Plane waves in conductors, dielectric-conductor boundaries

Problem:

A “tenuous” plasma consists of free electric charges of mass m and charge –e (where e is positive).  There are n charges per unit volume.  Assume that the density is uniform and that the interactions between the charges may be neglected.  Also assume that the charges can be treated classically.
A linearly-polarized electromagnetic wave of frequency ω is incident on the plasma.
Let the electric field component of the plane wave be E = E0exp(i(kx - ωt)).
(a)  Solve the equation of motion for a single charge and find the current density j and the conductivity σ of the plasma as a function of ω.
(b)  Assume a plane wave of the form E = E0exp(i(kx - ωt)) propagate in the plasma with conductivity σ.  Find the dispersion relation —the relation between k and ω— for the electromagnetic wave in the plasma and the index of refraction as a function of ω.

Solution:

Problem:

In a medium with conductivity σ but no net charge, write down Maxwell's equations, and derive the wave equation for the electric field, E, in this medium.

Solution:

Problem:

Starting from Maxwell's equations, prove that a plane EM wave propagates in a good conductor such that the electric and magnetic fields are out of phase by 45o.

Solution:

Problem:

An electromagnetic wave with frequency of 106 Hz "travels" along the z-axis in an aluminum medium located at z ≥ 0.  The conductivity of aluminum is 38.2∙ 106 (Ωm)-1 and its relative permeability is km = 1.  Just inside the conductor at z = +0, the electric field amplitude is E0 i

(a)  Write down an expression for the electric field inside the conductor.
(b)  Find the skin depth, wave velocity, and wavelength of the wave in aluminum.
(c)  Determine the corresponding magnetic field.
(d)  Find the phase difference between the electric and magnetic fields at each fixed location in aluminum.

Solution:

Problem:

An electric field E = i E0exp(-iωt)is applied at the interface of a vacuum (z < 0) and a conductor (z > 0) of real average conductivity σc.  Assume ε = ε0, μ = μ0, in the conductor.
(a)  For σc >> ε0ω, calculate how deeply the electric field penetrates into the conductor, i.e. calculate the depth at which the electric field amplitude has decreased to 1/e of its value at the surface.
(b)  Calculate dW/dt =  j∙EdV, the rate at which work is done by the field on the charges in a volume dV, as a function of z.

Solution: