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How Does a Kibble Balance Weigh a Mass Without Knowing the Strength of Its Magnet?

Category: Technology

For well over a century, the kilogram was one polished metal cylinder kept in a vault near Paris. In November 2018 the international scientific community voted to stop using it as the definition, and since 2019 the kilogram has been tied to a fixed number for a constant of nature instead [1].

That raises a tricky question. A number is not something you can put on a scale. So how do laboratories turn it back into a real lump of metal with a known mass? One answer is a machine called a Kibble balance. It measures the same magnetic effect twice, so the quantity nobody can measure well cancels out of the maths.

When the Kilogram Became a Number, Labs Needed a Way to Turn It Back Into a Weight

The new definition fixes the value of the Planck constant, written h, at exactly 6.626 070 15 × 10−34 joule seconds (J s) [2]. Together with the definitions of the second and the metre, that number is enough, in principle, to say what a kilogram is.

The official guidance for putting the definition into practice says something important: the definition does not require any particular experiment. Any method that can produce a mass value traceable to the fixed constants can be used, and the guidance lists the methods that are easiest to carry out or give the smallest uncertainty [2]. The Kibble balance is one of the best known of these.

The Kibble Balance Compares a Weight With an Electrical Force

It is named after Bryan Kibble of the UK’s National Physical Laboratory (NPL), who conceived the idea in 1975 [3]. NPL says he developed the apparatus there in the 1970s [4]. At first it was called a watt balance, because it measures an electric current and a voltage whose product is a power in watts, and it was renamed in his honour later [3][4].

Like any balance, it works by setting one force against another. Here the weight of the test mass is cancelled by an upward force on a coil of wire, hung in the field of a strong magnet and carrying an electric current. The size of that force is controlled by adjusting the current [3].

In the First Experiment, an Electric Current Holds the Mass Up

In what NIST calls the weighing or force mode, the test mass sits on a pan attached to the coil. The current is adjusted until the upward magnetic force exactly balances the downward weight [3]. In symbols, weight (mass m times the strength of gravity g) equals current I times the magnet’s field strength B times the length of wire L:

m g = I × B × L

Current is something laboratories can measure very well, and gravity can be measured separately. The catch is the product of field strength and wire length, BL. NIST says it is extremely hard to measure directly to the accuracy needed [3]. Without it you cannot get the mass, so one experiment alone is stuck.

In the Second Experiment, Moving the Coil Reveals the Same Hidden Number

In the velocity or calibration mode, the test mass is taken off and the current is switched off. Now the same coil is moved through the magnetic field at a carefully controlled, steady speed v. A coil moving through a magnetic field has a voltage V induced in it, and that voltage can be measured [3]:

V = v × B × L

The same awkward BL appears. But this time the voltage and the speed can both be measured accurately, and dividing the voltage by the speed gives BL. So the second experiment tells you what BL is, in terms of things you can measure. Put the two results together and, in NIST’s words, the problematic B and L cancel out, leaving I × V = m × g × v [3]. Solved for the mass:

m = I × V / (g × v)

In a Japanese-language summary of the official guidance, the same step is described as setting a mechanical power equal to an electrical power, even though the two never happen at the same moment, which is why the power is sometimes called “virtual” (see the note under Sources). One thing has to stay true for the trick to work: the coil must have the same BL in both experiments, so the machine is built and run to keep it steady.

The Machine Measures Voltage, Current, Speed and Gravity, and Each Has Its Own Method

That leaves four things to measure very carefully. NIST says the electrical quantities are measured with quantum electrical effects, gravity with an absolute gravimeter, and the coil’s velocity with laser interferometry [3]. NPL likewise says the instrument lets mass be realised through two quantum electrical units, the Josephson voltage and the quantum Hall resistance, together with the fixed Planck constant [4]. That is how a fixed number for h ends up connected to a real weight.

These machines are large. NIST describes its NIST-4 balance as about 2.5 metres tall inside a metal vacuum enclosure, with a 4 kg moving coil wound with about 1.4 km of wire and a permanent magnet system of about 1,000 kg producing 0.55 tesla [3].

Before the Redefinition, a Kibble Balance Had to Measure the Constant Itself, and It Did So Very Precisely

There is one more twist. Before the number for h could be fixed, someone had to measure it, using the old kilogram. NIST’s team did this with NIST-4, making more than 10,000 weighings of masses from 0.5 kg to 2 kg between 2015 and 2017, and reported the Planck constant with a relative uncertainty of 13 parts in a billion [5]. After the new definition, the logic runs the other way: h is fixed, and the balance is used to measure mass.

