Page 441 - Electromagnetics

˜
where A h is given in (6.48). Using
ˆ
ˆ
ˆ r × ˆ x = φ cos θ cos φ + θ sin φ
we find that
e − jkr cos(π X) sin(πY)
˜ ˆ ˆ
E = jk 0 abE 0 θ sin φ + φ cos θ cos φ 2 2 .
r π − 4(π X) πY
˜
˜
The magnetic field is merely H = (ˆ r × E)/η.

6.4 Problems
6.1 Beginning with the Lorentz reciprocity theorem, derive (6.8).

6.2 Obtain (6.8) by substitution of (6.7) into Faraday’s law.
6.3 Show that (6.8) returns the null result when evaluated within the excluded regions.

6.4 Show that under the condition kr 1 the formula for the magnetic field of a
dipole antenna (6.28) reduces to (6.32), while the formulas for the electric fields (6.29)
and (6.30) reduce to (6.31).
6.5 Consider the dipole antenna shown in Figure 6.4. Instead of a standing-wave
current distribution, assume the antenna carries a traveling-wave current distribution
˜
˜ i
J (r,ω) = ˆ zI(ω)e − jk|z| δ(x)δ(y), −l ≤ z ≤ l.
Find the electric and magnetic fields at all points away from the current distribution.
Specialize the result for kr 1.
6.6 A circular loop of thin wire has radius a and lies in the z = 0 plane in free space.
A current is induced on the wire with the density
δ(θ − π/2)
ˆ ˜
˜
J(r,ω) = φI(ω) cos [k 0 a(π −|φ|)] δ(r − a) , |φ|≤ π.
r
Compute the far-zone fields produced by this loop antenna. Specialize your results for
the electrically-small case of k 0 a 1. Compute the time-average power radiated by, and
the radiation resistance of, the electrically-small loop.
6.7 Consider a plane wave with the fields
˜
E 0
˜
˜
˜
E = E 0 ˆ xe − jkz , H = ˆ ye − jkz ,
η
normally incident from z < 0 on a square aperture of side a in a PEC ground screen
at z = 0. Assume that the field in the aperture is identical to the field of the plane
wave with the screen absent (this is called the Kirchhoff approximation). Compute the
far-zone electromagnetic fields for z > 0.
6.8 Consider a coaxial cable of inner radius a and outer radius b, opening into a PEC
ground plane at z = 0. Assume that only the TEM wave exists in the line and that no
higher-order modes are created when the wave reflects from the aperture. Compute the
far-zone electric and magnetic fields of this aperture antenna.

© 2001 by CRC Press LLC