Prehistory

Nearly all descriptions of electric and magnetic phenomena before the 19th century were qualitative. Instruments available to researchers like B. Franklin and A. Volta were designed to display phenomena or to impress rather than to measure. References to what today is called the difference of potentials were often made in terms of sensations such as sound, shocks or spark appearances. Electric batteries were tested and compared by their taste to the tongue when two electrodes were applied. Electric currents were oflen rated in terms of redness of thin wires through which they passed.

Even in the mid-19th century, when telegraphy became a practical reality, potential differences were still rated in numbers of battery cells and resistances in terms of miles of the so-called "standard" wire. The standard in England was copper wire #16, while in Germany it was iron wire #8, and individual researchers often worked with their own peculiar units. All this was very inconvenient and complicated international cooperation in science and commerce. Thus, for the sake of progress, various makeshift units had to be replaced by universal units derived from both theoretical and practical considerations.

History

It was the famous German mathematician and scientist, F. Gauss, who was the first to show (1833) that all electromagnetic quantities, with which he had to deal in a study of magnetism, could be expressed in units derived from three fundamental unitsÑlength, mass, and time. It is interesting that the fundamental units chosen by Gauss, and by his successor in this area, W. Weber, were millimeter, milligram, and second. English scientists, headed by W. Thomson (Lord Kelvin) and by J. Clerk Maxwell, adopted the Gauss-Weber methodology, but proposed to use centimeter, gram, and second as fundamental units. That is how the famous CGS system was conceived in the 1860's. It is interesting to record that an alternative system based on foot, grain, and second was also considered by the British scientists, but the metric CGS system was selected as superior.

It became clear that there were at least two systematic ways of developing CGS units for the electromagnetic quantities, one called electrostatic (es), and another called electromagnetic (em). In both cases, equations would be written sequentially in such a way that each successive equation contained only one new quantity. Thus the units for all quantities could be derived from the equations, one after another. The first equation in the es approach was Coulomb's law for electric charges:

where k was taken as a dimensionless quantity equal to one. It was used to define the unit of electric charge called franklin. One franklin was defined as a charge which acted on another charge, of the same magnitude, with a force of one dyne when the distance between the charges, r, was one centimeter. Once the charge unit was determined, the unit of current could be defined from the I = Q/t formula, etc. All other units of a system would be derived in a similar way.

In the em approach, the first equation was originally Coulomb's law for magnetic charges. Later, however, the concept of magnetic charge (also called pole strength) was discredited, and Ampere's law became the first formula in the em sequence of definitions. This formula

gives the force per unit length, F/L, between two parallel currents I₁ and I₂, separated by distance d. The Factor 2 enters naturally from the formula for the magnetic field at a distance and from a long straight wire. The coefficient was chosen to be dimensionless and equal to one. The unit of electric current defined by the above formula is a current which acts on another current of the same magnitude with a force of one dyne per centimeter when the distance d is one centimeter. In honor of a French scientist, Biot, this unit was named biot. One biot was found to be 3 x 10¹⁰ times larger than the es unit of current, franklin per second. That finding was very significant because 3 x 10¹⁰ is the speed of light in cm/s. The appearance of c in magnetic relations was an important clue to Maxwell, who developed the theory of electromagnetic waves (1873), and subsequently to Einstein, who developed the theory of relativity (1905).

Both the es and em units are often referred to as absolute or theoretical units, because they are products of logical considerations not backed by any specific standards. Absolute units can be contrasted with practical units, which are defined in terms of agreedupon standards or procedures. To make this point clear, consider the origin of our unit of length, meter. At first a theoretical definition of meter was introduced; one meter was declared to be 10⁷ times smaller than the distance from the north pole to the equator along a particular meridian. This was a well-defined length, but the absolute definition was of no help to those who had to measure distances. What was needed was a ''yardstick" representing one meter. Such a stick was constructed, and many replicas of it were made and distributed among various countries. However, several years later, when more precise measurements of our planets became available, it was found that the man-made prototype, named meter, was actually 0.008% smaller than it was meant to be. Rather than building a new prototype, a decision was made to change the definition. From 1889 up to 1960, the meter was defined as a distance between the two scratches on the man-made prototype, called the standard international meter.

The story repeated itself in the development of practical units for electrical measurements. When units called ohm, ampere, and volt were introduced to satisfy practical needs of electrical engineers, they were meant to be integral multiples of the corresponding em units. Thus, one ohm was meant to be 10⁹ em units of R, one ampere was meant to be 10^-1 em units of I, and one volt was meant to be equal to 10⁸ em units of potential difference, U. But the manmade prototypes for these units turned out to be slightly different from what was intended, and consequently the prototypes became the definitions of the named units. The reason for defining practical units in terms of multiples of the em units, rather than in terms of the em units themselves, had to do with a preference for dealing with numbers that were neither too large nor too small in most common applications.

