Fifth Mundane Reality: The Electrical Impedance of Free Space

In order to generate coherent units for all the quantities related to the phenomena of electricity and magnetism, it is only necessary to choose a coherent unit for one electric or magnetic quantity. From that quantity, together with units for mechanics, we can derive coherent units for all the other quantities of electricity and magnetism. The difficulty is that, even though electromagnetism provides the underlying explanation for many “mundane” phenomena – including visible light, the entire field of chemistry, and even the everyday solidity of material objects – unfortunately the fundamentals of electromagnetism were not evident to our ancestors until very late in history (as late as the 19th_d Century | 11st_z Biquennium). So there is very little in direct, everyday, intuitive experience that human beings can draw upon to form the basis for a set of electrical and magnetic units.

The most fundamental quantity in this domain of physics is electric charge (q) or “quantity of (static) electricity”. A static electric charge generates an electric field, which is a vector field that causes an attractive or repulsive force on other static electric charges. Unfortunately, there is no particular amount of electric charge that could be deemed a “mundane reality” distinguished from any other, that could be chosen as a potential electrel for everyday use.

The elementary charge (), the amount of charge of an individual electron or proton, is a fundamental physical constant, but it is necessarily microscopic and therefore outside of “mundane” human experience. Once we have chosen a possible electrel suitable for human use at a macroscopic scale, we can give it a precision specification as some multiple of the elementary charge. For convenience, we could arrange for that number to be an exact whole number, but it would be a rather large and arbitrary-seeming value. This still leaves the actual choice of an electrel for everyday use an open question, so we have to approach that choice more indirectly.

A moving electric charge generates a magnetic field which is a vector field that produces an attractive or repulsive force on other moving electric charges or “magnetic poles”. This effect is a consequence of Einstein’s theory of Special Relativity applied to static electric charges when observed from an inertial reference frame in relative motion. When a quantity of electricity () is in motion with a given velocity () , we can define this as a “quantity of magnetism” (). This quantity is also known as magnetic pole strength. Therefore, given an electrel and a velocitel, we can easily derive a magnetel as the product electrel × velocitel, and use that as a coherent unit for this quantity. However, there is no specific quantity of magnetism that can provide us with a “mundane reality” either.

Electric current () is the time rate of flow of electric charge through a surface within a conductor. It can also be described as the linear density of the magnetism present along the length of the conductor. Hence, we can derive a currentel, a coherent unit of electric current, from either the quotient electrel/timel or the quotient magnetel/lengthel. Conversely, an electrel can be derived from the product currentel×timel and a magnetel can be derived from the product of currentel×lengthel. That said , there is no particular non-arbitrary amount of electric current that could be deemed a “mundane reality” that we can directly use as a currentel.

Indeed, SI’s own currentel, the ampere (A), was originally derived rather arbitrarily: It was defined as the amount of constant current in two infinitely-long parallel wires of negligible width set 1 meter apart necessary to generate a magnetic force of (2×10⁻⁷)_d newtons (N) – this tiny power of ten being an entirely arbitrary choice just to get the ampere to a reasonable size. SI’s electrel, the coulomb (C), was then defined as the amount of charge that would flow under a 1 ampere current in 1 second, i.e. coulomb = ampere×second. SI usually measures magnetic pole strength in ampere×meters, but note that this is equivalent to coulomb×meters/second. dHowever, Primel does not follow this pattern to derive its electromagnetic units.

Electric potential () is the amount of potential energy per unit of electric charge within an electric field, such as the electric field in a circuit. But once again, there is no particular amount of electric potential that could be deemed a “mundane reality.”

However, if we consider electrical resistance (), electrical reactance (), and electrical impedance (), there does exist a particular quantity with some special properties: namely, the impedance of free space or vacuum impedance (). This is a universal physical constant whose value falls comfortably within the range that human beings typically encounter at a macroscopic level: about 376.73_d ohms (). It turns out that if we use this quantity as Primel’s coherent unit of resistance, reactance, and impedance (i.e. as the ⚀resistancel, ⚀reactancel, and ⚀impedancel), the result is a system of units for electromagnetism that is generally well-balanced for human usage.

For instance, the ⚀electrel derived from this winds up being approximately 58.35_d micro·coulombs (μC). This might seem small, but in reality this is a reasonable amount of static electric charge that a human might encounter. A coulomb actually represents an unusually large amount of static charge.

The ⚀currentel winds up being approximately 2.0167_d milli·amperes (mA). That is also small but comfortably within the everyday range for small devices. Of course, like all Primel units, it is amenable to scaling up with SNN_z prefixes. For instance, the ⚀triqua·currentel (⚀t↑ctℓ) is approximately 3.48_d A. Standard household currents of 100_d to 200_d A would be approximately between 24.84_z and 49.48_z ⚀t↑ctℓ.

Primel’s coherent unit of electric potential, the ⚀electrelic·potentialel (or ⚀εpotel for short), comes out to approximately 0.7597667_d volts (V), which is an eminently reasonable size. An electric potential of 2 ⚀εpotels approximates the typical 1.5 V of an alkaline battery. The ⚀unqua·εpotel (⚀u↑εptℓ) is about 9.1172_d V, approximating the 9 V of a PP3 style battery. The ⚀biqua·εpotel (⚀b↑εptℓ) is approximately 109.4_d V. So a typical home electrical supply of 110_d or 220_d V could be neatly approximated by 1 or 2 ⚀b↑εptℓ (100_z or 200_z ⚀εptℓ).

The “balanced” nature of the resulting system becomes even more evident, once we examine the special relationships that the vacuum impedance bears with other universal constants that are not as close to human-scale, including the so-called electric constant () and the so-called magnetic constant (). These constants appear in Maxwell’s Equations as well as in Coulomb’s Force Law, Ampère's Force Law, the Biot-Savart Law, and the Lorentz Force Law. All of these constants also bear a special relationship with the speed of light in a vacuum (), yet another universal constant. The reciprocals of each of these quantities are also of some importance, although not all of them have been given convenient names. Primel offers the following alternative terminology for all of these fundamental constants, their reciprocals, and the physical quantities they manifest. It also offers a notation for deriving symbols for the reciprocal quantities by placing an overline on the symbol for the reciprocated quantity:

Notice that the vacuum elastivity (), the vacuum inductivity (), and vacuum lightspeed, are all the exact same multiple of their respective Primel unit. Likewise, the vacuum capacivity (), the vacuum reluctivity (), and the reciprocal vacuum lightspeed (), are all the exact same multiple of their respective Primel unit. This is no accident, but a consequence of the following universal relationships:

In short: (1) The vacuum elastivity constant () is the ratio between the influence of an electric field to the square quantity of electricity provided by the two electric charges involved. (2) The vacuum inductivity constant () is the ratio between the influence of a magnetic field to the square quantity of magnetism provided by the two moving electric charges involved. (3) Whereas the vacuum impedance constant () is is the geometric mean of both of these constants, representing the ratio of the influence of an electromagnetic field, to the product of both the quantity of electricity causing the electric aspect of the field, and the quantity of magnetism causing the magnetic aspect of the field. Thus vacuum impedance stands at the heart of electromagnetism, as it were.

Coulomb’s Law

where:

• = =

• = quantities of electricity (electric charges)

• = distance between them

• = surface area of sphere at radius

• = Coulomb’s constant

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