Problem 7.1 The magnetic eld of a wave propagating through a certain
nonmagnetic material is given by
H = z 30 cos(108t 0.5y) (mA/m)
Find the following:
(a) The direction of wave propagation.
(b) The phase velocity.
(c) The wavelength in the material.
(d)
Problem 3.35 Transform the following vectors into spherical coordinates and then
evaluate them at the indicated points:
(a) A = xy2 + yxz + z4 at P1 = (1, 1, 2),
2 + y2 + z2 ) z(x2 + y2 ) at P = (1, 0, 2),
(b) B = y(x
2
(c) C = r cos sin + z cos sin at P3
Problem 3.34 Transform the following vectors into cylindrical coordinates and
then evaluate them at the indicated points:
(a) A = x(x + y) at P1 = (1, 2, 3),
(b) B = x(y x) + y(x y) at P2 = (1, 0, 2),
(c) C = xy2 /(x2 + y2 ) yx2 /(x2 + y2 ) + z4 at P3 = (
Problem 3.28 A vector eld is given in cylindrical coordinates by
E = rr cos + r sin + zz2 .
Point P = (2, , 3) is located on the surface of the cylinder described by r = 2. At
point P, nd:
(a) the vector component of E perpendicular to the cylinder,
(b) t
Problem 3.26 Find the volumes described by
(a) 2 r 5; /2 ; 0 z 2,
(b) 0 R 5; 0 /3; 0 2 .
Also sketch the outline of each volume.
Solution:
z
z
5
2
2
x
5
y
y
x
(a)
(b)
Figure P3.26: Volumes described by Problem 3.26 .
(a) From Eq. (3.44),
v=
2
5
z=0 = /2 r
Problem 2.81 For the circuit of Problem 2.80, generate a bounce diagram for the
current and plot its time history at the middle of the line.
Solution: Using the values for g and L calculated in Problem 2.80, we reverse
their signs when using them to const
Problem 3.25 Use the appropriate expression for the differential surface area ds to
determine the area of each of the following surfaces:
(a) r = 3; 0 /3; 2 z 2,
(b) 2 r 5; /2 ; z = 0,
(c) 2 r 5; = /4; 2 z 2,
(d) R = 2; 0 /3; 0 ,
(e) 0 R 5; = /3; 0 2 .
Al
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
Problem 2.79 Suppose the voltage waveform shown in Fig. P2.77 was observed at
the sending end of a 50- transmission line in response to a step voltage introduced
by a generator with Vg = 15 V and an unknown series resistance Rg . The line is 1 km
in lengt
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
7 s
t
14 s
Figure P2.78: Volta
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 = 60
Problem 2.68 A 50- lossless line is to be matched to an antenna with
ZL = (75 j20) using a shorted stub. Use the Smith chart to determine the stub
length and distance between the antenna and stub.
1.0
1.2
1.4
1.6
7
3
0.1
1.8
0.6
0.3
0.2
2
0.4
0.2
40
0.3
3
Problem 2.63 A 50- lossless line 0.6 long is terminated in a load with
ZL = (50 + j25) . At 0.3 from the load, a resistor with resistance R = 30 is
connected as shown in Fig. P2.63. Use the Smith chart to nd Zin .
Zin
Z0 = 50
0.3
30
Z0 = 50
ZL
0.3
ZL =
Problem 2.61 Using a slotted line on a 50- air-spaced lossless line, the following
measurements were obtained: S = 1.6 and |V |max occurred only at 10 cm and 24 cm
from the load. Use the Smith chart to nd ZL .
1.2
1.0
6
0.3
1.4
4
0.1
1.6
7
0.3
3
0.1
1.8
0
Problem 2.58 A lossless 100- transmission line 3 /8 in length is terminated in
an unknown impedance. If the input impedance is Zin = j2.5 ,
(a) Use the Smith chart to nd ZL .
(b) Verify your results using CD Module 2.6.
1.2
1.0
1.4
0.1
1.6
7
60
0.3
3
0.1
Problem 2.53 A lossless 50- transmission line is terminated in a load with
ZL = (50 + j25) . Use the Smith chart to nd the following:
(a) The reection coefcient .
(b) The standing-wave ratio.
(c) The input impedance at 0.35 from the load.
(d) The input ad
Problem 4.12 Three point charges, each with q = 3 nC, are located at the corners
of a triangle in the xy plane, with one corner at the origin, another at (2 cm, 0, 0),
and the third at (0, 2 cm, 0). Find the force acting on the charge located at the origi
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
R2
R = y 0.1 zz
Problem 4.15 Electric charge is distributed along an arc located in the xy plane
and dened by r = 2 cm and 0 /4. If = 5 ( C/m), nd E at (0, 0, z) and
then evaluate it at:
(a) The origin.
(b) z = 5 cm
(c) z = 5 cm
Solution: For the arc of charge shown in F
Problem 6.18 An electromagnetic wave propagating in seawater has an electric
eld with a time variation given by E = zE0 cos t. If the permittivity of water is
810 and its conductivity is 4 (S/m), nd the ratio of the magnitudes of the conduction
current de
Problem 6.16 The parallel-plate capacitor shown in Fig. P6.16 is lled with a lossy
dielectric material of relative permittivity r and conductivity . The separation
between the plates is d and each plate is of area A. The capacitor is connected to
a time-v
Problem 6.14 The plates of a parallel-plate capacitor have areas of 10 cm2 each
and are separated by 2 cm. The capacitor is lled with a dielectric material with
= 40 , and the voltage across it is given by V (t) = 30 cos 2 106t (V). Find the
displacement
Problem 6.7 The rectangular conducting loop shown in Fig. P6.7 rotates at 6,000
revolutions per minute in a uniform magnetic ux density given by
B = y 50 (mT).
Determine the current induced in the loop if its internal resistance is 0.5 .
z
2c
m
B
3 cm
B
y
Problem 6.3 A coil consists of 100 turns of wire wrapped around a square frame
of sides 0.25 m. The coil is centered at the origin with each of its sides parallel to
the x- or y-axis. Find the induced emf across the open-circuited ends of the coil if the
Problem 6.6 The square loop shown in Fig. P6.6 is coplanar with a long, straight
wire carrying a current
I(t) = 5 cos(2 104t) (A).
(a) Determine the emf induced across a small gap created in the loop.
(b) Determine the direction and magnitude of the curre
Problem 6.1 The switch in the bottom loop of Fig. P6.1 is closed at t = 0 and
then opened at a later time t1 . What is the direction of the current I in the top loop
(clockwise or counterclockwise) at each of these two times?
R2
I
t = t1
t=0
R1
Figure P6.
Problem 5.34 In Fig. P5.34, the plane dened by x y = 1 separates medium 1 of
permeability 1 from medium 2 of permeability 2 . If no surface current exists on
the boundary and
B1 = x2 + y3 (T)
nd B2 and then evaluate your result for 1 = 52 .
Hint: Start by
Problem 5.32
The xy plane separates two magnetic media with magnetic
permeabilities 1 and 2 (Fig. P5.32). If there is no surface current at the interface
and the magnetic eld in medium 1 is
H1 = xH1x + yH1y + zH1z
nd:
(a) H2
(b) 1 and 2
(c) Evaluate H2 ,