Last revised 2000/01/05 |

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Due on Monday | 02/14 | 02/21 | 02/28 | 03/06 | 03/13 | 03/20 | 04/03 | 04/10 | 04/17 | 04/24 | 05/01 | 05/08 |

Chapter | 22 | 23 | 23, 24 | 25 | 26 | 27, 28 | 29 | 30 | 31 | 31 | 33 | 33 |

I will almost always have hints posted on the web beginning a week before the assignment is due. The day before the problem session I will try to post answers in the same location. I will frequently post solutions after the problem session is over.

In general, I have chosen problems like the more difficult ones in the chapter. It may be helpful to work a few of the simpler problems first, especially if you have trouble getting started on one of the assigned problems. If you have difficulty, see me in Sharp Lab 264, call x2013, or send EMail to mvb@udel.edu, preferably before turning the set in. It is as important to know why incorrect attempts do not work as it is to know what the correct solutions are. Please feel free to talk to me at any time either in person, by phone, or by E-MAIL.

The problems will be graded for the existence of diagrams, the use of units, and correct vector notation. Otherwise, a reasonable attempt will get full credit whether it is right or wrong. Since the problems are graded for technique and not for correctness, you can work with each other on the assignments. You should do so. You may also consult others outside the class, but I strongly advise against looking at their solutions before attempting the problems yourself. You must write up your own final copy, unless I have stated otherwise (no Xeroxes!).

Many of these problems are adapted from a book by Tipler, because I already have reliable answers and solutions for those problems. Similar problems appear in our text. I am also modifying some of the problems from the text so that they fit better with the other assigned problems. Whenever the assigned problem is very similar to one in the text, the assignment includes the statement "compare HRW # ...." If an assigned problem is nearly identical to one in the text, the assignment will say "see HRW # ...."

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**2**. Four charges,
each of magnitude *q*, are at the corners of a square of side
*L*. The two charges at one pair of opposite corners are
positive, and the other two charges are negative. Find the force
exerted on either of the negative charges by the three remaining
charges. See figure to the right.

**3**. (See HRW p. 551 #18 and p. 553 #41). A certain
charge *Q* is divided into two parts *alpha Q* and
*(1 - alpha) Q*, which are then separated by a certain
distance. What must *alpha* be in order to maximize the
electrostatic repulsion between the two charges? Draw a graph of the
electrostatic force *F* as a function of *alpha* and
verify the correctness of your answer.

**4**. (HRW p. 551 #20). In the figure on the right, two
tiny conducting balls of identical mass *m* and identical
charge *q* hang in equilibrium from nonconducting threads of
length *L*. Assume that *theta* is so small that
*tan(theta)* can be replaced by its approximate equal
*sin(theta)* [both are in fact essentially equal to
*theta*]. (a) Show that, for equilibrium,

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**5**. (Compare HRW p. 577
# 57) Two charged plates of length *L* are placed a distance
*d* apart as shown in the figure to the right. An electron is
injected between the plates, just clearing the lower plate as it
enters. It feels a constant electric field **E** pointed straight
upward due to the charges on the plates. If the electron's initial
velocity is v_{0} at an angle theta with respect to the lower
plate, for what values of theta will it not hit the upper plate? If
it clears the upper plate, how far does it get before hitting the
lower plate?

**6**. (Compare HRW p. 575 #22)
Four charges of equal magnitude are arranged at the corners of a
square of side *L* as shown in the figure to the right. Show
that the electric field at the midpoint of one of the sides of the
square is directed along that side, points toward the negative charge,
and has a magnitude *E* given by

**7**. A water molecule has its
oxygen atom at the origin, one hydrogen nucleus at
x = 0.077 nanometer, y = 0.058 nm, and the
other hydrogen nucleus at x = - 0.077 nm,
y = 0.058 nm. If the electrons from the hydrogen atoms
are transferred completely to the oxygen atom so that it has a charge
of *-2e*, what is the dipole moment of the water molecule?
[This characterization of the chemical bonds of water as being totally
ionic overestimates the dipole moment of a water molecule].

