A magnetic mirror trap has an axial field

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

Plasma Physics, Ch Waves in a non-uniform plasma: rays, reflection, mode conversion and radiofrequency heating

Problem

B(z) = B_0\left(1 + (z/z_m)^2\right)

, with

B_0 = 0.90\,\mathrm{T}

at the mid-plane

z=0

and

z_m = 1.0\,\mathrm{m}

to the throat, so the mirror ratio is

R_{\rm mir} = B(z_m)/B_0 = 2.0

. A right-hand circularly polarised wave of frequency

f = 28.0\,\mathrm{GHz}

is launched along the axis from beyond the throat, travelling inward towards the mid-plane, so that it meets the resonance from the high-field side, where no R-wave cutoff lies in its way. Take the gyrofrequency rule

f_{ce}(\mathrm{GHz}) = 28.0\,B(\mathrm{T})

(from

\Omega_e = eB/m_e

). ((a)) Because the field varies only with

z

and the wave is launched with

\mathbf{k}

along

z

(no transverse wavevector component), Hamilton's ray equations give a straight axial path. Cold-plasma theory (Chapter 9) shows that this R-mode wave resonates where

\omega = |\Omega_e(z)|

. Find the axial position

z_{\rm res}

of the resonance layer. ((b)) An electron leaves the mid-plane with pitch angle

\alpha_0 = 30^\circ

(measured from

\mathbf{B}

) and drifts along the field line towards the throat, conserving its magnetic moment

\mu = mv_\perp^2/2B

(Chapter 3) until it reaches

z_{\rm res}

. Find its pitch angle

\alpha_{\rm res}

there, and determine whether, in the absence of the wave, this electron would mirror before reaching the throat or would be lost through it. ((c)) State the minimum pitch angle the wave must give the electron at

z_{\rm res}

for it to become trapped (unable to reach the throat), and explain in a sentence why resonant absorption can change the pitch angle without conserving

\mu

, whereas the adiabatic drift in part (b) does conserve it.

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