Download η η η η η η η

Homework #4
P.8-16 There is a continuing discuss on radiation hazards to human health. The following
calculations will provide a rough comparison.
a) The U.S. standard for personal safety in a microwave environment is that the power density be
less than 10 (mW/cm2). Calculate the corresponding standard in terms of electric field intensity.
In terms of magnetic field intensity.
b) It is estimated that the earth receives radiant energy from the sun at a rate of about 1.3 (kW/m2)
on a sunny day. Assuming a monochromatic plane wave (which it is not), calculate the equivalent
amplitudes of the electric and magnetic field intensity vectors.
Ans:
2
E
 102 (W/cm2).
Pav 
20
a) E  0.020  2.75 (V/cm) = 275 (V/m).
H 
E
0
 7.28  103 (A/cm) = 0.728 (A/m).
2
E
b) Pav 
 1300 (W/m2).
20
|E| = 990 (V/m), |H| = 2.63 (A/m).
P.8-18 Assuming that the radiation electric field intensity of an antenna system is
E = aE + aE,
find the expression for the average outward power flow per unit area.
Ans:
Since E-field is expressed as above, the magnetic field can be found as
 1
 1
H  aˆ R  E   aˆ E  aˆ E  .




 
1
1
2
Pav  e  E  H    aˆ R
E  E
2
2
2
.
P.8-19 From the point of view of electromagnetics, the power transmitted by a lossless coaxial cable
can be considered in terms of the Poynting vector inside the dielectric medium between the inner
conductor and the outer sheath. Assuming that a d-c voltage V0 applied between the inner conductor
(of radius a) and the outer sheath (of inner radius b) causes a current I to flow to a load resistance,
verify that the integration of the Poynting vector over the cross-sectional area of the dielectric
medium equals the power V0I that is transmitted to the load.
Ans:


From Gauss’s law, E  aˆr  , where   is the line charge density on the inner conductor.
2 r

a 
V0
 
b
V0    E  dr   ln    E  aˆr
.
b
r ln(b / a )
2  a 

I
.
From Ampere’s circuital law, H  aˆ
2 r
  
V0 I
Poynting vector: P  E  H  aˆ z
.
2
2 r ln(b / a )
Power transmitted over cross-sectional area:
 
2 b  1 
V0 I
P   P  ds 
  rdrd  V0 I .
S
2 ln(b / a ) 0 a  r 2 
P.8-22 A uniform sinusoidal plane wave in air with the following phasor expression for electric
intensity
Ei(x, z) = ay10e – j(6x + 8z) (V/m).
is incident on a perfectly conducting plane at z = 0.
a) Find the frequency and wavelength of the wave.
b) Write the instantaneous expression for Ei(x, z; t) and Hi(x, z; t), using a cosine reference.
c) Determine the angle of incidence.
d) Find Er(x, z) and Hr(x, z) of the reflected wave.
e) Find E1(x, z) and H1(x, z) of the total field.
Ans:
P. 8-26 Determine the condition under which the magnitude of the reflection coefficient equals that
of the transmission coefficient for a uniform plane at normal incidence on an interface between two
lossless dielectric media. What is the standing-wave ratio in dB under this condition?
Ans:
For normal incidence: 1 +  = , where ||  1.
If || = ||:  < 0 and  1 –  2 = 2 2 →  1 = 3 2 → || = 0.5.
1 |  |
Thus, S 
 3 → SdB = 20 log103 = 9.54 (dB).
1 |  |
P.8-28 A uniform plane wave in air with Ei(z) = axE0 exp(– j0z) impinges normally onto the surface
at z = 0 of a highly conducting medium having constitutive parameters 0, , and  (/0 >> 1).
a) Find the reflection coefficient.
b) Derive the expression for the fraction of the incident power absorbed by the conducting
medium.
c) Obtain the fraction of the power absorbed at 1 (MHz) if the medium is iron.
Ans:
P.9-2 The electric and magnetic fields of a general TEM wave traveling in the +z-direction along a
transmission line may have both x- and y-components, and both components may be functions of
the transverse dimensions.
a) Find the relations among Ex(x, y), Ey(x, y), Hx(x, y) and Hy(x, y).
b) Verify that all the four field components in part (a) satisfy the two-dimensional Laplace’s
equation for static fields.
Ans:
a)   (aˆ x Ex  aˆ y E y )   j (aˆ x H x  aˆ y H y ).

  E   H , (1)
x
 y
   Ex   H y , (2)
 E
E
y

 x.
(3)
y
 x
  (aˆ x H x  aˆ y H y )  j (aˆ x Ex  aˆ y E y ).

  H   E , (4)
y
x

   H x   E y , (5)
 H
H x
y


. (6)
y
 x
From (1) and (5):     .
Ex

Hy
From (2) or (4):
Ey
From (1) or (5):
b) From (3):
2 Ey
yx
Hx


 2 Ex
.
y 2
From (4), (5) and (6):
(7)

  . (8)


  . (9)

(10)
E
2 Ey
Ex
 2 Ex
 y 


. (11)
x
y
x 2
xy
Combining (10) and (11), we have:
Similarly, we can obtain:
2 Ey
x 2

 2 Ex  2 Ex

 0.
x 2
y 2
2 Ey
y 2
 0,
2 H y 2 H y
2 H x 2 H x

 0.


0,
and
x 2
y 2
x 2
y 2
P.9-4 Consider a transmission line made of two parallel brass strips—c = 1.6  107 (S/m)—of width
20 (mm) and separated by a lossy dielectric slab— = 0,  r = 3,  = 10 – 3(S/m)—of thickness 2.5
(mm). The operating frequency is 500 MHz.
a) Calculate the R, L, G, and C per unit length.
b) Compare the magnitudes of the axial and transverse components of the electric field.
c) Find  and Z0.
Ans:
 r  0
Ez



 7.222  105
Ey
c
c
P.9-7 In the derivation of the approximate formulas of  and Z0 for low-loss lines in Subsection
9-3.1, all terms containing the second and higher powers of (R/L) and (G/C) were neglected in
comparison with unity. At lower frequencies, better approximations than those given in Eqs. (9-54)
and (9-58) may be required. Find new formulas for  and Z0 for low-loss lines that retain terms
containing (R/L)2 and (G/C)2. Obtain the corresponding expression for phase velocity.
Ans: