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Solved Problems-4

Problem-4
Consider a monochromatic plane wave, where the electric field is given by
Description: 37766.png
 
where E0 is an arbitrary constant vector and other symbols have their usual meanings.
 
(a) Show that the electric field vector lies in a direction perpendicular to the propagation.
 
(b) Determine the corresponding magnetic field.
 
(c) Calculate the wave impedance and show that this is equal to the intrinsic impedance of the medium.
Solution
Here, Description: 37757.png
 
By Maxwell’s equation,
 
 
Description: 37748.png
 
or, Description: 37740.png
 
or, Description: 37733.png
 
Comparing both sides, we get,
 
 
Hx = 0, Hz = 0
 
and
 
Description: 37725.png
 
Description: 37716.png
 
(a) Here, the electric field propagates in the x-direction and the magnetic field propagates in the y-direction whereas the wave propagates in the z-direction. Thus, we can say that the electric field vector lies in a direction perpendicular to the propagation.

Since the electric field is directed across unit vector Description: 37707.png and the magnetic field is directed across the unit vector Description: 37697.png, we conclude that the two fields are perpendicular to each other.

(b) The corresponding magnetic field is given as,
 
Description: 37688.png

(c) The wave impedance is given as,
 
 
Description: 37680.png
 
This is equal to the intrinsic impedance of the medium.
 





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