(a)
Find the wavelength and wave frequency of the wave in air and the transmitted wave in the dielectric medium.
(a)
Answer to Problem 60P
The wavelength and wave frequency of the wave in air are
The wavelength and wave frequency of the transmitted wave in the lossless dielectric medium are
Explanation of Solution
Calculation:
Write the expression for given incident electric field in medium-1 (air).
From the expression, the values of magnitude of incident electric field at
Consider the expression to find the wavelength of the wave in air (medium-1).
Substitute
Consider the expression for phase constant of medium-1.
Rewrite the expression for the given data.
Rearrange the expression for wave frequency.
Substitute
The wave frequency is same for both the waves in air and the lossless dielectric medium.
Consider the expression for phase constant of medium-2 (lossless dielectric medium).
Rewrite the expression for the given data.
Substitute
Consider the expression to find the wavelength of the transmitted wave in the lossless dielectric medium (medium-2).
Substitute
Conclusion:
Thus, the wavelength and wave frequency of the wave in air are
(b)
Find the incident magnetic field.
(b)
Answer to Problem 60P
The incident magnetic field is
Explanation of Solution
Calculation:
Consider the expression to find the incident magnetic field.
Here,
As the medium-1 is free space, the value of
Find the
Substitute
Conclusion:
Thus, the incident magnetic field is
(c)
Find the reflection and transmission coefficients.
(c)
Answer to Problem 60P
The reflection and transmission coefficients are
Explanation of Solution
Calculation:
Consider the expression to find the reflection coefficient.
Here,
Find the intrinsic impedance for medium-2.
Substitute
Consider for the expression for transmission coefficient.
Substitute
Conclusion:
Thus, the reflection and transmission coefficients are
(d)
Find the total electric field and time-average power in both the regions.
(d)
Answer to Problem 60P
The total electric field in the air and lossless dielectric medium are
The time-average power density in the air and lossless dielectric medium are
Explanation of Solution
Calculation:
Consider the expression to find the total electric field in the air.
Here,
Consider the expression for reflected electric field in medium-1.
Here,
Consider the expression for reflected electric field at
Substitute
Substitute
Substitute
The total electric field in the medium-2 is the transmitted electric field component.
Consider the expression for transmitted electric field.
Here,
Consider the expression to find the magnitude of transmitted electric field at
Substitute 0.732 for
Substitute
Consider the expression to find the time-average power in region 1 (medium-1).
From Equation (1), the vector
Substitute
Consider the expression to find the time-average power in region 2 (medium-2).
Substitute
Conclusion:
Thus, the total electric field in the air and lossless dielectric medium are
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Chapter 10 Solutions
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