Page 433 - Electromagnetics

This expression cannot be computed in closed form. For a short dipole we may use (6.33)
to approximate the power, but the result is somewhat misleading since the current on a
˜
short dipole is much smaller than I. A better measure of the strength of the current is
its value at the center, or feedpoint, of the dipole. This input current is by (6.26) merely
˜
˜ I 0 (ω) = I(ω) sin(kl). Using this we find
ˇ 2
l
|I 0 | 1 2 π 3 π
2
ˇ 2
P av ≈ η (kl) sin θ dθ = η |I 0 | .
4π 4 0 3 λ
This is exactly 1/4 of the power radiated by a Hertzian dipole of the same length and
current amplitude (5.95). The factor of 1/4 comes from the difference between the current
of the dipole antenna, which is zero at each end, and the current on the Hertzian dipole,
which is constant across the length of the antenna. It is more common to use a dipole
antenna that is a half wavelength long (2l = λ/2), since it is then nearly resonant. With
this we have through numerical integration the free-space radiated power

ˇ 2 π 2 π
|I 0 | cos 2 cos θ
ˇ 2
P av = η 0 dθ = 36.6|I 0 |
4π 0 sin θ
and the radiation resistance
2P av 2P av
R r = 2 = = 73.2 .
ˇ
ˇ 2
|I(z = 0)| |I 0 |

6.3 Fields in a bounded, source-free region
In § 6.2 we considered the first important special case of the Stratton–Chu formula:
sources in an unbounded medium. We now consider the second important special case
of a bounded, source-free region. This case has important applications to the study of
microwave antennas and, in its scalar form, to the study of the diffraction of light.

6.3.1 The vector Huygens principle

We may derive the formula for a bounded, source-free region of space by specializing
the general Stratton–Chu formulas. We assume that all sources of the fields are within
the excluded regions and thus set the sources to zero within V . From (6.7)–(6.8) we have

N
˜ ˜ ˜ ˜

E(r,ω) = (ˆ n × E) ×∇ G + (ˆ n · E)∇ G − jω ˜µ(ˆ n × H)G dS −
n=1 S n
N
1
˜

− c (dl · H)∇ G, (6.34)
jω˜
n=1
na + nb
and
N
˜ ˜ ˜ c ˜

H(r,ω) = (ˆ n × H) ×∇ G + (ˆ n · H)∇ G + jω˜ (ˆ n × E)G dS +
n=1 S n
© 2001 by CRC Press LLC