52), 53(55), 59(60), 61(62), 63(64)
20.41(42). Heat Pump. Aheat pump is a heat engine run in reverse. In winter it pumps heat from the cold air outside
into the warmer air inside the building, maintaining the building at a comfortable temperature. In summer it
pumps heat from the cooler air inside the building to the warmer air outside, acting as an air conditioner. (a) If
the outside temperature in winter is -5.00 C and the inside temperature is 170 C, how many joules of heat will the
heat pump deliver to the inside for each joule of electrical energy used to run the unit, assuming an ideal Carnot
cycle? (b) Suppose you have the option of using electrical resistance heating rather than a heat pump. How much
electrical energy would you need in order to deliver the same amount of heat to the inside of the house as in part
(a)? Consider a Carnot heat pump delivering heat to the inside of a house to maintain it at 680 F. Show that the
heat pump delivers less heat for each joule of electrical energy used to operate the unit as the outside temperature
decreases. Notice that this behavior is opposite to the dependence of the ef ciency of a Carnot heatengine on the
difference in the reservoir temperatures. Explainwhy this is so.
20.43(44). Heat Pump. As a budding mechanical engineer, you are called upon to design a Carnot engine that has 2.00
mol of a monatomic ideal gas as its working substance and operates from a high-temperature reservoir at 5000 C.
The engine is to lift a 15.0-kg weight 2.00 m per cycle, using 500 J of heat input. The gas in the engine chamber
can have a minimum volume of 5.00 L during the cycle. (a) Draw a pV-diagram for this cycle. Show in your
diagram where heat enters and leaves the gas. (b) What must be the temperature of the cold reservoir? (c) What is
the thermal ef ciency of the engine? (d) How much heat energy does this engine waste per cycle? (e) What is the
maximum pressure that the gas chamber will have to withstand?
p
20.45(46). What is the thermal ef ciency of an engine that
operates by taking n moles of diatomic ideal gas through
the cycle 1 S 2 S 3 S 4 S 1 shown in Fig.?
2
3
p0
1
4
O
V0
2p0
20.47(48). Thermodynamic Processes for a Refrigerator. A refrigerator operates on the cycle shown in Fig. The compression (d S a) and expansion
(b S c) steps are adiabatic. The temperature, pressure, and volume of the
coolant in each of the four states a, b, c, and d are given in the table. (a)
In each cycle, how much heat is taken from inside the refrigerator into the
coolant while the coolant is in the evaporator? (b) In each cycle, how much
heat is exhausted from the coolant into the air outside the refrigerator while
the coolant is in the condenser? (c) In each cycle, how much work is done
by the motor that operates the compressor? (d) Calculate the coefficient of
performance of the refrigerator.
28
CuuDuongThanCong.com
https://fb.com/tailieudientucntt
2V0
V
20.49(52). A Stirling-Cycle Engine. The Stirling cycle is similar to the Otto cyp
c
cle, except that the compression and expansion of the gas are done at constant
temperature, not adiabatically as in the Otto cycle. The Stirling cycle is used in
T2
external combustion engines (in fact, burning fuel is not necessary; any way of
d
producing a temperature difference will do - solar, geothermal, ocean temperab
ture gradient, etc.), which means that the gas inside the cylinder is not used
T1
a
in the combustion process. Heat is supplied by burning fuel steadily outside
V
the cylinder, instead of explosively inside the cylinder as in the Otto cycle. For
O Vb 5 Va/r
Va
this reason Stirling-cycle engines are quieter than Otto-cycle engines, since
there are no intake and exhaust valves (a major source of engine noise).
While small Stirling engines are used for a variety of purposes, Stirling engines for automobiles have not been
successful because they are larger, heavier, and more expensive than conventional automobile engines. In the
cycle, the working uid goes through the following sequence of steps (Fig.):
(i) Compressed isothermally at temperature T1 from the initial state a to state b, with a compression ratio r.
(ii) Heated at constant volume to state c at temperature T2
(iii) Expanded isothermally at T2 to state d.
(iv) Cooled at constant volume back to the initial state a.
Assume that the working uid is n moles of an ideal gas (for which CV is independent of temperature).