The first electrical yardstick was constructed in 1863 under supervision of Maxwell, and it was meant to represent a resistance equal to 10⁹ em units. It was made from a wire of Pt-Ag alloy of specified composition and named BAU (British Association Unit). But a significant discrepancy between the resistance of one BAU and what it was meant to represent was discovered, and this led to a construction of another prototype. This new prototype was made in the form of a mercury column, which at specified temperature had a length of 106.300 cm and a uniform cross section of one square millimeter. This definition of a practical ohm was declared official by the International Electrical Congress in Paris in 1889. Later the official international ohm was found to be about 0.05% larger than the absolute ohm, but the legal definition remained in force until 1948 when it was replaced by a new definition. The superiority of the mercury ohm standard over the wire ohm was in the simplicity of reproduction. No actual prototype was needed, an exact description was sufficient.

A practical unit of electric current, supposedly equal to one tenth of the em unit of I, was defined as a current which, when flowing through a water solution of AgNO₃, deposited silver at a rate of 1.118 mg per sec. This definition became legal after the International Electrical Congress in Chicago in 1893 and after a conference in London in 1908. The unit of potential difference, named Volt, was simply declared to be a drop of potential across one ohm when a current was one ampere. The electromotive force of a standard Weston type cell was found to be equal to 1.0183 V, at 20¡ C, and in practice it was often used to calibrate voltmeters or other instruments.

It is important not to confuse absolute ohm, ampere, and volt, which we use today, with the ohm, ampere, and volt described in the last two paragraphs, unless discrepancies of up to 0.05% are of no significance. To avoid confusion, earlier ohm, ampere, and volt are often labeled as "international" units, while today's units are usually referred to without this adjective. Although the CGS approach has been replaced by the SI, we can still look at our present ohm, ampere, and volt as exactly integral multiples of the corresponding em units.

Thus by the end of the 19th century, the international ohm and international ampere were legally defined and gained widespread popularity. Other practical units, such as volt and farad, were then defined in terms of ohm, ampere, centimeter, gram, and second. Instruments calibrated in these units became available and confusions created by makeshift units came to an end. But the situation was not totally satisfactory because practical units were slightly different from the absolute units, and because there were two mutually inconsistent CGS systems; es and em. For example the dimension of electric current in the es was g^1/2*cm^3/2/s² while in the em it was g^1/2*cm^1/2/s; the es dimension of the potential difference was g^1/2*cm^1/2/s while in the em it was g^1/2*cm^3/2/s. For the time being these deficiencies were tolerated, but eventually they led to further improvements which came in the form of our present SI system.

Toward the SI Approach to Electromagnetism

The history of the SI begins in 1892 with the work of a British scientist, O. Heaviside, and in 1902 with the unrelated work of an Italian electrical engineer, G. Giorgi. Heaviside was the first to observe that numerous appearances of the 4*PI factor in the formulas derived from (1) and (2) would disappear if k and gamma were equal to 1/(4*PI). By placing 4*PI in defining formulae, he tried to create a situation in which 4*PI would appear only in problems with spherical symmetry, and 2*PI only in problems with axial symmetry. This approach was called rationalization, but it did not become very popular, probably because the computational conveniences were not worth the effort of changing the units. But when the CGS was abandoned, as proposed by Giorgi, the rationalization was performed on the newly defined units. That is why we have 4*PI in Coulomb's law and in the Biot Savart formula today.

Before the proposals of Giorgi can be explained, let us answer the following question. Why was it necessary to define international ampere electrochemically rather than electrodynamically in terms of Ampere's law (2)? The answer is very simple. The values of F/L are usually very small, and the methods of measuring currents in terms of forces between the wires were not as accurate as those based on cumulative electrochemical effects. Consider for example two straight wires separated by 0.5 cm when I₁ and I₂ are each equal to one ampere (0.1 emu). According to formula (2) this would result in F/L = 0.04 dyn per cm, which is equivalent to the weight of 40 micrograms mass per unit length. A force of that magnitude could not be measured with the same degree of accuracy as the weight of over two grams of silver that would be deposited on a cathode of an electrochemical cell by a current of one ampere in 30 minutes.

Technological progress in the 1930s, however, led to better ways of measuring electric forces between wires, especially when straight wires were replaced by interacting coils, as in electric balances. That is why the old definition has been abandoned. The familiar SI definition of ampere is electrodynamical. The great advantage of defining a practical unit electrodynamically is the resulting identity of practical and absolute definitions. Both are now based on the same formula.

To understand the significance of Giorgi's contributions, it is important to realize that the joule, now widely accepted as a unit of energy, was not a CGS unit. In fact it was introduced in 1882 by W. Siemens to represent the amount of heat produced in one second by a current of one ampere flowing through a resistor of one ohm. In 1889 this unit was legally accepted together with watt, the unit of rate at which electrical work is done.

Giorgi was the first to recognize that if the three fundamental units were replaced by meter, kilogram, and second, then the joule, equal to 10⁷ ergs, would become a natural unit of work and energy in all areas. Most importantly, ohm, ampere, and volt would become natural units of the new system, and not just subunits of the em, defined in terms of multiplying factors 10⁹, 0.1, and 10⁸. This follows from the fact that joule is volt times ampere and that 10⁸*0.1 is equal 10⁷, which is the number of ergs in one joule.

