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Sample Problem 1 Figure 6 shows three
charged particles, held in place by forces not
shown. What electrostatic force, owing to
the other two charges, acts on q1? Take
q1=–1.2 mC, q2=+3.7 mC, q3=–2.3 mC,
r12=15 cm, r13=10 cm, and q=32°.
Sample Problem 2 A penny, being electrically neutral, contains equal amounts of
positive and negative charge. What is the
magnitude of these equal charges?
Sample Problem 1 A proton is placed in a uniform electric field E. What
must be the magnitude and direction of this field in the electrostatic force
acting on the proton is just to balance its weight?
Figure 6 The three charges exert three
pairs of action-reaction forces on each
other. Only the two forces acting on q1
Sample Problem 3 In Sample Problem 2 we saw that a cooper penny contains both positive and negative charges, each of a magnitude 1.37x105 C.
Suppose that these charges could be concentrated into two separate bundles, held 100 m apart. What attractive force would act on each bundle?
Sample Problem 4 The average distance r between the electron and the
proton in the hydrogen atom is 5.3x10–11 m. (a) What is the magnitude of the
average electrostatic force that acts between these two particles? (b) What is
the magnitude of the average gravitational force that acts between these
particles?
Sample Problem 5 The nucleus of an iron atom has a radius of about
4x10–15 m and contains 26 protons. What repulsive electrostatic force acts
between two protons in such a nucleus if they are separated by a distance of
one radius?
Sample Problem 2 In an ionized helium atom (a helium atom in which
one of the two electrons has been removed), the electron and the nucleus are
separated by a distance of 26.5 pm. What is the electric field due to the nucleus at the location of the electron?
Sample Problem 3
Figure 3 shows a
charge q1 of +1.5 mC and a charge q2 of +2.3
mC. The first charge is at the origin of an x
axis, and the second is at a position x=L,
where L=13 cm. At what point P along the x
axis is the electric field zero?
Figure 3 At point P, the electric fields
of the charges q1 and q2 are equal, so the
net field at P is zero.
Sample Problem 4 In Fig. 5, how does the
magnitude of the electric field vary with the
distance from the center of the charged body?
Sample Problem 5 A charge drop of oil of radius
R= 2.76 mm and density r=920 kg/m3 is maintained
in equilibrium under the combined influence of its
weight and a downward uniform electric field of magnitude E= 1.65´106 N/C (Fig. 13). (a) calculate the
magnitude and sign of the charge on the drop. Express the result in terms of the elementary charge e.
(b) The drop is exposed to a radioactive source that
emits electrons. Two electrons strike the drop and
are captured by it, changing its charge by two units.
If the electric field remains at its constant value, calculate the resulting acceleration of the drop.
Sample Problem 6 Figure 14 shows the deflecting
electrode system of an ink-jet printer. An ink drop
whose mass m is 1.3´10–10 kg carries a charge q of
–1.5´10–13 C and enters the deflecting plate system
with a speed v=18 m/s. The length L of these plates is
1.6 cm, and the electric field E between the
plates is 1.4´106 N/C. What is the vertical
deflection of the drop at the far edge of the
plates? Ignore the varying electric field at
the edges of the plates.
Sample Problem 7 A molecule of water
vapor (H2O) has an electric dipole moment
of magnitude p=6.2´10–20 C×m. (This large
dipole moment is responsible for many of
the properties that make water such an important substance, such as its ability to act
Chapter 27-1
Chapter 28-1
Figure 5 Lines of force
surrounding a positive
point charge. The direction
of the force on a positive
test charge, and thus the direction of electric field at
any point, is indicated by
the direction of the lines.
The relative spacing between the lines at any location indicates the relative
strength of the field at that
location. The lines are assumed to terminate on distant negative charges that
are not shown.
