Practice problems I - Staff.kmutt.ac.th

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18 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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Uniform Plane Wave


1. A wave with


= 6.0 cm in air is incident on a nonmagnetic, lossless liquid media.
In the liquid, the wavelength is measured as 1.0 cm. What is the wave’s frequency (a)
in air? (b) in the liquid? (c) What is the liquid’s relative
permittivity?


(a)

(b) the frequency doesn’t change with the media (the wavelength does) so
f

= 5 GHz

(c)


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2. Given


= 1.0x10
-
5

S/
m ,

r

= 2.0,

r

= 50., and
f

= 10. MHz, find

,

,

, and

.






Inserting these into the expressions for


and

,


___
__________________________________________________________________

3. Suppose in free space,
H
(
x,t
) = 100.cos(2

x10
7
t




x

+

/4)
a
z

mA/m. Find
E
(
x,t
).




Since free space is stated,


and then


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4. A 100 MHz wave in free space propagates in the y direction with an amplitude of 1
V/m. If the electric field vector for this wave has
only an
a
z

component, find the
instantaneous expression for the electric and magnetic fields.


From the given information we have

and

or
.

Now to find
H
.


So


or


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5. In a lossless, nonmagnetic material with

r

= 16,
H

= 100 cos(

t



10
y
)
a
z

mA/m.
Determine the propagation velo
city, the angular frequency, and the instantaneous
expression for the electric field intensity.







___________________
__________________________________________________

6. In a media with properties



= 0.00964 S/m ,

r

= 1.0,

r

= 100., and
f

= 100. MHz,
a 1.0 mA/m amplitude magnetic field travels in the +
x

direction with its field vector in
the
z

direction. Find the instantaneous form of the related electric field intensity.






Finally,


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7. In seawater, a propagating electric field is given by
E
(
z,t
) = 20.e
-

z

cos(

x


t




z

+ 0.5)
a
y

V/m. Assuming

’’
=0, find (a)


and

, and (b) the instantaneous form of
H
.


For seawater we have

r

= 72,



= 5, and

r

= 1.

So:







or with appropriate significant digits:


_____________________________________________________________________

8. For Nickel (


= 1.45 x 10
7
,

r

= 600), make a table of

,

,

, u
p
, and


for 1Hz,
1kHz, 1MHz, and 1 GHz.


For Ni we have


= 1.45x10
7
S/m,

r

= 600



= 1/




Table

f
(Hz)=

1

10
3

10
6

10
9


(Np/m)

185

5860

185x10
3

5.9x10
6


(rad/m)

185

5860

185x10
3

5.9x10
6



18e
j45º


570e
j45º


18e
j45º

m


0.57e
j45º




5.4mm

170

m

5.3

m

170nm

u
p
(m/s)

12x10
6

12x10
6

12x10
6

12x10
6

9. A semi
-
infinite slab exists for
z

> 0 with


= 300 S/m,

r

= 10.2, and

r

= 1.0. At
the surface (
z

= 0),


E
(0,
t
) = 1.0 cos(


x 10
6
t
)
a
x

V/m.

Find the instantaneous expressions for
E

and
H

anywhere in the slab.


The general expression for
E

is:



Here,




(i.e. it is a good conductor), so



So now we have


To find
B

we’ll work in phasors.





_____________________________________________________________________

10. A 600 MHz uniform plane wave incident in the
z

direction on a thick slab of
Teflon (

r

= 2.1,

r

= 1.0) imparts a 1.0 V/m amplitude
y
-
polarized electric field
intensity at
the surface. Assuming


= 0 for Teflon, find in the Teflon (a)
E
(
z,t
), (b)
H
(
z,t
) and (c)
P
av
.




Teflon:


= 0 so


= 0,

and

(a)

(b)


(c)

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11.
A 200 MHz uniform plane wave incident on a thick copper slab imparts a 1.0
mV/m amplitude at the surface.

How much power passes through a square meter at
the surface? How much power passes through a square meter area 10.

m beneath the
surface?



Cu:


Now at 10

m beneath the surface, we have



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________________________

12. Given
E
(
z,t
) = 10.cos
(

t
-

z)
a
x

-

20.cos
(

t
-

z
-
45

)
a
y

V/m, find the polarization
and handedness.


The field can be rewritten as
E
(
z,t
) = 10.cos
(

t
-

z)
a
x

+ 20.cos
(

t
-

z
-
45

-
180°
)
a
y

or
E
(
z,t
) = 10.cos
(

t
-

z)
a
x

+ 20.cos
(

t
-

z
+135°
)
a
y

_____________________________________________________________________

13. Given



we say that
E
y

leads
E
x

for 0


<


< 180

, and that
E
y

lags
E
x

when

180


<


< 0

.
Determine the handedness for each of these two cases.


For

0 <


< 180°, we have LHP


For 180° <


< 360°, we have RHP

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14. Suppose a UPW in air carrying an average power density of 100 mW/m
2

is
normally incident on a nonmagnetic material with

r

= 11. What is the time
-
averaged
power density of the reflected and transmitted waves?






______________________________________________________
_______________

15. A UPW in a lossless nonmagnetic

r

= 16 media (for
z

< 0) is given by

E
(
z,t
) = 10.cos
(

t
-

1
z)
a
x

+ 20.cos
(

t
-

1
z+

/3)
a
y

V/m.

This is incident on a lossless media characterized by

r

= 12,

r

= 6.0 (for
z

> 0). Find
the instantaneous exp
ressions for the reflected and transmitted electric field
intensities.









or
, so

.

_____________________________________________________________________

16. The wave
E
i

= 10.cos(2

x 10
8
t
-


1
z
)
a
x

V/m is incident from air onto a copper
conductor. Find
E
r
,
E
t

an
d the time
-
averaged power density transmitted at the
surface.


For copper we have


so

We find

So
E
r

=
-
10.cos(2

x 10
8
t +

1
z)
a
x

V/m


and


17. A wave specified by
E
i

= 100.cos(

x10
7
t
-

1
z
)
a
x

V/m is incident from air (at
z

< 0)
to a nonmagnetic media (
z

> 0,


= 0.050 S/m,

r

= 9.0). Find
E
r
,
E
t

and
SWR
. Also
find the average power densities f
or the incident, reflected and transmitted waves.



In this problem we find in medium 2 (
z
> 0) that


= 0.0025 and


= 0.05. These
values are too close to allow for simplifying assumptions. Using (5.13) and (5.31),
we calcula
te:

.

Then,



,

so

,

so





(check: 13.3 W/m
2

= 10.7 W/m
2

+ 2.6 W/m
2
)

_____________________________________________________________________

18. A 100 MHz TM polarized wave with amplitude 1.0 V/m is obliquely inci
dent
from air (
z

< 0) onto a slab of lossless, nonmagnetic material with

r

= 25 (
z

> 0). The
angle of incidence is 40

. Calculate (a) the angle of transmission, (b) the reflection
and transmission coefficients, and (c) the incident, reflected and transm
itted fields.


(a) The material parameters in this problem are the same as for P5.48. So, once again
we have

t

= 7.4°. Also,

1

= 2.09 rad/m and

2

= 10.45 rad/m.

(b)



(c)

Incident:





Reflected:





transmitted:




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