On the magnetic field line whose equatorial crossing lies at Earth radii

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

Plasma Physics, Ch Cold magnetised plasma waves: the dielectric tensor, cutoffs, resonances and the CMA diagram

Problem

On the magnetic field line whose equatorial crossing lies at

L=5

Earth radii, the electron density there is

n_e = 5.0\times10^{7}\,\mathrm{m^{-3}}

. Earth's equatorial surface field is

B_{surf}=3.1\times10^{-5}\,\mathrm{T}

, and a dipole field falls off along a field line's equatorial crossing as

B\propto L^{-3}

, so at this crossing

B = B_{surf}/L^3

. ((a)) Find the equatorial electron cyclotron frequency

f_{ce}

and plasma frequency

f_{pe}

on this field line. ((b)) Estimate the group velocity, as a fraction of

c

, of a whistler-mode wave of frequency

f=500\,\mathrm{Hz}

there, using

v_g \approx 2c(\omega|\Omega_e|)^{1/2}/\omega_{pe}

, valid for

\Omega_i\ll\omega\ll|\Omega_e|

and

\omega_{pe}\gg|\Omega_e|

. ((c)) Check that

\omega\ll|\Omega_e|

is a reasonable approximation here by quoting the ratio

\omega/|\Omega_e|