Problem 4.41 A cylindrical bar of silicon has a radius of 4 mm and a length of
8 cm. If a voltage of 5 V is applied between the ends of the bar and e = 0.13
(m2 /Vs), h = 0.05 (m2 /Vs), Ne = 1.5 1016 electrons/m3 , and Nh = Ne , nd the
following:
(a) The
Problem 4.13 Charge q1 = 4 C is located at (1 cm, 1 cm, 0) and charge q2
is located at (0, 0, 4 cm). What should q2 be so that E at (0, 2 cm, 0) has no
y-component?
Solution:
z
q2
4 cm
^
R1 = -x + y(2-1) = (-^ + ^ ) cm
xy
R2
1 cm
0
1 cm
q1
R1
^
^
R2 = (y2
Problem 4.14 A line of charge with uniform density = 8 ( C/m) exists in air
along the z-axis between z = 0 and z = 5 cm. Find E at (0,10 cm,0).
Solution: Use of Eq. (4.21c) for the line of charge shown in Fig. P4.14 gives
1
dl
l
R
,
40 l
R 2
R = y 0.1
Problem 4.22
Given the electric ux density
D = x2(x + y) + y(3x 2y) (C/m2 )
determine
(a) v by applying Eq. (4.26).
(b) The total charge Q enclosed in a cube 2 m on a side, located in the rst octant
with three of its sides coincident with the x-, y-, and
Problem 4.23 Repeat Problem 4.22 for D = xxy3 z3 (C/m2 ).
Solution:
(a) From Eq. (4.26), v = D =
(xy3 z3 ) = y3 z3 .
x
(b) Total charge Q is given by Eq. (4.27):
Q=
V
D dV =
2
2
z=0 y=0
xy4 z4
y z dx dy dz =
16
x =0
2
2
2
2
33
= 32 C.
x=0 y=0 z=0
(c) Us
Problem 4.26 In a certain region of space, the charge density is given in cylindrical
coordinates by the function:
v = 5rer
(C/m3 )
Apply Gausss law to nd D.
Solution:
z
r
L
Figure P4.26: Gaussian surface.
Method 1: Integral Form of Gausss Law
Since v var
Problem 4.29 A spherical shell with outer radius b surrounds a charge-free cavity
of radius a < b (Fig. P4.29). If the shell contains a charge density given by
v0
v = 2 ,
a R b,
R
where v0 is a positive constant, determine D in all regions.
r3
v
a
r1
r2
b
Problem 4.30 A square in the xy plane in free space has a point charge of +Q at
corner (a/2, a/2), the same at corner (a/2, a/2), and a point charge of Q at each
of the other two corners.
(a) Find the electric potential at any point P along the x-axis.
(b
Problem 4.34 Find the electric potential V at a location a distance b from the origin
in the xy plane due to a line charge with charge density and of length l . The line
charge is coincident with the z-axis and extends from z = l /2 to z = l /2.
Solution:
Problem 4.36 For each of the distributions of the electric potential V shown in
Fig. P4.36, sketch the corresponding distribution of E (in all cases, the vertical axis
is in volts and the horizontal axis is in meters).
Solution:
V
30
3
5
8
11 13
16
x
30
E
Problem 4.40 The xy plane contains a uniform sheet of charge with s1 = 0.2
(nC/m2 ). A second sheet with s2 = 0.2 (nC/m2 ) occupies the plane z = 6 m. Find
VAB , VBC , and VAC for A(0, 0, 6 m), B(0, 0, 0), and C(0, 2 m, 2 m).
Solution: We start by nding t
Problem 4.43 A 100-m-long conductor of uniform cross-section has a voltage drop
of 4 V between its ends. If the density of the current owing through it is 1.4 106
(A/m2 ), identify the material of the conductor.
Solution: We know that conductivity charact
Problem 4.46 A 2 103 -mm-thick square sheet of aluminum has 5 cm 5 cm
faces. Find the following:
(a) The resistance between opposite edges on a square face.
(b) The resistance between the two square faces. (See Appendix B for the electrical
constants of m
Problem 4.51 Figure P4.51 shows three planar dielectric slabs of equal thickness
but with different dielectric constants. If E0 in air makes an angle of 45 with respect
to the z-axis, nd the angle of E in each of the other layers.
z
E0
45
0 (air)
1 = 30
2
Problem 4.53 Dielectric breakdown occurs in a material whenever the magnitude
of the eld E exceeds the dielectric strength anywhere in that material. In the coaxial
capacitor of Example 4-12,
(a) At what value of r is |E | maximum?
