# Instrumentation & Measurement Magazine 26-1 - 6

```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
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
```

# Instrumentation & Measurement Magazine 26-1

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