fundamentalsmeasurement continued of write it, has the special name newton (symbol: N). A complete list of units with special names is found in Table 4 of [2]. It has been known for hundreds of years that the gravitational force between two spheres is proportional to the product of their masses and inversely proportional to the square of the distance r between their centers. For simplicity, suppose that the two spheres have equal mass m. Then F mr 22 / . In order for the proportionality to become a homogenous equation, a constant multiplier must be introduced. The multiplier is the Newtonian gravitational constant, G. The dimension of G is just what is needed to make the equation: F Gm r / 22 (2) homogeneous, i.e., for the dimension on both sides of the equation to be the same. The dimension of G must therefore be: dim G = dim Fr2/m2 = T−2 M1 L1 ·L2·M−2·I0=T−2 M−1 L3 In the SI, G has the coherent unit m3s−2 kg−1 I0 . This simple dimensional analysis merely tells us the dimension of G in the SI and its corresponding coherent SI unit. The analysis does not tell us the value of G, or that G might have a different dimension in a different unit system, or that G is a fundamental constant of nature. Ultimately, dimensions are " a choice that we make " [3] to establish a useful system of units. The various cgs systems of units are all based on the centimeter, gram, and second (cm, g, s). The dimension, dim Q, of any quantity Q can usually be expressed easily in terms of the dimensions and units of other systems. Thus, the value of each quantity Q can be expressed in different systems of coherent units. This means, for instance, that: QQ Q Q Q SI cgs SI cgs (3) where {Q}SI is the numerical value of Q when it is expressed in its SI unit [Q]SI, and similarly for the numerical value of the same quantity when it is expressed in a cgs system. For example, if the value of a force, F, is 1 N in SI, then F = in a cgs system is found . We know that 1 kg , which 1 kg m s−2 ; its numerical value {F}cgs from the relation 1 kg m s−2 = 103 g and 1 m = 102 = {F}cgs g cm s−2 cm. It then follows that {F}cgs = 105 means that 1 N is equivalent to 105 can say that there are 105 dyn; put another way, one dynes in one newton. The dimension of some quantities is simply 1 (such quantities are sometimes called " dimensionless " ). This will be the case for ratios of two quantities of the same kind, such as the ratio of two velocities, v1 and v2 [2]. The ratio of two given velocities must be the same number in all unit systems. Another example is the ratio of the proton mass to the electron mass, which is the same number, approximately 1863, in all unit systems. Ratios of the same kind of quantity have a special place in the following discussion. 6 " Thought " Experiments Follow the Rules of Dimensional Analysis As noted above, relations among coherent units follow the symbolic relationships among quantities, such as " force equals mass times acceleration. " Elaborate symbolic relations often must be used to model laboratory and industrial work so that all significant corrections are taken into account. However, even textbook examples where students must determine the results of simplified, unrealistic " thought " experiments must also obey the rules of dimensional analysis. One such thought experiment gave us the previous SI definition of the ampere, which was in force for 60 years, until May 20, 2019. This definition relied on a text-book problem: Two parallel wires of infinite length and negligible circular cross-section are maintained in vacuum; if the wires are separated by a distance a and a current I flows in each wire, find the force per unit length on each wire. Answer: In the SI, if a = 1 m and I = 1 A, the force per unit length on each wire is 2 × 10−7 enous because μ0 N/m. The force equation was homog, also known as the magnetic permeability of free space, had been defined to be exactly 4π × 10−7 N A−2 . This was sufficient to define the ampere as the base unit of the dimension I. How this definition of the ampere came into being is a long and interesting story in which electrical engineers played a key role [4]. Suffice it to say that this definition ensured that the ampere and all the other electrical and magnetic units that electrical engineers had adopted [4] became coherent in the MKSA system, which is the three-dimensional MKS system with a fourth dimension added for current. This was an important precedent that has since been followed to define, and then to redefine, base units of the SI. Maxwell's equations require the relation c2 act in all unit systems, where ε0 μ0ε0 =1 to be exis the electric constant, also known as the electrical permittivity of free space and c is the speed of light in vacuum. As an aside, Gaussian cgs units choose ε0 = 1 and μ0 = 1/c2 . The SI unit of current is the ampere, which is a base unit of the SI. The corresponding unit, the statampere (statA), is the Gaussian unit of current written in cgs base units as g1/2 cm3/2 SI s−2 shown that 1 A 10 c statA , where {c}SI . Making use of (3), it can be is given in Table 1. One way to derive this relation is to return to the textbook problem of the force per unit length on each of two infinitelylong, parallel wires in vacuum. If the distance between the wires is 1 m and the current in each wire is 1 A, the result in SI equaled to 2 × 10−7 N/m. The result calculated in the Gaussian system must be the same for the same wire separation, 100 cm, and the equivalent current in statA. The Gaussian result is expressed in dyn/cm, where 1 dyn/cm = 10−3 N/m. The identical force per unit length can only be achieved if the relation between the SI ampere and the Gaussian statampere is as stated above. IEEE Instrumentation & Measurement Magazine February 2023

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