350lect22 - 22 Dispersion and propagation in collisionless...

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22 Dispersion and propagation in collisionless plasmas TEM plane wave propagation in homogeneous conducting media can be described in terms of wavenumbers and intrinsic impedances k = ω ± μ ( ± + σ ) k ± - jk ± and η = ² μ ± + σ ≡| η | e as we have seen in Lecture 18. For real valued σ these relations imply complex valued k and η as well as an ω dependent propagation velocity v p = ω k ± = 1 Re { ³ μ ( ± + σ ) } . Having an ω dependent v p is a telltale sign that propagation of TEM waves in the medium will be dispersive ,mean ingthat the shapes of TEM signals waveforms other than co-sinusoids will be distorted as a consequence of propagation — the dis- tortion happens because diFerent co-sinusoid components of the signal having diFerent frequencies ω travel with diFerent velocities v p and thus fall out of synchronism! Dispersion in wave motions can be caused by a variety of reasons in- cluding the frequency dependence of the medium parameters as well as
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geometrical efects related to the dimensions oF the propagation region in relation to a wavelength. ±or an ohmic medium where σ is real — such as seawater or copper —wavepropagat ionisboth lossy and dispersive. An important propagation medium known to be dispersive but lossless is the “ collisionless plasma ”, an ionized gas in which collisions oF the charge carriers (with one another) are negligibly small — a collisionless plasma provides an ideal setting to explore and understand the wave dispersion efects without having to deal with complications arising From losses and dissipation. Aco l l is ion lessp lasmaisessent ia l lyaconduct ingmed iumw ithapure ly imaginary conductivity σ (or, equivalently, a dielectric with a relative permittivity less than one, as we will see). To develop the conductivity model For a collisionless plasma we en- vision a region oF volume in Free-space containing N Free electrons per unit volume along with N positive ions (e.g., O + in the ion- ized portions oF the upper atmosphere) which are also Free. Each oF these Free charge carriers with charge q and mass m respond to an alternating electric ²eld with a phasor ˜ E as dictated by Newton’s ²rst law: m d
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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350lect22 - 22 Dispersion and propagation in collisionless...

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