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Equation 2 · Part 6 · What a Particle Detector Actually Measures

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s=qBℓ28p,σpp  ∝  pBℓ2.s = \frac{qB\ell^{2}}{8p}, \qquad \frac{\sigma_p}{p} \;\propto\; \frac{p}{B\ell^{2}} .
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Momentum is not measured. Geometry is measured, and momentum is computed from it under an assumed field map. A charged particle in a solenoidal field follows a helix, and the deviation of its path from a straight line over a chord — the sagitta — carries the momentum information: s=qBℓ28p,σpp  ∝  pBℓ2s = \frac{qB\ell^{2}}{8p}, \qquad \frac{\sigma_p}{p} \;\propto\; \frac{p}{B\ell^{2}} . Here s is the sagitta, B the field, ℓ\ell the path length in the field and p the momentum [ 2 ] . Two consequences follow directly and are worth stating plainly, because they explain most of the architecture of a collider detector.

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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