Figure 13 A negatively charged drop is
placed in a uniform electric field E. The
drop moves under the combined influence of its weight mg and the electric
force qE.
as an almost universal solvent.) Figure
18 is a representation of this molecule,
showing the three
nuclei and the surrounding
electron
clouds. The electric
dipole moment p is
represented by a
vector on the axis of
symmetry. The dipole moment arises
because the effective
center of positive Figure 14 (a) The essential features of an ink-jet printer. An input sigcharge does not coin- nal from a computer controls the charge given to the drop and thus the
cide with the effec- position at which the drop strikes the paper. A transverse force from
the electric field E is responsible for deflecting the drop. (b) A detail of
tive
center
of the deflecting plates. The drop moves in a parabolic path while it is benegative charge. (A tween the plates, and it moves along a straight line after it leaves the
contrasting case is plates.
that of a molecule of
carbon dioxide, CO2. Here the three atoms are joined in a straight line, with
a carbon in the middle and oxygens on either side.
The center of positive charge and the center of
negative charge coincide at the center of mass of
the molecule, and the electric dipole moment of
CO2 is zero.) (a) How far apart are the effective
centers of positive and negative charge in a molecule of H2O? (b) What is the maximum torque on a
molecule of H2O in a typical laboratory electric
field of magnitude 1.5´104 N/C? (c) Suppose the
dipole moment of a molecule of H2O is initially
Figure 18 A molecule of H2O,
pointing in a direction opposite ti the field. How
showing the three nuclei, the
electron clouds, and the electric much work is done by the electric field in rotating
dipole moment vector p.
the molecule into alignment with the field?
Sample Problem 1 Consider the closed surface of Fig. 1e, which shows a
volume enclose by five surfaces (1, 2, and 3, which are parallel to the surfaces of Figs. 1a, 1b, and 1c, along with 4 and 5, which are parallel to the
streamlines). Assuming the velocity field is uniform, so that it has the same
magnitude and direction everywhere, find the total flux through the closed
surface.
Figure 1 A wire loop of area A is immersed in a flowing stream, which we represent as a velocity field.
(a) The loop is at right angles to the flow. (b) The loop is turned through an angle q; the projection of
the area perpendicular to the flow is A cos q. (c) When q=90°, none of the streamlines pass through the
plane of the loop. (d) The area of the loop is represented by a vector A perpendicular to the plane of the
loop. The angle between A and the flow velocity v is q. (e) A closed surface made of five plane surfaces.
The area A of each surface is represented by the outward normal.
Sample Problem 2 Figure 3 shows a hypothetical closed cylinder of radius R immersed in a uniform electric field E, the cylinder axis being parallel
to the field. What is FE for this closed surface?
Sample Problem 3
The electric field just
above the surface of the
charged drum of a photocopy machine has a
magnitude E of 2.3x105
N/C. What is the surface charge density on
the drum if it is a conductor?
Sample Problem 4 Figure 3 A closed cylinder is immersed in a uniform electric field E
The magnitude of the parallel to its axis.
average electric field
normally present in the Earth's atmosphere just above the surface of the
Earth is about 150 N/C, directed downward. What is the total net surface
charge carried by the Earth? Assume the Earth to be a conductor.
Sample Problem 5 A plastic rod, whose length L is 220 cm and whose radius R is 3.6 mm, carries a negative charge q of magnitude 3.8´10-7 C,
spread uniformly over its surface. What is the electric field near the midpoint of the rod, at a point on its surface?
Chapter 28-2
Chapter 29-1
Sample Problem 6 Figure 13a shows
portions of two large sheets of charge
with uniform surface charge densities of
s+=+6.8 mC/m2 and s-=-4.3 mC/m2.
Find the electric field E (a) to the left of
the sheets, (b) between the sheets, and
(c) to the right of the sheets.
Sample Problem 1 Two protons in the nucleus of a
238U atom are 6.0 fm apart. What is the potential energy associated with the electric force that acts between these two particles?
Sample Problem 2 In the system shown in Fig. 3,
assume that r12=r13= r23=d=12 cm, and that
q1=+q, q2=-4q, and q3=+2q,
where q=150 nC. What is the potential energy of the
system?
