 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/2012/s19/1/a/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 / 2012 / S19 / 1 / a / Solution by

Codex  0 2026-10-25

Use transverse coordinates

(x, x^{'})

and a signed focusing strength

K

in Hill's equation,

x^{''} + K x = 0

. A drift of length

ℓ

has beam transfer matrix

D (ℓ) = (10 ℓ 1) .

For a uniform focusing quadrupole magnet with

K = κ^{2} > 0

, and a defocusing one with

K = - κ^{2}

,

Q_{F} (ℓ) = (cos κ ℓ - κ sin κ ℓ sin (κ ℓ) / κ cos κ ℓ), Q_{D} (ℓ) = (cosh κ ℓ κ sinh κ ℓ sinh (κ ℓ) / κ cosh κ ℓ) .

The focusing plane has oscillatory trajectories, while the defocusing plane has hyperbolic growth and decay. A magnetic quadrupole focuses one transverse plane and defocuses the other; reversing its gradient interchanges their roles.

If

∣ K ∣ ℓ^{2} ≪ 1

and the element's internal drift can be neglected or assigned separately, its kick may be concentrated at one position. With

1/ f = ∣ K ∣ ℓ

,

F = (1 - 1/ f 01), G = (1 1/ f 01) .

Both kicks preserve position and change the slope linearly with displacement. All these ideal uncoupled maps have determinant one.

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