The proposal made by Giorgi was to replace the CGS systems by one new system. This new system would be based on four, rather than three basic units: meter, kilogram, second, and one electrical unit. In the original recommendation, the electric unit to become basic was ohm. The replacement of two mutually contradictory systems of units by one was certainly a very desirable step, but somehow no attention was paid to Giorgi's proposal until the 1930's when it finally caught the eye of the International Electrical Commission (IEC). The commission supported the proposal but decided to use ampere rather than ohm as the fourth basic unit. Consequently the Giorgi system became known as MKSA, where M stands for meter, K for kilogram, S for second, and A for ampere. In 1935, and again in 1938, the MKSA was officially endorsed by the IEC as a new world system to become effective in 1940. World War II delayed the implementation of tnese decisions, and only in 1954 did the 10th General Conference of Weights and Measures formally legalize the MKSA system. In the rationalized form it became part of our present Sl system, which became official in 1960; Sl is the acronym for the full French name "Systeme International d'Unites."

The Sl Units

Unlike the CGS systems, which were based on three fundamental units, the MKSA is based on four. Ampere is now operationally defined by formula (2) with gamma equal to 10^-7. Thus in the SI one ampere is a current which, in a vacuum, acts on an identical current, one meter away, with a force per unit length equal to 2 x 10^-7 newtons per meter. There is nothing magic in the 10^-7 factor; it is simply a means of making one ampere equal to 0.1 emu, as was originally intended. To show this, one should convert the distance between the wires into 100 cm, 2 x 10^-7 N/m into 2 x 10^-4 dynes/cm and reexamine the official SI definition in terms of formula (2) using the old em value of gamma = 1. The elevation of ampere to the rank of a fundamental quantity has an important consequence. It means that in the SI, gamma in (2) must have a dimension, which is N/A². Otherwise, the defining formula would be dimensionally unbalanced.

Definitions of units for other electric and magnetic quantities of the Sl are obtained in the same way as was done in the CGS systems. For example, the Sl unit of charge, called coulomb, is nothing else but As. Similarly, the SI unit of potential difference, volt, is nothing else but J/C, while ohm is simply V/A, etc. Although the approach and philosophy behind the Sl were in many ways very different from the earlier approaches, the sizes of electrical units in the Sl remained essentially the same as in the earlier practical system, based on the em. Thus all the laboratory instruments purchased and calibrated before the introduction of the Sl could still be used after that system became official. Only in very rare instances would small differences between the earlier international and new absolute units become significant

It is important to keep in mind that in the SI unit of charge, coulomb, is defined without any reference to Coulomb's law. It means that one is not free to assign an arbitrary value to k in (1). The dimension of k must be N m²/C² and its numerical value must be determined from experiments, more or less like the value of the universal gravitational constant in Newton's law. It turns out that k is approximately equal to 9 x 10⁹ N*m2/C².

The original MKSA system was not rationalized, but the Sl is. This means that k and gamma in formulas (1) and (2), for a vacuum, must be written as x/(4*PI) and y/(4*PI), where x and y play the role of new coefficients. There is nothing else involved in the rationalization. In the SI, x is usually written as 1/eps_zero while y is written as mu_zero. Thus the value of eps_zero must be about 8.85 x 10^-12*C²/(N*m²) and that of mu_zero must be 4*(PI) N/A². Coefficients eps_zero and mu_zero are called permittivity and permeability of empty space. A discussion of the origin of these names is beyond the scope of this short article. It is important to note, however, that the ratio of k/gamma, which can also be written as l/(eps_o*mu_zero), is exactly equal to the square of the speed of light in a vacuum, both numerically and dimensionally.

Final Comment

It is very unlikely that the SI will be changed or modified drastically in the near future. But this does not mean that the Sl approach to electromagnetism is the best of all possible approaches. Elementary physics teachers who have to deal with the l/(4*PI*eps_o) factor in Coulomb's law and with mu_zero would certainly have something to say about this subject. Published records show that the physics community was deeply divided in the period of transitions to the Sl and many physicists opposed the innovations. Their opposition was not always unjustified.

A vivid and very readable account of animosities which divided physicists at the times of official recognition of the MKSA, can be found in an article entitled, "Systems of UnitsÑTheir Past and Their Probable Future," written nearly half a century ago (1). In reading this article and other publications of those times, it is quite clear that the effect of the new approach on teaching elementary electromagnetism, was not a primary argument for or against the MKSA. This became an issue 10 years later with the publication of the Coulomb's law committee of the A.A.P.T. entitled, "The Teaching of Electricity and Magnetism at the College Level.''(2) It would certainly be very desirable to evaluate the effects of the SI on teaching elementary electromagnetism in the last two or three decades to see what extent the expectations of the report were fulfilled.

References

1. R. N. Varney, Am. J. Phys. 8, 222, 1940. 
2. Am. J. Phys. 18, 1, 1950.