1) Derive (vector*) wave equations for the following cases. (Note that the induced conduction current in medium due to c

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1) Derive (vector*) wave equations for the following cases.
(Note that the induced conduction
current in medium due to conductivity is 𝐽⃗𝑐 = 𝜎𝐸⃗⃗ and 𝐽⃗ is the
independent electric current density,
where 𝐽⃗ ≠ 𝐽⃗𝑐 . Do not forget to include 𝜎𝐸⃗⃗ in Maxwell equations
for lossy medium.)
Hint: Derive the below equations starting from the Maxwell
equations and taking the curl of curl
equations as done in lecture notes while deriving wave equation in
homogeneous lossless medium.
a) In frequency domain for linear, inhomogeneous, lossy (𝝁(𝒓⃗⃗),
𝜺(𝒓⃗⃗), 𝝈(𝒓⃗⃗)) and isotropic
medium assuming that 𝜌≠0, J≠0. . (You can derive either for
electric field or magnetic field
vector)
b) In frequency domain for linear, inhomogeneous, lossy (𝝁(𝒓⃗⃗),
𝜺(𝒓⃗⃗), 𝝈(𝒓⃗⃗)) , isotropic and
source free (𝜌 = 0, 𝐽⃗ = 0) medium. (You can derive either for
electric field or magnetic field
vector and you can make use of the equation derived in (1a))
c) In frequency domain for linear, homogeneous (𝝁 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕, 𝜺 =
𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕), lossless (𝝈 =
𝟎) and isotropic medium assuming that 𝜌≠0, J≠0. (You can derive
either for electric field
or magnetic field vector and you can make use of the equations that
you derived previously.)
d) In frequency domain for linear, homogeneous (𝝁 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕, 𝜺 =
𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕), lossy (𝝈 =
𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕), isotropic and source free (𝜌 = 0, 𝐽⃗ = 0) medium. (You
can derive either for
electric field or magnetic field vector and you can make use of the
equations that you derived
previously.)