Answer To: Please solve the two problems provide. Make sure to include all steps and details leading to the...
Bhaumik answered on Oct 07 2024
Advanced Electromagnetic Fields and Waves I
Solutions
Solution for Part 1:
a) Expressing the constitutive relations in terms of new parameters :
Given:
(1)
(2)
The following must be expressed:
(3)
(4)
The process is about identifying how to translate the parameters into
By comparing, we can set:
· (from the term in ),
· (from the term in ),
· (from the term in ).
Thus, we have the relations:
, , and
b) Showing
Using the modified form:
As there are no free charges in the region, Gauss’s law for the electric displacement field is:
Substituting into this equation:
Since the medium is homogeneous, the parameters and are constants, so the divergence of this expression becomes:
Now, using the facts in the absence of magnetic monopoles, we have:
Thus, the equation reduces to:
Since for a non-trivial medium, we conclude:
This shows that the electric field is solenoidal (divergence-free), consistent with the problem’s assumptions.
c) Deriving the wave (Helmholtz) equation for
From Maxwell’s equation, we know that:
Substituting into this:
Since time-harmonic fields are assumed (), this becomes:
Similarly, for the curl of H:
Now, let’s derive the wave equation,
Taking the curl of
Using the vector identity and knowing , this simplifies to:
To obtain Helmholtz equation for using the constitutive relations and and assuming time-harmonic fields (), we substitute Maxwell’s curl equations.
d) Verifying the plane wave solution:
Given:
First, calculate Since, the wave is propagating in the z-direction, we only need to take the second...