Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S7 / 1 / e

Table of contents
- Solution e

Solution

Expand

A (r) = A (r_{0}) + A^{'} (r_{0}) s + O (s^{2})

about

q A (r_{0}) = p_{z}

. Then

p_{z} - q A ≃ - q A^{'} (r_{0}) s

and conservation of energy gives

\overset{s}{˙}^{2} + Ω^{2} s^{2} = \frac{2 H}{m} = Ω^{2} ρ^{2}, ρ = \frac{2 H / m}{∣Ω∣}, s (t) = ρ cos (∣Ω∣ t + ϕ) .

Here

Ω = q A^{'} (r_{0}) / m

can be signed, but the physical oscillation frequency and amplitude are nonnegative. The small-energy condition ensures that this local expansion samples only a narrow range about

r_{0}

, provided the field varies smoothly there.

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