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Equation 4 · The Last Artefact: Redefining the Kilogram

What does this equation mean?

KJ=2eh,RK=he2,KJ2RK=4hK_{\mathrm J} = \frac{2e}{h}, \qquad R_{\mathrm K} = \frac{h}{e^{2}}, \qquad K_{\mathrm J}^{2} R_{\mathrm K} = \frac{4}{h}

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 with2e
Divide byh
This relates toK_mathrm J
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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KJK_{\mathrm J}

Symbol K_mathrm J

KmK_mathrm J is part of the quantity the equation computes from the expression on the right.

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ee

Symbol e

e is an input to the expression that computes the quantity on the left.

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hh

Symbol h

the fixing.

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RKR_{\mathrm K}

Symbol R_mathrm K

RmR_mathrm K is an input to the expression that computes the quantity on the left.

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e2e^{2}

Symbol e^2

The square of e: multiply e by itself.

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KJ2K_{\mathrm J}^{2}

Symbol K_mathrm J^2

KmK_mathrm J2J^2 is an input to the expression that computes the quantity on the left.

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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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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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2e2e

Numerator: 2e

The complete quantity above the fraction bar.

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44

Numerator: 4

The complete quantity above the fraction bar.

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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

The remaining question is what the electrical quantities are referred to. The current is obtained from a voltage drop across a stable resistor; both voltages are measured against the Josephson constant, taken as KJ = 2e/h, and the resistance against the von Klitzing constant, taken as RK = h/e², so that KJ²RK = 4/h and the mass reduces to h multiplied by an experimental frequency term and divided by the product of local gravity and coil velocity [ 4 ] : KJ=2eh,RK=he2,KJ2RK=4hK_{\mathrm J} = \frac{2e}{h}, \qquad R_{\mathrm K} = \frac{h}{e^{2}}, \qquad K_{\mathrm J}^{2} R_{\mathrm K} = \frac{4}{h}. This is where the quantum electrical standards earn their place. Since 1990 the practical realization of the volt and the ohm had rested on the Josephson and quantum Hall effects, but with conventional values assigned to…
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The remaining question is what the electrical quantities are referred to. The current is obtained from a voltage drop across a stable resistor; both voltages are measured against the Josephson constant, taken as KJ = 2e/h, and the resistance against the von Klitzing constant, taken as RK = h/e², so that KJ²RK = 4/h and the mass reduces to h multiplied by an experimental frequency term and divided by the product of local gravity and coil velocity [ 4 ] : KJ=2eh,RK=he2,KJ2RK=4hK_{\mathrm J} = \frac{2e}{h}, \qquad R_{\mathrm K} = \frac{h}{e^{2}}, \qquad K_{\mathrm J}^{2} R_{\mathrm K} = \frac{4}{h}. This is where the quantum electrical standards earn their place. Since 1990 the practical realization of the volt and the ohm had rested on the Josephson and quantum Hall effects, but with conventional values assigned to the two constants — a decision Taylor and Witt set out when the new reference standards took effect on 1 January 1990, intended to “improve significantly the international uniformity of electrical measurements and their consistency with the SI” [ 9 ] . The consequence was a parallel electrical unit system, close to the SI but not identical to it [ 3 ] . Fixing h and e together dissolved the parallel system: KJ and RK are now calculated exactly from the defining constants, and the conventional values were retired [ 3 ] . A Kibble balance is therefore not merely an accurate balance. It is a mechanical quantity expressed entirely in quantum-referenced electrical terms, which is why the electrical metrology community wanted the kilogram and the ampere redefined in the same act.

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