The Standard Unit of Mass has been defined as the mass of metal cylinder since 1799 and today the mass is being re-defined in terms of several fundamental constants of the universe.

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Recent History of the kilogram

In 1799, an all-platinum kilogram prototype was fabricated.  This prototype kilogram was designed to equal the mass of one cubic decimeter of water at 4 °C. The prototype was formally ratified as the kilogramme des Archives(Kilogram of the Archives) and the kilogram standard became equal to this prototype mass. 

Since the earlier definition of the kilogram was defined by an amount of water at its maximum density, it is interesting to note that  a cubic decimeter of distilled water at its point of maximum density is only 25 parts per million less massive than the prototype kilogram.

It was discovered through extremely careful measurement that copies of the prototype stored throughout the globe were diverging in mass from the prototype over time. And this led to the effort to redefine the definition of the standard kilogram mass.

The Redefined Basis for the Standard kilogram

Derek Muller's Veritasium video, "How we're redefining the kg", explains that we -- the international scientific community -- is redefining the standard unit of mass in terms of unchanging constants of the universe.

Here's the theory of the Watt Balance or Kibble Balance used to define Planck's constant.

A wire of length, L, that carries a current, I, perpendicular to a magnetic field of strength, B, will be directly affected by a Lorentz force equal to `B*L*I`.  See Lorentz Force Law and 

  eq 1: `F = B*L*I`

In the Kibble balance, the current is varied so that this force exactly counteracts the weight, w,  of a standard mass, m. This is also the principle behind the ampere balance. The mass's weight is given by the mass, m,  multiplied by the local gravitational acceleration, g.  See the vCalc equation:  Weight

So,

  eq 2: `w =m*g=B*L*I`

The Kibble balance avoids the problems of measuring the field density, B, and the length of wire, L, with a second calibration step. The same wire (in practice, a coil of wire) is moved through the same magnetic field at a known speed, s. By Faraday's law of induction, a potential difference, V, is generated across the ends of the wire, which equals `B*L*s`. Thus

  eq 3: `V = B*L*s`   This is the equation defining the velocity mode discussed in Derek Muller's video, "How we're redefining the kg".

The unknown product `B*L` is eliminated from the equations by combining the two equations for velocity mode and weighing mode, so first you rearrange equation 2:

  eq 4: `B*L = (m*g)/I`

and this `B*L` is then substituted into the velocity mode equation 3:

  eq 5: `V = ((m*g)/I)*s`

Rearranging equation 5 gives:

  eq 6: `V*I = m*g*s`

`V*I` on the left side of the equation gives you electrical power and `m*g*s` on the right side gives you mechanical power. Both sides of the equation have the dimensions of power, measured in watts and so the device discussed here based on these principles was originally named the "watt balance".

V, I, g, and s can be accurately measured in the laboratory and this equation, in turn, then gives an accurate value for the mass, m:

  eq 7: `m = (V*I) / (g*s)`

As Derek Muller's video explains further you can more accurately measure the voltage in this analysis using a macroscopic quantum effect of Josephson Junctions, where we know that the voltage is defined:

  eq 8: `V = (n*h*f)/(2e)`

where h is Planck's constant, f is the frequency of the microwave radiation applied to the Josephson Junction,  n is the number of Josephson Junctions, and e is the well-known charge on an electron.  See the vCalc equation:  Voltage on a Josephson Junction

Since the current, I, is difficult to measure precisely, equation 7 is converted to:

  eq9:  `m = (V*(V/R)) / (g*s)` where R is the resistance

Using the Quantum Hall Effect, you can relate the resistance as: 

  eq 10:  `R = h / (p*e^2)` where p is an integer

Substituting equation 8 and equation 10 into equation 9, we get a definition of mass in terms of  Planck's constant:

eq 11: `m = (((n*h*f)/(2e))*((n*h*f)/(2e)) )* (((p*e^2)/h) / (g*s))` and simplifying:

eq 12: `m = ((n^2*h^2*f^2)/(4e^2)) * (((p*e^2)/h) / (g*s))` 

eq13:  `m = (p*n^2*h*f^2)/(4* g*s)` 

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