Kibble balances were not the only route. A second method counts atoms in a nearly perfect silicon sphere to find the same constant. Japanese metrology researchers at AIST describe results from both methods being combined in the 2017 international adjustment of the constants, and that agreement between independent methods helped make the redefinition possible [6].

Why the Old Cylinder Could Not Be Trusted to Stay the Same

The reason for all this effort is that a single object is a fragile definition. NIST notes that the standard masses compared against the cylinder, and the cylinder itself, were found to drift a little over time, and that making comparisons among them was extremely laborious [1]. If the definition is the object, you cannot even say whether the object changed. AIST puts the long-term stability of the cylinder at roughly 50 micrograms, from surface contamination and similar effects, though this is an estimate, not a measured loss [6].

A constant of nature has no surface to get dirty, and the official guidance says no particular experiment is required, so any laboratory that can trace its mass to the fixed constants can in principle realise the kilogram [2]. That is why the answer to the lead’s question mattered: with a method like the Kibble balance, realising the kilogram no longer means comparing against the cylinder kept near Paris.

Think It Through: Cancel an Unknown With Two Measurements

You can feel the idea behind the trick with pencil and paper. Suppose a friend gives you a mystery ruler whose centimetres are secretly some unknown size k times too long. You cannot see k.

  1. Measure once. Measure a book’s length with the mystery ruler. You get a reading that is the true length divided by k.
  2. Measure a known length. Now measure something whose true length you already know, such as a line of exactly 10 cm drawn with a trusted ruler. That second reading is that known length divided by k.
  3. Combine them. Divide the first reading by the second. The unknown k cancels. The ratio tells you how many times longer the book is than the known length, without ever knowing k.

A Kibble balance does the same thing with magnetism instead of rulers. To go further, read NIST’s page on how the machine works and the abstract of the NIST-4 paper listed below [3][5].

Sources

I read the NIST pages (introduction and the Kibble balance page), the NPL page and the arXiv abstract of the NIST-4 paper through a fetch tool, and quoted figures only from what I could read. The NIST pages were read as text excerpts, not in full. For the BIPM guidance I could read only the opening part of the PDF (the definition and the statement that no particular experiment is required); the passage on the “virtual” power comes from the same guidance as summarised in a Japanese-language article I used for fact-checking, not re-read in the original. The cylinder’s stability estimate (about 50 micrograms) and the 2017 agreement between methods come from AIST (the National Institute of Advanced Industrial Science and Technology, Japan), a Japanese-language document that I did not re-open, so treat those two points as less firmly checked. The NIST pages did not give a figure for how far the cylinder drifted. I used no Wikipedia numbers.

  1. NIST, “Kilogram: Introduction.” https://www.nist.gov/si-redefinition/kilogram-introduction (the cylinder as the definition, November 2018 vote, drift and laborious comparisons; text excerpt)
  2. BIPM, SI Brochure, 9th edition, Appendix 2: mise en pratique for the definition of the kilogram. https://www.bipm.org/utils/en/pdf/si-mep/SI-App2-kilogram.pdf (fixed value of h, no particular experiment required; opening section only)
  3. NIST, “Kilogram: The Kibble Balance.” https://www.nist.gov/si-redefinition/kilogram/kilogram-kibble-balance (force mode, velocity mode, cancellation of BL, 1975, renaming, NIST-4 size; summary of the page text)
  4. NPL, “Kibble balance.” https://www.npl.co.uk/research/mass-mechanical-measurement/kibble-balance (developed in the 1970s, quantum electrical units, renamed from watt balance; excerpt)
  5. D. Haddad et al., “Measurement of the Planck constant at the National Institute of Standards and Technology from 2015 to 2017,” Metrologia 54 (2017) 633–641; arXiv:1708.02473. https://arxiv.org/abs/1708.02473 (over 10,000 weighings, 0.5–2 kg, relative uncertainty 13 parts in a billion; abstract only)
  6. AIST (Japan), N. Kuramoto, “Kagaku to Kogyo” 72-1 (Japanese-language). https://unit.aist.go.jp/riem/mass-std/化学と工業.pdf (cylinder stability estimate, silicon-sphere method, 2017 agreement; not re-read for this article)

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