Page 567 - Engineering Electromagnetics, 8th Edition

CHAPTER 14 ELECTROMAGNETIC RADIATION AND ANTENNAS 549

current element of differential length d, oriented along the z axis, and
centered at the origin.
1
14.6 Evaluate the time-average Poynting vector, <S>= Re E s × H ∗ s for

2
the Hertzian dipole, assuming the general case that involves the field
components as given by Eqs. (10), (13a), and (13b). Compare your result to
the far-zone case, Eq. (26).
14.7 A short current element has d = 0.03λ. Calculate the radiation resistance
that is obtained for each of the following current distributions: (a) uniform,
I 0 ;(b) linear, I(z) = I 0 (0.5d −|z|)/0.5d;(c) step, I 0 for 0 < |z| < 0.25d
and 0.5I 0 for 0.25d < |z| < 0.5d.
14.8 Evaluate the time-average Poynting vector, <S>= (1/2)Re E s × H ∗ s for

the magnetic dipole antenna in the far zone, in which all terms of order 1/r 2
4
and 1/r are neglected in Eqs. (48), (49), and (50). Compare your result to
the far-zone power density of the Hertzian dipole, Eq. (26). In this
comparison, and assuming equal current amplitudes, what relation between
loop radius, a, and dipole length, d,would result in equal radiated powers
from the two devices?
14.9 A dipole antenna in free space has a linear current distribution with zero
current at each end, and with peak current I 0 at the enter. If the length d is
0.02λ, what value of I 0 is required to (a) provide a radiation-field amplitude
of 100 mV/m at a distance of 1 mi, at θ = 90 ;(b) radiate a total power of
◦
1W?
14.10 Show that the chord length in the E-plane plot of Figure 14.4 is equal to
b sin θ, where b is the circle diameter.
14.11 A monopole antenna extends vertically over a perfectly conducting plane,
and has a linear current distribution. If the length of the antenna is 0.01λ,
what value of I 0 is required to (a) provide a radiation-field amplitude of
100 mV/m at a distance of 1 mi, at θ = 90 ;(b) radiate a total power of 1
◦
W? Assume free space above the plane.
14.12 Find the zeros in θ for the E-plane pattern of a dipole antenna for which (a)
= λ;(b)2 = 1.3λ. Use Figure 14.8 as a guide.
14.13 The radiation field of a certain short vertical current element is
E θs = (20/r) sin θ e − j10πr V/m if it is located at the origin in free space.
(a) Find E θs at P(r = 100, θ = 90 , φ = 30 ). (b) Find E θs at
◦
◦
P(100, 90 , 30 )if the vertical element is located at A(0.1, 90 , 90 ).
◦
◦
◦
◦
(c) Find E θs at P(100, 90 , 30 )if identical vertical elements are located at
◦
◦
A(0.1, 90 , 90 ) and B(0.1, 90 , 270 ).
◦
◦
◦
◦
14.14 Fora dipole antenna of overall length 2 = λ, evaluate the maximum
directivity in decibels, and the half-power beamwidth.
14.15 Fora dipole antenna of overall length 2 = 1.3λ, determine the locations in
θ and the peak intensity of the sidelobes, expressed as a fraction of the main