**8**. A positive point charge
*+Q* is at the origin. A dipole of moment **p** is a
distance *r* away from the charge and points in the radial
direction, as shown in the diagram to the right.

(a) Show that the force exerted by the electric field of the point charge on the dipole is attractive and has a magnitude of approximately

(b) Now assume that the dipole is centered at the origin and that the
point charge *Q* is a distance *r* away along the line
of the dipole. From your result in part (a) and Newton's Third Law,
show that the magnitude of the electric field of the dipole along the
line of the dipole a distance *r* away is approximately

**9**. (HRW p. 575 # 26) *Electric Quadrupole
*. The figure shows an electric quadrupole. It consists of two
dipoles with dipole moments that are equal in magnitude but opposite
in direction. Show that the value of the magnitude of the electric
field *E* on the axis of the quadrupole for points a distance
*z* from its center (assume *z >> d*) is given by

in which Q (=2 q d^{2}) is known as the *quadrupole
moment* of the charge distribution. [This is not the only way to
get a quadrupole moment.]

**10**. (Compare HRW p. 576 #33) (a)
A thin, nonconducting rod carries a charge of constant linear charge
density *lambda* and lies on the *x*-axis from *x =
0* to *x = a*. Show that the *y*-component of the
electric field at any point on the *y*-axis is given by

Go to solution for part (a)

(b) Show that if the rod extends from *x = - b* to
*x = a*, the *y*-component of the electric field at the same
point is given by

(c) HRW asks for the special case *a = b = L/2*. Show that in
this case you get

(d) Show that if and only if *a = b* the x-component of the electric
field vanishes.

**11**. A butterfly net is attached to a
ring of radius R, and the ring is held in a plane perpendicular to a
constant electric field of magnitude E. What is the electric flux

through the net? The angle *theta* is the angle between the
normal to the net and the electric field.

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**13**. A disk of radius
*R* carries a uniform charge density *sigma*. (a)
Compare the approximation *E = sigma / (2 epsilon _{0}
)* with the exact expression for the electric field on the axis of
the disk by computing the neglected term as a percentage of

**14**. (Hint: compare HRW p. 576 #34) In the figure to
the right, two nonconducting rods of length *L* have charge
*q* uniformly distributed along their lengths. (a) What is the
linear charge density of each rod? (b) What is the electric field
*E* due to the left-hand rod at a distance *a* beyond
its right-hand end? (c) What is the force that one of the rods exerts
on the other? (d) Show that when the distance *d* between the
rods becomes large compared to the length of the rods, the force tends
toward the expected result *q ^{2} / d^{2}*.

**16**. (HRW p. 599 #52). A solid nonconducting sphere of
radius *R* has a nonuniform charge distribution of * rho =
rho _{s} r / R*, where

**17**. (Compare HRW p. 598
#31) An infinitely long nonconducting cylindrical shell of inner
radius *a* and outer radius *b* has a uniform volume
charge density *rho* . Find the electric field everywhere.

**18**. An infinitely long nonconducting
solid cylinder of radius *R* has a charge density of *B /
r* , where *r* is the distance from the axis of the
cylinder. Find the electric field everywhere.

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**20.** Three
equal charges lie in the *xy* plane. Two are on the
*y*-axis at *y = - a* and *y = + a* , and the
third is on the x axis at *x = a* . (a) What is the potential
*V(x)* due to these charge at any point on the *x*-axis?
(b) Find *E _{x}* along the

**21**. (HRW p. 624 #41) The figure shows a plastic rod
of length *L* and uniform positive charge *Q* lying on
an *x* axis. With *V = 0* at infinity,
find the electric potential at point P_{1} on the axis, at
distance *d* from one end of the rod.

**22.**
Four equal charges *Q* are at the corners of a square of side
*L*. The charges are released one at a time proceeding
clockwise around the square. Each charge is allowed to reach its
final speed a long distance from the square before the next charge is
released. What is the final kinetic energy of (a) the first charge
released, (b) the second charge released, (c) the third charge
released, and (d) the fourth charge released?

