Potential Simplification of Charge and the Coulomb Force without Affecting Predictions

We are taking a deeper look at charge and the Coulomb force and other electric properties. There is an embedded 10 −7 in the Coulomb constant that we will claim is “only” needed to cancel out an embedded 10 7 in the charge squared. We suggest three alternatives to redefine the charge and the Coulomb constant that give considerable simplification. The Coulomb constant is not needed as a separate constant as, in the new suggested framework, it can be replaced with simply the speed of light without affecting predicted output values. We also point out potential issues with the 2019 redefinition of the Coulomb constant and elementary charge. This is not meant conclusive but to open up for further discussion on how one potential can simplify parts of physics.


Discussion of the Coulomb's Law and Charge
The Coulomb's [1] force is, in modern physics papers and university text books, [2] given as: 1 2 2 e q q F k r = (1) 1 4 e k ε = π (2) This way of writing the Coulomb constant is, we will claim, confusing as it conceals what the Coulomb constant actually represents. It looks like the Coulomb constant is a separate constant needed, which is not the case in our view.
In the Coulomb constant, the 0 ε is the so-called vacuum permittivity given by Here, there will be an uncertainty in the elementary charge that comes from the uncertainty in the measurements of the fine structure constant α . Since 2019, the elementary charge has been exactly defined as 1.602176634 × 10 −19 Coulomb (C) (see [3]), something we will soon get back to.
Based on the definition of elementary charge given by Equation (4)  We suggest the Coulomb formula can be rewritten as: 2 2 qq F c r = (7) but now with the 7 10 taken out of the charge. For elementary charges this would now give: in other words, the same end result as before from Equation (5). Perhaps even simpler would be to re-define the elementary charge as: and then the Coulomb constant as c, which would give: In other words, this is the same as before, but with a strong simplification of the formulation. This would naturally lead to one also needing to redefine other electric properties accordingly.
The Planck charge is given by: which gives a Coulomb's force of: The Planck charge would be based on our new suggestion, where e k is redefined as 2 c and the charge have to be re-defined from or in the case in which we set e k c = , we have to re-define the Planck charge as simply  . This would still give the same result as before, as we would have: which is the same end result what one gets in the standard theory.
Column one in Table 1 shows the standard way (the way before 2019) to express some electric properties as well as the Coulomb force. Normally, the Coulomb constant is written as π , something we will claim is confusing as it is   , then some curious researchers will start to wonder why the 10 −7 is there, and why not only 2 c . We think we possi- bly have the answer to this; namely, that the embedded 10 7 and 10 −7 are only due to human convention and that the 10 −7 is only there indirectly to cancel out the 10 7 that is embedded in the elementary charge as well as the Planck charge. And the reason for this is likely an arbitrary choice of units before the theory had attained much depth. We therefore suggest re-formulating both the charge and the Coulomb constant and thereby the Coulomb force. The first alternative is shown in Table 1, where we just got rid of the 10 7 and 10 −7 , while alternatives two and three simplify this further. The Coulomb force in alternative 3 is similar to the Gauss unit system, which was frequently used in older papers and is still used today. In the Gauss unit system, for example, the Coulomb force is given by ; that is, with no Coulomb force constant. See, for example [4].

Short Discussion of the Coulomb Constant and Elementary Charge Using the 2019 Definition
In the 2019 redefinition of the SI base units, the value of the vacuum permeabil- Further, the elementary charge was redefined to be an exact value. We doubt this was optimal or even a very Logical consistent redefinition of SI units. In our E. G. Haug Journal of Applied Mathematics and Physics view, the 10 −7 is simply needed to cancel the 10 7 embedded in the elementary charged squared as described in the section above. Making the Coulomb constant a non-exact constant seems to be far from the optimal way to move around the uncertainty in the measured Coulomb force. The uncertainty in the Coulomb force should, in our view, come from the uncertainty in the charge that again comes from the uncertainty in the fine structure constant. The elementary charge can be, as described in the previous section, expressed as: 7 10 e c α = ×  (16) but this is not compatible with the new 2019 SI definition.
We think it would have been more consistent with fixing the Planck charge and instead let the elementary charge have uncertainty related to the uncertainty in the fine structure constant as also discussed by [6]. The Planck charge should be fixed . This requires that one keep the vacuum permeability Since the discovery of charge, there have naturally been many papers discussing how to measure it most accurately; see, for example, [7] [8]. However, since 2019, the elementary charge has been defined as exact [9]. We know c has already been defined as exact and this makes sense as the round-trip speed has been well tested to be a constant, and isotropic, and the same in any reference frame. Further, the one-way speed of light is the same as the round-trip speed of light when using Einstein's synchronized clock (per definition). In rotating systems, it could be more complicated; see [10] [11]. Further, the Planck constant is a constant. The fine structure constant is not exact, so to get this formula now to be exact based on the 2019 assumptions, then 10 7 cannot be exactly 10 7 , but must be slightly different, and not only that, the uncertainty in 10 7 must exactly offset the uncertainty in α . That is, the errors in the two must cancel exactly. This makes no sense in our view, in particular, since 10 7 will always cancel with 10 −7 embedded in the Coulomb constant. We think it has been a mistake to define the elementary charge exactly in 2019, as this likely leads to more problems than it solves, in particular in terms of lack of logic. The decision to not have the Planck charge constant and to make the elementary charge constant was partly favored by arguments from super string theory: "The advantage of the second choice is the possibility of accommodating a space-time varying fine structure constant, which appears then as a property of the vacuum and, in this case, easier implementation of gauge invariance. Advanced string theory points in this direction for the future [12]." From [6] page 585.
Superstring theory has had very limited success as is very well illustrated by the book by Conlon [13] Table 1 seem to simplify physics compared to the 2019 standard, where the columns marked alternative 1 to 3 again seem to be preferable. One then has the elementary charge as a function of the fine structure constant, and thereby uncertain.

