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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/2015/s19/3/c/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 / 2015 / S19 / 3 / c / Solution by

Codex  0 2026-10-25

In the cold unmagnetized linear plasma model, the refractive-index relation is

n^{2} = 1 - ω_{p}^{2} / ω_{L}^{2}

, with

ω_{p}^{2} = \frac{n _{e} e ^{2}}{ϵ _{0} m _{e}} .

The critical density occurs at

ω_{p} = ω_{L} = 2 π c / λ_{L}

. Thus

n_{crit} = \frac{ϵ _{0} m _{e}}{e ^{2}} (\frac{2 π c}{λ _{L}})^{2} ≃ 1.12 \times 1 0^{27} m^{- 3} = 1.12 \times 1 0^{21} c m^{- 3},

for

λ_{L} = 1 μ m

. The corresponding frequency is

ω_{p} ≃ 1.88 \times 1 0^{15} rad s^{- 1}, f_{p} = ω_{p} / (2 π) ≃ 3.0 \times 1 0^{14} Hz .

A propagating conventional laser wake generally uses an underdense plasma, rather than this cutoff density. Strong relativistic laser motion, collisions or a magnetized plasma modify this simple critical-density condition.

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