**23.** (Compare HRW p. 624 #
46) The electric potential in a region of space is given by

Find the electric field at all points *(x, y, z)*

**24**. Consider two
concentric spherical metal shells of radii *a* and *b*
where *b > a* . The outer shell has a charge *Q* , but
the inner shell is grounded [it is at the same potential as a point at
infinity]. This means that the inner shell is at zero potential and
implies that electric-field lines leave the outer shell and go to
infinity but that other electric-field lines leave the outer shell and
end on the inner shell. Find the charge on the inner shell.

**25**.
A nonconducting sphere of radius *R* has a volume charge
density *rho = rho _{0} r / R * , where

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**27**. Consider two
parallel-plate capacitors, C_{1} and C_{2}, that are
connected in parallel. The capacitors are identical except that
C_{2} has a dielectric inserted between its plates. A voltage
source of *V* volts is connected across the capacitors to
charge them and is then disconnected. (a) What is the charge on each
capacitor? (b) What is the total stored energy of the capacitors? (c)
The dielectric is removed from C_{2}. What is the final
stored energy of the capacitors? (d) What is the final voltage across
the two capacitors?

**28**. (Compare HRW p. 650 #
63) A parallel-plate capacitor of area *A* and separation
*d* is charged to a potential difference *V* and is then
removed from the charging source. A dielectric slab of constant
*kappa*, thickness *d*, and area *A/2* is
inserted along one edge of the capacitor as shown in the figure. Let
*sigma _{1}* be the free charge density at the
conductor-dielectric surface and

and that the new potential difference is *
2 V / ( kappa + 1 )
*

**29**. HRW p. 650 # 68: The space between two
concentric conducting spherical shells of radii *b* and
*a* (where *b* > *a*) is filled with a
substance of dielectric constant *kappa*. A potential
difference *V* exists between the inner and outer shells.
Determine (a) the capacitance of the device, (b) the free charge
*q* on the inner shell, and (c) the charge *q'* induced
along the surface of the inner shell.

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**30**. The wires in a house
must be large enough in diameter so that they do not get hot enough to
start a fire. Suppose a certain wire is to carry a current of 20 A,
and it is determined that the joule heating of the wire should not
exceed 2 W/m. What diameter must a copper wire have to be "safe" for
this current?

**31**. HRW p. 672 #59: A linear accelerator produces a
pulsed beam of electrons. The pulse current is 0.50 A, and the pulse
duration is 0.10 microsecond. (a) How many electrons are accelerated
per pulse? (b) What is the average current for a machine operating at
500 pulses/s? (c) If the electrons are accelerated to an energy of 50
MeV, what are the average and peak powers of the accelerator?

**32**. HRW p. 671 #41: A resistor has the shape of a
truncated right-circular cone (see figure 27-25 in text). The end
radii are *a* and *b*, and the altitude is *L*.
If the taper is small, we may assume that the current density is
uniform across any cross section. (a) Calculate the resistance of
this object. (b) Show that your answer reduces to
*rho L / A* for the special case of zero taper
(that is, for *a* = *b*).

**33**. A resistor of size *R _{1}* is
placed in series with a parallel combination of resistors

**34**. For the circuit shown
in the figure, find (a) the current in each resistor, (b) the power
supplied by each EMF, and (c) the power dissipated in each
resistor. (d) Is energy conservation obeyed?

**35**. Nine resistors of
resistance *R* are connected as shown in the figure, and a
potential difference *V* is applied between points *a*
and *b*. What is the equivalent resistance of this network? (b)
Find the current in each of the nine resistors.

**36**. Find the current through each of the elements in
the diagram to the right. The use of a symbolic-algebra computer
program is recommended. [When you have correct answers, keep them for
use in the chapter on AC circuits.]

