Past exam of the particle physics course of the University of Oxford / 2022 / 4 / a / ii

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- Solution ii

Solution

The sagitta is the maximum distance between the circular track and its chord. The perpendicular bisector of a chord of length

L

gives a right triangle with hypotenuse

R

and sides

L /2

and

R - s

. Hence

s = R - R^{2} - L^{2} /4 = \frac{L ^{2}}{8 R} + O (L^{4} / R^{3}) ≃ \frac{∣ q ∣ B L ^{2}}{8 p} .

At fixed electric charge, magnetic field and lever arm,

p = ∣ q ∣ B L^{2} / (8 s)

, so differentiation and propagation of uncertainty give

\frac{σ ( p )}{p} = \frac{σ ( s )}{s} = \frac{8 p σ ( s )}{∣ q ∣ B L ^{2}} .

Thus a fixed spatial measurement uncertainty produces a relative momentum uncertainty growing linearly with

p

. If momentum is expressed in

GeV / c

,

B

in tesla,

L, s

in metres and

z = q / e

, restore the usual conversion

p = 0.2998∣ z ∣ BR

.

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