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Equation 2 · What a Particle Detector Actually Measures

What does this equation mean?

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

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withqBell^2
Divide by8p
This relates tos
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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ss

Symbol s

the sagitta.

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qq

Symbol q

q occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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BB

Symbol B

the field.

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pp

Symbol p

the momentum [ 2 ].

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σp\sigma_p

Symbol sigma_p

sigmapa_p occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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=

=

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

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fraction

fraction

Divide the expression above the line by the one below it.

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∝

∝

Proportional to; the scale factor is not shown.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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qBℓ2qB\ell^{2}

Numerator: qBell^2

The complete quantity above the fraction bar.

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8p8p

Denominator: 8p

The complete quantity below the fraction bar; it must be nonzero for this division.

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Bℓ2B\ell^{2}

Denominator: Bell^2

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

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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Sources cited in the surrounding passage

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