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**38**. A particle of charge
*q* and mass *M* moves in a circle of radius *r*
and with angular velocity *omega*. (a) Show that the average
current is
*I = q omega / (2 pi )* and
that the magnetic moment has the magnitude
*mu = (1/2) q omega r ^{2}*.
(b) Show that the angular momentum of this particle has the magnitude

**39.**
A current-carrying wire is bent into a semicircular loop of radius
*R*, which lies in the *xy* plane. There is a uniform
magnetic field
**B** = *B* **1 _{z}**
perpendicular to the plane of the loop. Show that the force acting on
the loop is

**40**. A nonconducting rod
of mass *M* and length *l* has a uniform charge per unit
length *lambda* and rotates with angular velocity
*omega* about an axis through one end and perpendicular to the
rod. (a) Consider a small segment of the rod of length *dx* and
charge *dq = lambda dx* at a distance *x*
from the pivot. Show that the magnetic moment of this segment is
(1/2) lambda omega x^{2} dx. (b)
Integrate your result to show that the total magnetic moment of the
rod is
*mu = (1/6) lambda omega l ^{3} *
(c) Show that the magnetic moment

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**42**: HRW p. 747 # 38:
In the figure to the right (HRW figure 30-55), the long straight wire
carries a current of 30 A and the rectangular loop carries a current
of 20 A. Calculate the resultant force acting on the loop. Assume
that the distance *a* = 1.0 cm,
*b* = 8.0 cm, and
*L* = 30 cm.

**43**: HRW p. 749 #50. Figure 30-64 of the text shows a
cross section of a long cylindrical conductor of radius *a*
containing a long cylindrical hole of radius *b*. The axes of
the cylinder and hole are parallel and are a distance *d*
apart; a current *i* is uniformly distributed in the conductor
outside the hole. (a) Use superposition to show that the magnetic
field at the center of the hole is

mu_{0}i d B = ------------------------ 2 pi (a^{2}- b^{2})

(b) Discuss the two special cases *b* = 0 and
*d* = 0. (c) Use Ampere's Law to show that the
magnetic field in the hole is uniform.

(*Hint*: Regard the cylindrical hole as filled with two equal
currents moving in opposite directions, thus canceling each other.
Assume that each of these currents has the same current density as
that in the actual conductor. Then superimpose the fields due to two
complete cylinders of current, of radii *a* and *b*,
each cylinder having the same current density.)

**44**. A solenoid has *n* turns per unit length
and radius *R* and carries a current *I*. Its axis is
along the *x* axis with one end at *x = -L/2*
and the other end at *x = +L/2*, where *L* is
the total length of the solenoid. Show that the magnetic field
*B* at a point on the axis outside the solenoid is given by

B = (1/2) mu_{0}n l (cos theta_{1}- cos theta_{2})

where

x + (1/2) L cos theta_{1}= ------------------------- [ R^{2}+ (x + L/2)^{2}]^{1/2}

and

x - (1/2) L cos theta_{2}= ------------------------- [ R^{2}+ (x - L/2)^{2}]^{1/2}

**45**. In Problem 44, a formula for the magnetic field
along the axis of a solenoid is given. For *x* >> L
and L > R, the angles *theta _{1}* and

R tan (theta_{1}) = ------ x + L/2

and

R tan (theta_{2}) = ------- x - L/2

(b) Show that the magnetic field at a point far from either end of the solenoid can be written

mu_{0}q_{m}q_{m}B = ---- [ -- - -- ] 2 pi r_{1}^{2}r_{2}^{2}

where *r _{1} = x - L/2* is the distance to
the near end of the solenoid,

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This is the priciple of the commercial alternating-current generator. (b) Design a loop that will produce an emf of 150 V when rotated at 60.0 rev/s in a magnetic field of 0.500 T.

**47**. HRW p. 780 # 29: A rectangular loop of wire with
length *a*, width *b*, and resistance *R* is
placed near an infinitely long wire carrying current *i*, as
shown in Fig. 31-53 of the text. The distance from the long wire to
the center of the loop is *r*. Find (a) the magnitude of the
magnetic flux through the loop and (b) the current in the loop as it
moves away from the long wire with speed *v*. [Neglect the
magnetic field generated by the current in the loop itself. - MVB]

**48**. HRW p. 781 # 38:
In Fig. 31-57 of the text a conducting rod of mass *m*,
resistance *R*, and length *L* slides without friction
on two long, horizontal, resistanceless rails. A uniform vertical
magnetic field **B** fills the region in which the rod is
free to move. The generator G supplies a constant current *i*
directed as shown in the figure. (a) Find the velocity of the rod as a
function of time, assuming it to be at rest at t = 0. The
generator is now replaced by a battery that supplies a constant
EMF. (b) Show that the velocity of the rod now approaches a constant
terminal value *v* and give its magnitude and direction. (c)
What is the current in the rod when this terminal velocity is reached?
(d) Analyze this situation and that with the generator from the point
of view of energy transfers.

