Propagation of Plane Electromagnetic Waves in Low Pressure Ionized Gases

Plasma is made up of electrons of positive ions. The electrons are free to move under the influence of the electric and magnetic fields associated with the electromagnetic wave incident upon the medium. The positive ions are, however, heavy compared to the electrons so that are relatively immobile. The electron motion produced a current which influences the propagation of the wave.

Consider a plane electromagnetic wave traveling in the positive direction along the z-axis with the E and H vectors respectively parallel to the x-and the y-axis. An ion and charge Q, mass m, velocity u, situated at x, y, z, is subjected to a Lorentz force
        F = Q {E + (u x B)}                                ... (1)
        = Q {Ei – B dz/dt i + Bdx/dt k}.                            ... (2)
     d²x/dt² = Q/m (E – B dz/dt),                                ... (3)
Now,     E/B = c.

. :    d²x/dt² = Q/m (1 – 1/c dz/dt),                                ... (4)

Similarly, d²y/dt2 = 0,                                        ... (5)
     d²z/dt² = Q e/mc dx/dt,                                    ... (6)

Let us assume that
    1/c |dz/dt| << 1.                                         ... (7)

Then,        d²x/dt² = Q/m E = Q/m E₀ cos ωt.                             ... (8)

. :        x = -1/ω² Q/m E₀ cos ωt.                                ... (9)
        dx/dt = 1/ω Q/m E₀ sin ωt.                                ... (10)

Substituting in Eq. (6),
        d²z/dt² = Q² E²₀ /2ωm² C sin 2ωt,                            ... (11)
dz/dt = Q² E²₀ /4ωm² C cos 2ωt,                             ... (12)
z = - Q²E²₀ /8ωm² C sin 2ωt.                            ... (13)



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