Figure 13 (a) Two large parallel sheets of
charge carry different charge distributions
s+ and s-. The fields E+ and E- would be set
up by each sheet if the other were not present. (b) The net fields in the nearby regions
to the left (L), center (C), and the right (R) of
the sheets, calculated from the vector sum
of E+ and E- in each region.
Figure 3 An assembly of
three charges.
Sample Problem 3 An alpha particle (q=+2e) in a nuclear accelerator
moves from one terminal at a potential of Va=+6.5´106 V to another at a
potential of Vb= 0 V. (a) What is the corresponding
change in the potential energy to the system? (b) Assuming the terminals and their charges do not move
and that no external forces act on the system, what is
the change in kinetic energy of the particle?
Sample Problem 4 In Fig. 6 let a test charge q0 be
moved from a to b over the path abc. Compute the potential difference between a and b.
Sample Problem 5 What must be the magnitude
of an isolated positive point charge for the electric potential at 15 cm from the charge to be +120 V?
Sample Problem 6 What is the electric potential
at the surface of a gold nucleus? The radius is
7.0´10-15 m, and the atomic number Z is 79.
Sample Problem 7 Calculate
the potential at point P, located at
the center of the square of point
charges shown in Fig. 9a. Assume
that d=1.3 m and that the charges
are q1=+12 nC, q2=+31 nC,
q3=-24 nC, q4=+17 nC.
Sample Problem 8 An electric
quadrupole consists of two electric
dipoles so arranged that they almost, but not quite, cancel each
other in their electric effects at distant points (see Fig. 12). Calculate
V(r) for points on the axis of this
quadrupole.
Figure 6 A test charge q0
moves along the path abc
through the uniform electric field E.
Figure 9 (a) Four charges are held at the corners of
a square. (b) The curve connects points that have the
same potential (350 V) as the point P at the center of
the square.
Sample Problem 9 Calculate the potential at a point on the axis of a circular plastic disk of radius R, one surface of which carries a uniform charge
density s.
Chapter 29-2
Chapter 30-1
Sample Problem 1 A storage capacitor on a random access memory
(RAM) chip has a capacitance of 55 fF. If it is charged to 5.3 V, how many excess electrons are there on its negative plate?
Sample Problem 2 The plates of a parallel-plate capacitor are separated
by a distance d= 1.0 mm. What must be the plate area if the capacitance is to
be 1.0 F?
Sample Problem 3 The space between the conductors of a long coaxial cable, used to transmit TV signals, has an inner radius a= 0.15 mm and an
outer radius b= 2.1 mm. What is the capacitance per unit length of this cable?
Figure 18 A dipole is located
at the origin of the xz system.
Figure 22
A small charged
sphere is suspended inside a
larger charged spherical shell.
Figure 12 An electric quadrupole, consisting of two oppositely directed electric dipoles.
(
)
s
R 2 + z 2 - z for the potential on
2e 0
the axis of a uniform charged disk, derive an expression for the electric field
at axial points.
Sample Problem 10 Using V =
Sample Problem 11 Figure 18 shows a (distant) point P in the field of a
dipole located at the origin of an xz coordinate system. Calculate E as a function of position.
Sample Problem 12 Calculate the potential difference between the two
spheres illustrated in Fig. 22.
Sample Problem 4 What is the capacitance of the Earth, viewed as an isolated conducting sphere of radius 6370 km?
Sample Problem 5 (a) Find the
equivalent capacitance of the combination shown in Fig. 7a. Assume
C1= 12.0 mF, C2= 5.3 mF, and
C3=4.5 mF.
(b) A potential difference V=12.5 V
is applied to the terminals in Fig.
7a. What is the charge on C1?