**49**. A long solenoid has
*n* turns per unit length and carries a current given by
I = I_{0} sin (omega t). The solenoid
has a circular cross section of radius *R*. Find the induced
electric field at a radius *r* from the axis of the solenoid
for (a) *r < R* and (b) *r > R*.

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**51**. Show that the
inductance of a toroid of rectangular cross section, half of which is
shown in the figure, is given by

where *N* is the total number of turns, *a* is the
inside radius of the toroid, *b* is its outside radius, and
*h* is its height.

**52**. HRW p. 783 # 69. In figure 31-65 of the text,
E = 100 V, R_{1} = 10.0 Ohms,
R_{2} = 20 Ohms,
R_{3} = 30 Ohms, and
L = 2.00 H. Find the values of *i _{1}* and

**53**.
For the circuit in the figure, (a) find the rate of change of current
in each inductor and in the resistor just after the switch is
closed. (b) What is the current through the resistor after a long time
has passed?

**54**. In the circuit in the
figure, the switch is closed at *t = 0*. From time
*t* to time *L/R*, find (a) the total energy that has
been supplied by the battery, (b) the total energy that has been
dissipated in the resistor, and (c) the energy that has been stored
in the inductor. *Hint:* Find the rates as functions of time
and integrate from *t = 0* to
*t = L/R*

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**56**. A resistance
*R* and a 1.4-H inductance are in series across a 60-Hz AC
voltage. The voltage across the resistor is 30 V and the voltage
across the inductor is 40 V. (a) What is the value of the resistance?
(b) What is the AC input voltage?

**57**. A coil draws 15 A
when connected to a 220-V AC 60-Hz line. When it is in series with a
4-ohm resistor and the combination is connected to a 100-V battery,
the battery current after a long time is observed to be 10 A. (a)
What is the resistance in the coil? (b) What is the inductance of the
coil?

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**59**. The circuit shown in
the figure is called an RC high-pass filter because high input
frequencies are transmitted with greater amplitude than low input
frequencies. (a) If the input voltage is

V_{in} = V_{0} cos(omega t),

show that the output voltage [the potential difference between the two black dots in the figure] is

Vand determine the value of_{0}cos (omega t + phi) V_{out}= ------------------------- [ 1 + (omega R C)^{-2}]^{1/2}

**60**. A resistor and an
ideal inductor are connected in parallel across an emf
V = V_{0} cos(omega t). Show that

(a) the current in the resistor is
I_{R} = (V_{0}/R) cos(omega t),

(b) the current in the inductor is
I_{L} = (V_{0}/X_{L})cos(omega t - pi / 2),

(c) I = I_{R} + I_{L}
= I_{max}cos(omega t - delta) where
tan(delta) = R/X_{L} and
I_{max} = V_{0}/Z with
Z^{ -2} = R^{ -2} + X_{L} -2.

**61**. Solve for the amplitudes of all the currents in a
circuit which has a capacitor in series with a parallel combination of
a resistor and an ideal inductor. Use the values R=10 ohms,
|X_{L}|=30 ohms, |X_{C}|=10 ohms, and
|V_{0}|=20 volts.

**62**. Find the amplitudes of all the currents in the
3-loop circuit in the figure. Use
R_{1} = 10 ohms,
R_{2} = 40 ohms,
R_{3} = 20 ohms,
X_{L} = 30 ohms,
X_{C} = 10 ohms, and
V = 15 Volts. Do the calculation numerically or use a
computer-algebra program to work the problem analytically.