The Composite View of the Gravity Constant Simplifies Many Formulas Related to Planckian Electricity
Max Planck [14] [15] in 1899 introduced the Planck length: there were three important universal constants: G, c and  . In 1984, Cahill [16] already suggested that one could express the Newton gravitational constant as a function of the Planck mass:  . This is simply the Planck mass formula solved with respect to G. However, Cohen [17] in 1987 pointed out that this way to express G made little sense as it seemed impossible to find the Planck units without first knowing G, so it seemed to just lead to a circular problem, a view held by the physics community until at least 2016. In 2017, Haug [18] was able, for the first time, to demonstrate that the Planck length could actually be found without any knowledge of G, and later he showed how the Planck length could be found without any knowledge of G and  ; see [19] [20] and even without any knowledge off G, h and c; see [21] [22]. This means we can express the gravity constant as a function of the Planck length of the form the Planck length formula solved with respect to G. That is the composite view of G as recently discussed in detail by Haug [23]. However, now, as we can find p l independently of G, this no longer leads to a circular problem so we can replace G in a series of Planck-related electric properties and see that this leads to simpler and more intuitive formulas; see also [24].
Electric properties have been added in relation to the Planck scale since Max Planck introduced the Planck units. In 1961, de Beauregard [25] suggested that "In terms of inter-actions c  plays the same role in the case of gravity as 2 e does in the case of electromagnetism", and he links this to the Planck mass. This means that the Planck mass particle charge is p q c =  . Today, the Planck charge has been re-defined as: Further, this can be re-written as: There are no Planck units embedded in the Planck charge; however, it is assumed this is the charge of a Planck mass particle. The difference is that there is no embedded fine structure constant in the Planck charge, in contrast to the elementary charge (of the proton and electron).
Planck current and Planck voltage, on the other hand, both contain G and this means they indirectly contain Planck units. The Planck voltage is given by (see Lundgren [26] and also Buzcyna et al. [27]): We can now replace G with In other words, the Planck charge is directly linked to the Planck length, or it even looks like it is linked to the Planck frequencies as we have p p c f l = . Still, there is little intuition in having something that is Planck frequency times the square root of c times  and again multiplied by the square root of 10 7 . We therefore highly suspect that the Planck voltage is something quite arbitrarily defined. It is clearly rooted in something very fundamental: c, p l and even  , but we suspect it is a derivative of a deeper reality, something we plan to return to in another paper we hope to write.
Planck impendence is also linked to G and thereby also indirectly to the Planck scale and is given by: In other words, the Planck impendence is nothing more than the speed of light times 10 −7 . Also, when it comes to so-called Planck electric properties, we think the 10 −7 and 10 7 always cancel out against each other in relation to direct observables. For example, energy can be observed. In general, Planck electric energy is considered much higher than we even can observe in LHC, but still it is at least theoretically something that can be observed, as it is simply a frequency multiplied by the minimum quantity of energy  , and it is given by: Journal of Applied Mathematics and Physics stand absolutely still, as we have already suggested before, based on a different angle of reasoning; see [21] [28].
The Planck magnetic field is given by: We see that the ratio of the electric field divided by the magnetic field is simply the speed of light.

Similarities between the Coulomb Force and the Newton Force?
The Coulomb Planck force is The end result is also the same as the other alternatives above. This means the  [32], Pilot [33] and Sharafiddinov [34]. These and other possibilities should be investigated further based on the simplification of the Coulomb force and electric units as suggested above. A new theory that unifies gravity and quantum mechanics related to the Planck scale known as collision space-time [35] should also be interesting to investigate with respect to this.

Conclusions
We have suggested how to reformulate the elementary and Planck charge as well as the Coulomb constant to simplify and demystify the Coulomb force. In this view, there is no need for a separate constant, for we can simply replace the 1 Our view have somewhat changed since our working paper [24] in 2016, where we had not taken into account that the Newton gravity force not is valid for two equal masses when on its standard should be discussed and investigated carefully before any choices are made.
In light of this, further work should also investigate possible relations between the Coulomb force and the Newtonian gravitational force.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.