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Vgc ÞB ¼ ÀB  ½ðm=eÞðvgck Þ2 r? BðrÞ=B þ ðm=eÞrBðrÞ; so that vgc? ¼ ð2Wk þ W? ÞrBðrÞ Â B=ðeB3 Þ: ð3:32Þ We have written this in the conventional form (see, for example, Krall and Trivelpiece [4]), but it must be kept in mind that Eq. 3 Electron Drift Motions velocity. Generally speaking it is necessary to average the expression in Eq. 32) along the magnetic line of force followed by the guiding center. For electrons trapped in a magnetic mirror, this average is conveniently carried out using the action invariant [5], Ð J ¼ p||ds, where the integral is taken around a complete bounce cycle.
3. If the plasma is overdense, the slow wave propagates only within a cone centered on the direction of the magnetic field with its half angle given by qcrit, the angle for which the coefficient of n4 in Eq. 11) vanishes: qcrit ¼ arctan[À2K33/ (K11 þ K22)]1/2. 4. 5 the indices are shown when the density is just below the value for the left-hand cutoff. For densities above this value, the left-hand circularly polarized wave is evanescent while the right-hand circularly polarized whistler continues to propagate until the wave reaches gyroresonance.
V ¼ À(e/m)E where the matrix M is given by 0 1 Àiw þ n Wo 0 A: Àiw þ n 0 ð4:4Þ M ¼ @ ÀWo 0 0 Àiw þ n Transforming M into circular coordinates, we diagonalize M and obtain 0 1 Àiw þ nÀiWo 0 0 À1 A; 0 0 Àiw þ v þ iWo UMU ¼ @ 0 0 Àiw þ n ð4:5Þ resulting in the following three decoupled scalar dynamical equations: ðÀiw þ nÀiWo Þv þ ¼ Àðe=mÞE þ ðÀiw þ n þ iWo ÞvÀ ¼ Àðe=mÞEÀ ð4:6Þ ðÀiw þ nÞvz ¼ Àðe=mÞEz : For the present cold-plasma model, the perturbed current density, j ¼ À enov, and we obtain for the circular components of the current density j þ ¼ ie0 wpe 2 ðw þ Wo þ inÞÀ1 E þ ¼ s11 E þ jÀ ¼ ie0 wpe 2 ðwÀWo þ inÞÀ1 EÀ ¼ s22 EÀ À1 jz ¼ ie0 wpe ðw þ inÞ Ez ¼ s33 Ez : 2 ð4:7Þ j39 j 4 Wave Propagation and Cyclotron Damping in Magnetized Plasmas 40 Since from Eq.