Module 1 Assignment
Problem 1:
An electric field propagating in a lossless non-magnetic media is characterized by
𝐸(𝑦, 𝑡) = 100.0 𝑐𝑜𝑠(4𝜋 × 106 𝑡 − 0.1257𝑦) 𝑎𝑧
𝑉
𝑚
Find:
A. The wave amplitude, frequency, propagation velocity, wavelength, and relative
permittivity of the media.
B. Write the expression for H(y, t), including the direction (unit vector) and units.
Solution:
𝑉
A. Using the general form 𝐸(𝑦, 𝑡) = 𝐸0 cos(𝜔𝑡 − 𝛽𝑦) 𝑎𝑧 𝑚
𝑉
i.
Wave amplitude = 𝐸0 = 100 𝑚
ii.
Frequency = 𝑓 = 2𝜋 =
𝜔
4𝜋×106
2𝜋
2𝜋
2𝜋
= 2 𝑀𝐻𝑧
iii.
Wavelength = 𝜆 = 𝛽 = 0.1257 = 49.986𝑚
iv.
Propagation velocity = 𝑣𝑝 = 𝜆 ∙ 𝑓 = 49.986𝑚 ∙ 2 ∙ 106 𝑠 = 99.97 ∙ 106 𝑠
v.
Relative permittivity = 𝑣𝑝 =
1
𝐸̅
̅ (𝑦, 𝑡) = =
B. 𝐻
𝜂
100.0
𝜂
𝑐2
𝑐
(3∙108 )
√ 𝑟 𝑟
𝑝
𝑐𝑜𝑠(4𝜋 × 106 𝑡 − 0.1257𝑦) 𝑎𝑥 𝑚 =
√𝜂0
100.0 ∙ √3
𝑐𝑜𝑠(4𝜋 × 106 𝑡 − 0.1257𝑦) 𝑎𝑥
𝐴
=
𝑚
𝐴
=
𝑚
√377
𝐴
𝐻(𝑦, 𝑡) = 0.796 𝑐𝑜𝑠(4𝜋 × 106 𝑡 − 0.1257𝑦) 𝑎𝑥
𝑚
𝑐𝑜𝑠(4𝜋 × 106 𝑡 − 0.1257𝑦) 𝑎𝑥
Problem 2:
A magnetic field propagating in free space is given by
𝐻(𝑧, 𝑡) = 20. 𝑠𝑖𝑛(𝜋 × 108 𝑡 + 𝛽𝑧) 𝑎𝑥
Find f, λ.
Solution:
2
, 𝜇𝑟 = 1, 𝜖𝑟 = 𝑣2 = (99.97∙106 )2 = 9.005
𝜇 𝜖
𝐴
100.0 ∙ √𝜖
𝑚
𝐴
𝑚
𝜔
𝜋×108
𝑐
2𝜋
3∙108
i.
Frequency = 𝑓 = 2𝜋 =
ii.
Wavelength = 𝜆 = 𝑓 = 50∙106 = 6𝑚
= 50 𝑀𝐻𝑧