Find the power densities in both media.
Answer to Problem 59P
The power densities in medium-1 and medium-2 are
Explanation of Solution
Calculation:
Write the expression for incident electric field in medium-1.
Consider the expression for phase constant
Rewrite the expression for the given data.
Substitute
Substitute
Consider the expression to find the total electric field in medium-1.
Here,
Consider the expression for reflected electric field in medium-1.
Here,
Consider the expression for reflected electric field at
Here,
Write the expression to find the reflection coefficient.
Here,
As medium-2 is free space, the intrinsic impedance is
Find the intrinsic impedance for medium-1.
Substitute
Substitute
Substitute
Consider the expression to find the incident magnetic field in medium-1.
Find the vector
Substitute
Consider the expression to find the reflected magnetic field in medium-1.
Find the vector
Substitute
From Equation (1), the total electric field in medium-1 is the sum of incident and reflected electric fields. Similarly, the total magnetic field in medium-1 is the sum of incident and reflected magnetic fields.
Write the expression to find the power density in medium-1.
Find the cross product
Find the cross product
Substitute
Consider the expression for transmitted electric field in medium-2.
Here,
Consider the expression for phase constant
Substitute
Substitute
Consider the expression to find the magnitude of transmitted electric field at
Here,
Consider for the expression for transmission coefficient.
Substitute
Substitute
Substitute
Consider the expression to find the transmitted magnetic field in medium-2.
Find the vector
Substitute
Write the expression to find the power density in medium-2.
Conclusion:
Thus, the power densities in medium-1 and medium-2 are
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