(a) Calculate Q, W, and ΔU for each of the processes a S b, b S c, c S d, and d S a. (b) In the Stirling cycle,
the heat transfers in the processes b S c and d S a do not involve external heat sources but rather use regeneration: The same substance that transfers heat to the gas inside the cylinder in the process b S c also absorbs heat
back from the gas in the process d S a. Hence the heat transfers Q bS c a nd Q dSa do not play a role in
determining the ef ciency of the engine. Explain this last statement by comparing the expressions
for Q bS c a nd Q dSa calculated in part (a). (c) Calculate the ef ciency of a Stirling cycle engine in terms of the
temperatures T1 and T2. How does this compare to the ef ciency of a Carnot-cycle engine operating between these
same two temperatures? (Historically, the Stirling cycle was devised before the Carnot cycle.) Does this result violate the second law of thermodynamics? Explain. Unfortunately, actual Stirling-cycle engines cannot achieve this
ef ciency due to problems with the heat-transfer processes and pressure losses in the engine.
20.53(55). Automotive Thermodynamics. A Volkswagen Passat has
a six-cylinder Otto-cycle engine with compression ratio r = 10.6.
The diameter of each cylinder, called the bore of the engine, is
82.5 mm. The distance that the piston moves during the
compression in Fig. 20.5, called the stroke of the engine, is 86.4 mm.
The initial pressure of the air–fuel mixture (at point a in Fig. 20.6) is
8.50 * 10 4 Pa, and the initial temperature is 300 K (the same as
the outside air). Assume that 200 J of heat is added to each cylinder
in each cycle by the burning gasoline, and that the gas has CV
= 20.5 J>mol # K and g = 1.40. (a)) Calculate the total work done
in one cycle in each cylinder of the engine, and the heat released
when the gas is cooled to the temperature of the outside air. (b))
Calculate the volume of the air– fuel mixture at point a in the
cycle. (c)) Calculate the pressure, volume, and temperature of the
gas at points b, c, and d in the cycle. In a pV-diagram, show the
numerical values of p, V, and T for each of the four states. (d)
Compare the efficiency of this engine with the efficiency of a
Carnot-cycle engine operating between the same maximum and
minimum temperatures.
Otto cycle
p
2 Heating at constant
c
volume (fuel combustion)
QH
3 Adiabatic expansion
b
(power stroke)
W
0QC0
O
a
rV
V
V
1 Adiabatic compression
(compression stroke)
29
CuuDuongThanCong.com
d
https://fb.com/tailieudientucntt
4 Cooling at constant volume
(cooling of exhaust gases)
20.59(60). A TS-Diagram. (a) Graph a Carnot cycle, plotting Kelvin temperature vertically and entropy horizontally.
This is called a temperature–entropy diagram, or TS-diagram. (b) Show that the area under any curve representing a reversible path in a temperature–entropy diagram represents the heat absorbed by the system. (c) Derive
from your diagram the expression for the thermal ef ciency of a Carnot cycle. (d) Draw a temperature–entropy
diagram for the Stirling cycle described in Problem 20.49. Use this diagram to relate the ef ciencies of the
Carnot and Stirling cycles.
20.61(62). To heat 1 cup of water (250 cm3) to make coffee, you place an electric heating element in the cup. As the
water temperature increases from 20°C to 78°C, the temperature of the heating element remains at a constant
120°C. Calculate the change in entropy of (a) the water; (b) the heating element; (c) the system of water and heating element. (Make the same assumption about the speci c heat of water as in Example 20.10 in Section 20.7, and
ignore the heat that ows into the ceramic coffee cup itself.) (d) Is this process reversible or irreversible? Explain.
20.6
63(64). Consider a Diesel cycle that starts (at point a in Fig.) with air
at temperature Ta. The air may be treated as an ideal gas. (a)) If the
temperature at point c is Tc, derive an expression for the efficiency of
the cycle in terms of the compression ratio r. (b)) What is the effciency if Ta = 300 K, Tc = 950 K, g = 1.40, and r = 21.0?
Diesel cycle
2 Fuel ignition, heating at constant
p
QH
b
pressure (fuel combustion). This
c is a significant difference between
the Diesel and Otto cycles.
3 Adiabatic expansion
(power stroke)
W
d
a
O V
rV
0 QC 0
V
1 Adiabatic compression
(compression stroke)
4 Cooling at constant volume
(cooling of exhaust gases)
30
CuuDuongThanCong.com
https://fb.com/tailieudientucntt