Sample Problem 6 A 3.55-mF ca- Figure 7 (a) A combination of three capacitors. (b)
pacitor C1 is charged to a potential The parallel combination of C1 and C2 has been replaced by its equivalent, C12. (c) The series combinadifference V0=6.30 V, using a bat- tion of C and C has been
replaced by its equiva12
3
tery. The charging battery is then lent, C123.
removed, and the capacitor is connected as in Fig. 9 to an uncharged
8.95-mF capacitor C2. After the switch S is closed,
charge flows from C1 to C2 until an equilibrium is
established, with both capacitors at the same potential difference V. (a) What is the common potential difference? (b) What is the energy stored in Figure 9 Capacitor C1 has prethe electric field before and after the switch S in viously been charged to a potential difference V0 by a battery
Fig. 9 is thrown?
that has been removed. When
switch S is closed, the initial
Sample Problem 7
An isolated conducting the
charge q0 on C1 is shared with
sphere whose radius is R 6.85 cm carries a charge C2.
q=1.25 nC. (a) How much energy is stored in the
electric field of this charged conductor? (b) What
is the energy density at the surface of the sphere? (c) What is the radius R0 of
a spherical surface such that one-half of the stored potential energy lies
within it?
Sample Problem 8 A parallel-plate capacitor whose capacitance C0 is 13.5
pF has a potential difference V=12.5 V between its plates. The charging battery is now disconnected and a porcelain slab (ke=6.5) is slipped between the
Chapter 30-2
Chapter 31-1
plates as in Fig. 11b. What is
the stored energy of the unit,
both before and after the
slab is introduced?
Sample Problem 1 One end of an aluminum wire whose diameter is 2.5
mm is welded to one end of a copper wire whose diameter is 1.8 mm. The
composite wire carries a steady current i of 1.3 A. What is the current density in each wire?
Sample Problem 9 Figure
16 shows a parallel-plate ca- Figure 11 (a) An originally uncharged, empty capacitor is
charged by a battery, which is then removed. The voltmeter
pacitor of plate area A and shows the potential difference between the plates. (b) The
plate separation d. A poten- region between the plates is filled with dielectric. The
tial difference V0 is applied charge remains constant, but the potential difference decreases.
between the plates. The battery is then disconnected,
and a dielectric slab of thickness b and dielectric constant ke is placed between the plates as shown. Assume
A=115 cm2, d=1.24 cm, b=0.78 cm, ke=2.61, V0=85.5 V.
(a) What is the capacitance C0 before the slab is inserted? (b) What free
charge appears on the plates? (c) What is the electric field E0 in the gaps between the plates and the dielectric slab? (d) Calculate the electric field E in
the dielectric slab. (e) What is the potential difference between the plates after the slab has been introduced? (f) What is the capacitance with the slab in
place?
Sample Problem 2 What is the drift sped of the conduction electrons in
the copper wire of Sample Problem 1?
Figure 16 A parallel-plate capacitor contains a dielectric that only partially fills
the space between the plates
Chapter 31-2
Sample Problem 3 A strip of silicon, of width w=3.2 mm and thickness
d=250 mm, carries a current i of 190 mA. The silicon is an n-type semiconductor, having been "doped" with a controlled amount of phosphorus impurity. The doping has the effect of greatly increasing n, the number of charge
carriers (electrons, in this case) per unit volume, as compared with the value
for pure silicon. In this case, n= 8.0´1021 m-3. (a) What is the current density in the strip? (b) What is the drift speed?
Sample Problem 4 A rectangular block of iron has dimensions 1.2 cm ´
1.2 cm ´ 15 cm. (a) What is the resistance of the block measured between
the two square ends? (b) What is the resistance between the two opposing
rectangular faces? (c) The resistivity of iron at room temperature is
9.68´10-8 W×m.
Sample Problem 5 (a) What is the mean free time t between collisions for
the conduction electrons in copper? (b) What is the mean free path l for
these collisions? Assume an effective speed v of 1.6´106 m/s.
Sample Problem 6 You are given a length of heating wire made of a
nickel-chromium-iron alloy called Nichrome; it has a resistance R of 72 W.
It is to be connected across a 120-V line. Under which circumstances will the
wire dissipate more heat: (a) its entire length is connected across the line, or
(b) the wire is cut in half and the two halves are connected in parallel across
the line?
Chapter 32-1