(b) What is the breakdo
Problem 4.58 The capacitor shown in Fig. P4.58 consists of two parallel dielectric
layers. Use energy considerations to show that the equivalent capacitance of the
overall capacitor, C, is equal to the series combination of the capacitances of the
individ
Problem 4.60 A coaxial capacitor consists of two concentric, conducting,
cylindrical surfaces, one of radius a and another of radius b, as shown in Fig. P4.60.
The insulating layer separating the two conducting surfaces is divided equally into
two semi-cy
Problem 4.9 A circular beam of charge of radius a consists of electrons moving
with a constant speed u along the +z-direction. The beams axis is coincident with
the z-axis and the electron charge density is given by
v = cr2
(c/m3 )
where c is a constant a
Problem 4.6 If J = y4xz (A/m2 ), nd the current I owing through a square with
corners at (0, 0, 0), (2, 0, 0), (2, 0, 2), and (0, 0, 2).
Solution: Using Eq. (4.12), the net current owing through the square shown in Fig.
P4.6 is
2
I=
S
J ds =
2
2
x=0 z=0
Problem 4.4 If the line charge density is given by l = 24y2 (mC/m), nd the total
charge distributed on the y-axis from y = 5 to y = 5.
Solution:
24y3
24y dy =
Q=
l dy =
3
5
5
5
5
5
2
= 2000 mC = 2 C.
5
Problem 1.8 Two waves on a string are given by the following functions:
y1 (x, t ) = 4 cos(20t 30x) (cm)
y2 (x, t ) = 4 cos(20t + 30x) (cm)
where x is in centimeters. The waves are said to interfere constructively when their
superposition |ys | = |y1 + y2
Problem 1.12 Given two waves characterized by
y1 (t ) = 3 cos t
y2 (t ) = 3 sin( t + 60 ),
does y2 (t ) lead or lag y1 (t ) and by what phase angle?
Solution: We need to express y2 (t ) in terms of a cosine function:
y2 (t ) = 3 sin( t + 60 )
t 60 = 3 co
Problem 1.13 The voltage of an electromagnetic wave traveling on a transmission
line is given by (z, t ) = 5e z sin(4 109t 20 z) (V), where z is the distance in
meters from the generator.
(a) Find the frequency, wavelength, and phase velocity of the wave.
Problem 1.14 A certain electromagnetic wave traveling in seawater was observed
to have an amplitude of 98.02 (V/m) at a depth of 10 m, and an amplitude of 81.87
(V/m) at a depth of 100 m. What is the attenuation constant of seawater?
Solution: The amplitu
Problem 1.15 A laser beam traveling through fog was observed to have an intensity
of 1 ( W/m2 ) at a distance of 2 m from the laser gun and an intensity of 0.2
( W/m2 ) at a distance of 3 m. Given that the intensity of an electromagnetic
wave is proportio
Problem 1.27 Find the instantaneous time sinusoidal functions corresponding to
the following phasors:
(a) V = 5e j /3 (V)
(b) V = j6e j /4 (V)
(c) I = (6 + j8) (A)
(d) I = 3 + j2 (A)
(e) I = j (A)
(f) I = 2e j /6 (A)
Solution:
(a)
V = 5e j /3 V = 5e j( /3
Problem 2.75 Generate a bounce diagram for the voltage V (z, t ) for a 1-mlong
lossless line characterized by Z0 = 50 and up = 2c/3 (where c is the velocity of
light) if the line is fed by a step voltage applied at t = 0 by a generator circuit with
Vg = 6
Problem 2.77 In response to a step voltage, the voltage waveform shown in
Fig. P2.77 was observed at the sending end of a lossless transmission line with
Rg = 50 , Z0 = 50 , and r = 2.25. Determine the following:
(a) The generator voltage.
(b) The length
Problem 2.78 In response to a step voltage, the voltage waveform shown in
Fig. P2.78 was observed at the sending end of a shorted line with Z0 = 50 and
r = 4. Determine Vg , Rg , and the line length.
V(0, t)
12 V
3V
0
0.75 V
7s
t
14 s
Figure P2.78: Voltag
Problem 2.80 A generator circuit with Vg = 200 V and Rg = 25 was used to
excite a 75- lossless line with a rectangular pulse of duration = 0.4 s. The line
is 200 m long, its up = 2 108 m/s, and it is terminated in a load RL = 125 .
(a) Synthesize the volt