The Idea of Charge

1The idea of charge in the form of static electricity was known to the Greeks via rubbing pieces of amber. However, the first electrostatic generator was not built until around 1650 and later followed, in about 1744, by a device that could store static charge like a capacitor, which was called a Leyden Jar. Out of this historical context there grew a perception that charge was a ‘substance’ that flowed  like a liquid. However, the previous discussion, related to the idea of energy, raised doubts as to whether the concept of a matter particle, i.e. something made of physical substance, could have any meaning at the most fundamental level of reality. If so, then doubts might also be raised against the idea of charge being a property of a charged particle like the electron and proton. While, today, this subject is described in terms of electromagnetism, these fields are still said to emanate from charged particles, so we will begin by focusing on some fundamental ideas by making reference to Ampere’s force law:

[1]      1

This equation above is an expression of the magnetic force between 2 wires of length [s] carrying currents [I1] and [I2] respectively and separated by a distance [r]. Often, this equation drops the variable [s] and simply quantifies the results in terms of a force per unit length, but in this discussion, we want to take a closer look at all the units involved. At this point, we will simply define [KB] as a constant that has to resolve the units on both sides of [1]. Now, the definition of current [I], in MKS units, is defined in terms of charge per second, where the fundamental unit of charge is called the Coulomb [C]. As such, we have a specification all the units of all quantities in [1], except [KB], which we can now solve:

[2]      2

As per the previous discussion related to energy, it seems that the concept of mass is a deeply rooted concept in terms of the definition of units. In this context, we can see that the definition of force [F], derived from energy [U], must ultimately determine the units in [1]:

[3]      3

However, the fundamental units of [KB], in [2], do not really tell us much about the nature of this constant, so we will switch the focus to Coulomb’s law, which expresses the electric force between charges [q1,q2]:

[4]      4

Again, we can initially define [KE] as a constant required to resolve the units on both sides of [4] as follows:

[5]      5

Like [KB], the fundamental units of [KE] do not tell us much about the nature of this constant, but clearly it is also required within the SI system of units, although both of these constants are not usually presented as [KB, KE], but rather as shown below:

[6]      6

In the context of SI units, as shown in [6], the electric constant [ε] is also described in terms of a concept called ‘permittivity’, while the magnetic constant [μ] is described in terms of a concept called ‘permeability’.

In electromagnetism, permittivity is described as a measure of the resistance that is encountered when forming an electric field in a medium, while permeability is the measure of a material to support the formation of a magnetic field. However, these descriptions can also be extended to the vacuum of space, although we might still wish to question how these concepts are supported by a vacuum, if only empty space.

However, at this stage, we might reflect on the fact that [1] and [4] appeared to be linked by the SI definition of current [I] and charge [Q], where current equals charge/second. As such, assigning a value to one, must also define the other. The values of all the variables and constants, in [1] and [4], were initially determined by experiments performed by Weber and Kohlrausch, in 1856, which pointed to a relationship central to Maxwell’s equations , i.e. where [c] is the speed of light in a vacuum.

[7]      7

While we have define the magnetic force [FB] in [1] and the electric force [FE] in [4], they are combined in a more convenient form in Lorentz force equation:

[8]      8

As such, we may use the components of [8] to define the units of the electric [E] field and magnetic [B] field in SI units.

[9]      9

In a similar fashion to [7], we might consider the ratio of the electric field [E] to the magnetic field [B], which may again suggest a velocity, although this inference may have to be considered further.

[10]    10

As touched upon in earlier discussions, the SI/MKS units have been an international standard since the early 1960’s and one generally used throughout this website. However, the Gaussian system dates back to 1881 and was the original de-facto standard adopted by pioneers, such as Hertz and Maxwell. Therefore, this means that many original scientific papers were written in the Gaussian system and so cross-references to these papers still tend to make use of both systems, which also raise some interesting questions about the definition of the electric [E] and magnetic [B] fields and the meaning of the speed of light [c] in many equations. For the purposes of the following part of this discussion, it might be useful to provide a comparison of some of the key units in both the SI and Gaussian systems in fundamental units, i.e. length [m], time [s], mass [kg], charge [q]:

Quantity

Symbol SI Gaussian
Charge q q 1
Current I 2 3
Electric Field E 4 5
Permittivity ε 6 1
Magnetic Field B 7 8
Permeability μ 9 1
Energy U 10 11
Energy Density h 12 13
Force F 14 15

So, in the table above, we see the different definitions that arise in electromagnetism due to the more fundamental difference in the definition of charge [q], i.e. the SI system defines charge in terms of the Coulomb, while the Gaussian system uses the Statcoulomb. However, in practice, these names do not really tell us anything until presented in the base units of the MKS system.  However, it is useful to note that the Gaussian system, unlike the SI system, only defines charge in terms of the 3 base units, i.e. MKS, which stems from the fact that the Gaussian system has no normalising constants related to permittivity [KE,ε] and permeability [KB,μ].

