 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s19/3/d/solution

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S19 / 3 / d / Solution by

Codex  0 2026-10-25

The mean witness gradient must be

112.5 GeV /1.1 m ≃ 102 GV m^{- 1}

. To turn this into a density estimate, assume an accelerating wake amplitude of order the cold plasma wave-breaking scale,

E_{w} \sim \frac{m _{e} c ω _{p}}{e}, ω_{p}^{2} = \frac{n _{0} e ^{2}}{ε _{0} m _{e}} .

(1)

Eliminating

ω_{p}

gives

n_{0} \sim \frac{ε _{0} E _{w}^{2}}{m _{e} c ^{2}} ≃ 1.1 \times 1 0^{24} m^{- 3} ≃ 1.1 \times 1 0^{18} c m^{- 3} .

(2)

The associated plasma wavelength is about

31

micrometres, illustrating why a short driver structure is required. The given length and energy gain do not uniquely fix density without an assumption about wake amplitude relative to the plasma scale. Driver shape, beam loading, relativistic wave-breaking corrections and matching to a Positron witness are substantive additional issues; the estimate is a scale calculation, not a completed accelerator design.

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