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Equation 2 · Part 11 · What a Particle Detector Actually Measures

Numerator: qBell^2

s=qBℓ28p,σpp  ∝  pBℓ2.s = \frac{qB\ell^{2}}{8p}, \qquad \frac{\sigma_p}{p} \;\propto\; \frac{p}{B\ell^{2}} .
qBℓ2qB\ell^{2}

What this part means

The complete quantity above the fraction bar.

Its job in the formula

qBell2l^2 occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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