Homework #3
P.8-3 A Doppler radar is used to determine the speed of a moving vehicle by measuring the
frequency shift of the wave reflected from the vehicle.
a) Assuming that the reflecting surface of the vehicle can be represented by a perfectly
conducting plane and that the transmitted signal is a time-harmonic uniform plane wave
of a frequency f incident normally on the reflecting surface, find the relation between the
frequency shift f and the speed u of the vehicle.
b) Determine u both in (km/hr) and in (miles/hr) if f = 2.33 (kHz) with f = 10.5 (GHz).
Ans:
Assume that the vehicle moves with a velocity u in the +z-direction, which is the direction of
propagation of the incident wave.


j ( t  kz )
, Er   aˆ x E0e j ( t  k z )
a) Ei  aˆ x E 0 e
 
Ei  Er  0 must be satisfied on reflecting surface for all t,
z = ut: (  ku )t  (   k u )t.
u
   
      (k  k )u    u  (   ) .
c
c c 

u   
 f  1  u / c
2u

 1  1  .



 1  , for u  c.

c  

f 1 u / c
c
 f  f   f  
2u
f.
c
b) If f = –2.33 (kHz) with f = 10.5 (GHz): 
u
cf
 33.2857 (m/s)  119.83 (km/hr).
2f
P.8-4 For a harmonic uniform plane wave propagating in a simple medium, both E and H vary in
accordance with the factor exp(–jk  R) as indicated in eq. (8-26). Show that the four Maxwell;s
equations for uniform plane wave in a source-free region reduce to the following:
k  E   H,
k  H   E,
k  E  0,
k  H  0.
Ans:

 j 
t
 
 
 
 


Phasors for the field: E  E0 e  jk R , H  H 0 e  jk R , where E0 and H 0 are constant vectors.
Harmonic field time dependence: e j t ;
 
 
 
 
 



Now,  e  jk R  e  jk R   jk  R  e  jk R   j  k x x  k y y  k z z     j  ax k x  a y k y  az k z  e jk R



 
  jke  jk R .
Thus the Maxwell’s equations:

 
 




  E   e  jk R  E0   j H  k  E   H ;


 
 




  H   e  jk R  H 0  j E  k  H   E;


 
 


  E   e  jk R  E0  0  k  E  0;


 
 


  H   e  jk R  H 0  0  k  H  0.


P.8-6 The E-field of a uniform plane wave propagating in a dielectric medium is given by
E(t , z )  a x 2 cos(108 t  z / 3)  a y sin(108 t  z / 3)
(V/m).
a) Determine the frequency and wavelength of the wave.
b) What is the dielectric constant of the medium?
c) Describe the polarization of the wave.
d) Find the corresponding H-field.
Ans:

Phasor: E ( z )  aˆ x 2e  jz /
3
 aˆ y je  jz /
3
(V/m).
a)   108 (rad/s)  f  108 2  1.57 107 (Hz)
  1 3 (rad/m)    2   2 3 (m).
b) u 
c
r



 c 
 r  
  3.
 
2
c) Left-hand elliptically polarized.
d)  
 120 120


().

r
3


1
H ( z )  aˆ z  E ( z ) 


H (t , z ) 

3
aˆ y 2e  jz /
120
3
 aˆ x je  jz /
3
,
3 
aˆ x sin(108 t  z / 3)  aˆ y 2 cos(108 t  z / 3)  (A/m).

120
P.8-8 Prove the following:
a) An elliptically polarized plane wave can be resolved into right-hand and left-hand circularly
polarized waves.
b) A circularly polarized plane wave can be obtained from a superposition of two oppsitely
directed elliptically polarized waves.
Ans:
a) Let an elliptically polarized plane wave be represented by the phasor (with propagation factor
e –jkz omitted):

E  aˆ x E1  aˆ y E2 e j ,
where E1, E2, and  are arbitrary constants.

Right-hand circularly polarized wave can be expressed as: Erc  Erc (aˆ x  jaˆ y );

and left-hand circularly polarized wave can be expressed as: Elc  Elc (aˆ x  jaˆ y ).
1
1
E1  jE2 e j  and Elc   E1  jE2 e j  ,

2
2
 

then E  Erc  Elc . #
If Erc 
b) A right-hand circularly polarized wave is expressed as:

Erc  Erc (aˆ x  jaˆ y ),
which can be decoupled as



Erc  Erc (aˆ x 12  aˆ y j 2)  Erc (aˆ x 12  aˆ y j )  E   E  ,


where E  and E  are right-hand and left-hand elliptically polarized waves, respectively.
Similarly, for a left-hand circularly polarized wave:



Elc  Elc (aˆ x  aˆ y j )  Elc (aˆ x 12  aˆ y j 2)  Elc (aˆ x 12  aˆ y j )  E  E .
P.8-11 A 3 (GHz), y-polarized uniform plane wave propagates in the + x-direction in a nonmagnetic
medium having a dielectric constant 2.5 and a loss tangent 10 – 2.
a) Determine the distance over which the amplitude of the propagating wave will be cut in
half.
b) Determine the intrinsic impedance, the wavelength, the phase velocity, and the group
velocity of the wave in the medium.
c) Assuming E  a y 50sin(6 109 t   / 3) (V/m) at x = 0, write the instantaneous expression for
H for all t and x.
Ans:
f  3  109 (Hz),  r  2.5 , tan  c 
a) Eq. (8-48):  
 
 102.

         r
  
 0.497 (Np/m).
2  2    c
e  z 
1
2
 x
1

b) Eq. (8-50): c 
ln 2  1.395 (m).
1
r
0
0
  

o
1  j    238(1  j 0.005)  2380.29 (),
2 

 1     2 
Eq. (8-49):     1      31.6 (rad/m),
 8     

2

 0.063 (m),

 1.8973  108 (m/s).

up 
2
1
c  1     
8
1



     1.8975 10 (m/s).
d
 r  8    
d

c) At x = 0, E  aˆ y e j /3 ,
ug 
 1

H  aˆ x  E  aˆ z 0.210e j ( /30.0016 ) ,
c

H ( x, t )  aˆ z 0.210e 0.497 x sin(6 109 t  31.6 x  0.332 ) (A/m).
P. 8-13 Given that the skin depth for graphite at 100 (MHz) is 0.16 (mm), determine
a) the conductivity of graphite, and
b) the distance that a 1 (GHz) wave travels in graphite such that its field intensity is reduced by
30 (dB).
Ans:
a)  
1
 f 
 
1
 f 
2
 0.99 105
b) At f = 109 (GHz),    f   1.98 104
20 log10 e  z  30 (dB)  z 
(S/m).
(Np/m).
1.5
 1.75 104 (m)  0.175 (mm).
 log10 e