In a region of Earth's ionosphere the electron density is

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Waves in plasmas

Modes, Packets and Media, Ch Water waves and de Broglie waves

Problem

In a region of Earth's ionosphere the electron density is ne=2.0×1011 m−3n_e = 2.0\times10^{11}\,\mathrm{m^{-3}}. Radio waves there obey the plasma dispersion relation ω2=ωp2+c2k2\omega^2 = \omega_p^2 + c^2k^2, with plasma frequency ωp=nee2/(ε0me)\omega_p = \sqrt{n_e e^2/(\varepsilon_0 m_e)}, e=1.602×10−19 Ce = 1.602\times10^{-19}\,\mathrm{C}, ε0=8.854×10−12 F m−1\varepsilon_0 = 8.854\times10^{-12}\,\mathrm{F\,m^{-1}}, me=9.109×10−31 kgm_e = 9.109\times10^{-31}\,\mathrm{kg}, ℏ=1.0546×10−34 J s\hbar = 1.0546\times10^{-34}\,\mathrm{J\,s}, c=2.998×108 m s−1c = 2.998\times10^{8}\,\mathrm{m\,s^{-1}}. ((a)) Show that this relation has the same form as the relativistic matter-wave relation ω2=ωC2+c2k2\omega^2 = \omega_{\rm C}^2 + c^2k^2 for a particle of effective mass m=ℏωp/c2m = \hbar\omega_p/c^2, and evaluate mm. ((b)) For a wave of wavenumber k=0.10 rad m−1k = 0.10\,\mathrm{rad\,m^{-1}}, find ω\omega, the phase velocity vpv_p and the group velocity vgv_g, and confirm that vpvg=c2v_pv_g = c^2.

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