Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S16 / 1 / 4

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- Solution 4

Solution

At

v_{*} = V - σ_{b}

, the beam's positive slope is

n_{b} e^{- 1/2} / (2 π σ_{b}^{2})

, while the background slope has magnitude

n_{p} v_{*} e^{- v_{*}^{2} / (2 σ_{p}^{2})} / (2 π σ_{p}^{3})

. Thus a nonnegative total slope at this velocity requires

\frac{n _{b}}{n _{p}} \geq e \frac{v _{*} σ _{b}^{2}}{σ _{p}^{3}} e^{- v_{*}^{2} / (2 σ_{p}^{2})} .

Approximating the background slope by its value at

V

yields

\frac{n _{b}}{n _{p}} > C \frac{m _{e}}{k _{B} T _{p}} \frac{V T _{b}}{T _{p}} exp (- \frac{m _{e} V ^{2}}{2 k _{B} T _{p}}), C = e .

Equality is the local marginal-slope condition; strict inequality gives positive resonant drive in this approximation. The numerical

C

is of order unity and can change when the resonant velocity is optimized or the full wave dispersion is retained.

Replacing

v_{*}

by

V

in the exponential is controlled only if

V σ_{b} / σ_{p}^{2} ≪ 1

, in addition to

σ_{b} ≪ σ_{p}

. The two width inequalities alone do not guarantee that extra condition far out on a Maxwellian tail. The preceding unapproximated slope inequality remains available when it is not satisfied.

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