 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/2016/s16/3/2/solution

When

∣ k v_{z} / ω ∣ ≪ 1

over the populated velocities, expand the denominator in powers of

k v_{z} / ω

. Odd Gaussian moments vanish. The first two nonzero terms use

⟨ v_{z}^{2} ⟩ = v_{t h}^{2}

and

⟨ v_{z}^{4} ⟩ = 3 v_{t h}^{4}

, yielding

1 - ω_{p}^{2} / ω^{2} - 3 ω_{p}^{2} k^{2} v_{t h}^{2} / ω^{4} + \dots = 0

. Solving consistently to order

k^{2}

gives

ω^{2} ≃ ω_{p}^{2} + 3 k^{2} v_{t h}^{2} .

This is a long-wavelength/high-phase-speed approximation, not an exact Maxwellian dispersion relation at arbitrary

k

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