So how do these units change the description of electromagnetism?

Clearly, the definition of charge [q] in these 2 systems of units leads to some fundamental differences, which will be outlined below. However, what appears to be a fairly trivial aspect is the presence of [4π] in many electromagnetic equations, due to the definition of permittivity and permeability, which in the Gaussian system do not exist. In essence, these constants merge and get replaced by the speed of light [c], as per the following relationship, which might be seen as a conversion factor between the 2 systems:

[11]    11

While the SI units could also use [11] to eliminate one of the three constants [ε,μ,c], this could not be done in a symmetrical manner and therefore all three constants are retained in the SI system. Therefore, the description of permittivity and permeability of ‘free space’ may be a misleading artefacts of the SI system of units rather than any fundamental physical property of free space. We might also rationalise the definition of charge in the Gaussian system by the exclusion of these parameters when reformulating [4]:

[12]    12

What might also be noticed, in the table above, is that the electric field [E] and the magnetic field [B] end up having the same units in the Gaussian system, such that the ratio [E/B] originally presented in SI units in [10] no longer implies a velocity [c], but just a ratio between the field strengths:

[13]    13

The effect of [13] will also affect the form of Maxwell’s time-dependent equations, when presented in either SI and Gaussian units, shown in a simplified free-space form below:

[14]    14

Of course, while both the SI and Gaussian system have to be consistent in their definition of any composite quantity, it seems that the system of units adopted might come to change how we describe physical phenomena. As indicated, the idea has been raised that mass [kg] might only be a manifestation of energy, which might in-turn be described in terms of some sort of waveform. If so, there may be some general link to Planck’s energy equation [E=hf], where [f] corresponds to the wave frequency. Now, it is clear from [14] that Maxwell’s equations of electromagnetism are making reference to EM waves, as opposed to photons, such that we might question how the electric [E] and magnetic [B] field components of an EM wave might relate to energy. So, in the context of [14], [E] and [B] generally comply to the description of an ‘amplitude’ of a wave, which in classical wave mechanics might be associated with energy via the following proportional relationship:

[15]    15

So let us pursue a possible interpretation of the electric [E] and magnetic [B] field amplitudes, first in terms of their definition in Gaussian units:

[16]    16

At first, there may not be any obvious interpretation of the units on the right until we cross-reference the units of energy density in the table above. Given that [16] reflects the same units as energy density, it might be suggested that these fields also reflect the energy within some volume of space. We can repeat [16] in SI units, although we now have to introduce the permittivity and permeability constants required by the SI system:

[17]    17

In part, this discussion started out as an extension of the idea that mass might have no substance at the most fundamental level of reality. On the basis of this deductive premise, it was suggested that Einstein’s mass-energy equation had to be transposed, such that mass was an emergent form of energy, not the other way round. However, it was also recognised that energy, as a scalar quantity would require some mechanism to move in space and time, e.g. some form of wave. At this stage, no description of what this waveform might look like has been forwarded, but it was also recognised that in replacing the idea of mass, in the form of particles, the idea of a charged particle would also be lost. Therefore, let us table one further idea that may have emerged out of this discussion, i.e.

What is charge?

While it is often convenient to talk of a charged particle, in practice, all classical formulations involving charge really require 2 charge particles, even if one becomes the test charge of the measuring equipment. As such, we might wish to examine the fundamental units of charge within the Gaussian system, when aggregated as a relationship between 2 particles:

[18]    18

Of course, in line with the no mass premise, we would also have to replace the definition of mass [kg] with its energy [U] counterpart:

[19]    19

However, it is possible that [20] is simply reflecting the potential energy field that classical physics normally describes in terms of the charge between 2 particles separated by some distance [x], e.g.

[20]   20

While we have not really addressed the last question above, it is possible that physical particles are not required to support the idea of charge. As such, we might also want to take another look at the idea of a photon.