The time-averaged energy density of an electromagnetic plane wave in a conducting medium can be calculated using the Poynting vector. The Poynting vector describes the flow of energy per unit area and is given by:
S = (1/μ) * E x B
where S is the Poynting vector, E is the electric field vector, B is the magnetic field vector, and μ is the permeability of the medium.
For a plane wave, the electric field and magnetic field vectors are orthogonal to each other and also orthogonal to the direction of propagation. Let's assume the wave is propagating in the z-direction. The electric field can be written as:
E = E_0 * exp(i(kz - ωt))
where E_0 is the amplitude of the electric field, k is the wave vector, ω is the angular frequency, z is the direction of propagation, and t is time.
The magnetic field can be related to the electric field by the wave impedance Z:
B = (1/Z) * E
In a conducting medium, the wave impedance is given by:
Z = (μ/σ)^0.5
where σ is the conductivity of the medium.
Substituting the expressions for E and B into the Poynting vector equation, we get:
S = (1/μ) * (E x B) = (1/μ) * (E_0 * exp(i(kz - ωt)) x (1/Z) * E_0 * exp(i(kz - ωt)))
Since E_0 and Z are constants, we can simplify the equation:
S = (1/μZ) * E_0^2
The time-averaged energy density, , is given by the magnitude of the time-averaged Poynting vector:
= || = (1/2) * (1/μZ) * E_0^2
Therefore, the time-averaged energy density of an electromagnetic plane wave in a conducting medium is proportional to the square of the electric field amplitude and inversely proportional to the permeability and wave impedance of the medium.