Page 418 - Electromagnetics

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(d) E and H may be obtained from the potential Π e = ˆ x(E 0 /k )e − jkz ;
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(e) E and H may be obtained from the potential Π e = ˆ z( j E 0 x/k)e − jkz ;
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(f) E and H may be obtained from the potential Π h = ˆ z( j E 0 y/kη)e − jkz .
5.22 Prove the orthogonality relationships (5.149) and (5.150) for the longitudinal
ˇ
ˇ
fields in a lossless waveguide. Hint: Substitute a = ψ e and b = ψ h into Green’s second
identity (B.30) and apply the boundary conditions for TE and TM modes.
5.23 Verify the waveguide orthogonality conditions (5.151)-(5.152) by substituting the
field expressions for a rectangular waveguide.
5.24 Showthat the time-average power carried by a propagating TE mode in a lossless
waveguide is given by
1 2
ˇ ˇ
∗
P av = ωµβk c ψ h ψ dS.
h
2 CS
5.25 Showthat the time-average stored energy per unit length for a propagating TE
mode in a lossless waveguide is
2 2
ˇ ˇ
∗
W e /l = W m /l = (ωµ) k c ψ h ψ dS.
h
4 CS
5.26 Consider a waveguide of circular cross-section aligned on the z-axis and filled with
a lossless material having permittivity and permeability µ. Solve for both the TE and
TM fields within the guide. List the first ten modes in order by cutoff frequency.
5.27 Consider a propagating TM mode in a lossless rectangular waveguide. Show that
the time-average power carried by the propagating wave is
1 2 2 ab
= ω β nm k |A nm | .
P av nm c nm
2 4
5.28 Consider a propagating TE mode in a lossless rectangular waveguide. Show that
the time-average power carried by the propagating wave is
1 2 2 ab
= ωµβ nm k |B nm | .
P av nm
2 c nm 4
5.29 Consider a homogeneous, lossless region of space characterized by permeability µ
and permittivity . Beginning with the time-domain Maxwell equations, show that the
θ and φ components of the electromagnetic fields can be written in terms of the radial
components. From this give the TE r –TM r field decomposition.

5.30 Consider the formula for the radar cross-section of a PEC sphere (5.193). Show
that for the monostatic case the RCS becomes

2
2 ∞ n
"
σ = λ (−1) (2n + 1) .
(2)
ˆ
ˆ
(2)
H n (ka)H n (ka)
4π
n=1
5.31 Beginning with the monostatic formula for the RCS of a conducting sphere given
in Problem 5.30, use the small-argument approximation to the spherical Hankel functions
to showthat the RCS is proportional to λ −4 when ka 1.
© 2001 by CRC Press LLC