Solving the Problem of Infinities in Electrodynamics (and Mechanics) and a New Formula for the Electrostatic Potential

1. Introduction

Theoretical physics is faced with the task of constructing a consistent fundamental theory in which all physical quantities would be finite. A significant obstacle on this path is the problem of infinite values (divergences) that is existed in electrodynamics. Its essence is as follows.

The electric field of a motionless point electric charge q is described by the electrostatic potential, the modulus of which is determined in vacuum by the formula

ϕ = ϕ(r) = kq/r,

(1)

where k is a proportionality coefficient, r is the distance from the charge to the point for which the potential is determined.

According to modern concepts, the force (electric or gravitational) field surrounding the charge or body is limitless, therefore:

- at the infinite distance from the charge, the potential is zero: ϕ(r=∞) = 0;

- as the distance decreases, the potential increases;

- at the point where the charge itself is located, the potential goes to infinity: ϕ(r=0) = ∞.

Current theory considers an electron as a point particle (with a point charge), i.e., as a material object without extension [1]. Thus, a resting point-like electron must have the infinite self-energy and, therefore, the infinite mass. The meaninglessness of this result shows that electrodynamics becomes internally contradictory when moving to sufficiently small distances.

Electrodynamic equations can be written in different measurement systems.

It should be recalled that the International System of Units (SI) is a composite system that includes, in particular, the m-kg-s system of mechanical units (MKS system) and the m-kg-s-A system of electromagnetic units (MKCA system). The second system differs from the first primarily in that, along with the existing three basic units (meter, kilogram, and second), it has a fourth basic unit – ampere (A). For example, in the MKSA system, the elementary electric charge e = 1.6×10⁻¹⁹ C, and the coefficient k = 9×10⁹ N∙m²/C².

Although the educational literature on electrodynamics focuses on the SI, however, it usually does not indicate that the SI is a composite system. As a result, many do not understand that the MKS and MKSA sub-systems are the separate systems.

In 2018, an article [2] was published that showed that the electromagnetic units of the MKSA system (the ampere, coulomb, ohm, volt, etc.) can be converted using the basic units of the MKS system: m, kg, s. In the paper, it was shown that in the MKS system

е = 1.6×10^–25 kg∙m/s,

(2)

k = с²/F₁ = 9×10²¹ m/kg,

(3)

where c = 3×10⁸ m/s is the speed of light in vacuum, and F₁ = 10^–5 kg∙m/s² (or 1 g∙cm/s² – the unit of force in the СGS system).

Since (in the MKS system) the charge has the dimension of momentum, [e] = kg∙m/s, the ratio of the charge of an elementary particle to its mass has the dimension of velocity, [e/m] = [υ] = m/s. Therefore, we can write:

e/m = υ, e = mυ.

(4)

Obviously, the value of e/m is different for different particles; for example,

for an electron (its mass m_e = 0.9109×10^–30 kg) e/m_e = υ_е = 1.759×10⁵ m/s,

for a proton (its mass m_p = 1.672×10^–27 kg) e/m_p = υ_p = 95.79 m/s.

In 1874, Irish physicist G. Stoney (he is most famous for introducing the term electron as the “fundamental unit quantity of electricity”) gave a lecture to the British Association, in Belfast, which was subsequently published [3]. In his report, he first proposed the natural units of mass, length, and time, built on the universal constants c, е, and G (where G = 6.674×10^–11 m³∙kg^–1∙s^–2 is the gravitational constant).

Modern meaning of the Stoney mass

m_s = (ke²/G)^1/2 = 1.859×10^–9 kg.

(5)

Hence,

ke² = Gm_s².

(6)

In 2020, it is shown that the constant

G = kе²/m_s² = kυ_g²,

(7)

or

G = υ_g²с²/F₁,

(8)

where the quantity υ_g = (G/k)^1/2 = е/m_s = υ_s = 0.8617×10^–16 m/s is the elementary speed, i.e., the minimum speed of movement in nature, υ_g = υ_min [4].

In addition, it is shown that the Stoney mass, m_s, is boundary of the macrocosm and microcosm; in other words, this mass is the lower limit for the masses of ordinary bodies and the upper limit for the masses of elementary particles.

Using these results, we propose a method for solving the divergence problem.

2. Method

Each fundamental interaction has its own coupling constant.

Electromagnetic interactions of the charged elementary particles are characterized by one universal coupling constant α ≈ 1/137, which is determined by the formula

α = ke²/ћc,

(9)

where ћ = 1.054×10^–34 J∙s is the reduced Planck constant.

Using the equality ћ = ƛmc (where ƛ is the reduced Compton wavelength of a particle), we obtain:

ke²= αћc = αƛmc² = R₀mc²,

(10)

where R₀ = αƛ = ke²/mc².

At the beginning of the XX century, the quantity R₀ = ke²/m_еc² = 2.81×10^–15 м was considered the radius of the electron, but then it turned out that this is not so.

For example, for a proton the value R₀ = ke²/m_pc² = 1.53×10^–18 m, and its radius (as experiments have shown) is about 10^–15 m [5]. Thus, modern physics cannot explain the physical essence of the quantity R₀ and calls it the classical radius of a charged elementary particle.

However, this quantity is easy to understand.

According to Coulomb’s law, the absolute value of the force of electrostatic interaction of two point elementary charges e at a distance r, F = ke²/r², and the potential energy of interaction (its modulus) U = F∙r = ke²/r.

The absolute values of the force and energy of interaction between charges increases with decreasing distance between them, and in nature there are no electric charges without their carriers-particles. In addition, the speed of light in vacuum is the maximum speed of movement in nature, c = υ_max. Therefore, the expression

U_max = ke²/R₀ = mc²

(11)

determines the maximum potential energy of electrostatic interaction between two charged elementary particles of the mass m at the minimum distance R₀ between them (R₀ = r_min).

The interaction between charged particles is carried out by means of their electromagnetic fields. Consequently, the quantity R₀ is the minimum (internal) radius of the electromagnetic field of a charged elementary particle.

The equation R₀ = ke²/mc² can be written like this:

R₀ = km(e/mc)² = kmn,

(12)

where a dimensionless factor

n = (e/mc)² = υ²/c².

(13)

The factor n decreases with increasing particle mass.

For example, for an electron n_e = (e/m_ec)² = υ_е²/c² = 3.44×10^–7, for a proton n_p = (e/m_pc)² = υ_p²/c² = 1.02×10^–13, for a charged elementary particle with mass m_s the factor n has the smallest value, n_s = (e/m_sc)² = υ_g²/c² = n_g = 0.826∙10^–49.

Using the factor n, we write the equation ke² = R₀mc² as:

ke² = (R₀/n)mc²n = R_f mυ²,

(14)

where R_f = R₀/n. Therefore, the expression

U_min = ke²/R_f = mυ²

(15)

determines the minimum potential energy of electrostatic interaction between two charged elementary particles of the mass m at the maximum distance R_f between them (R_f = r_max).

Consequently, the quantity R_f is the maximum radius of action of the electromagnetic forces of a charged particle, i.e., is the outer radius of its electromagnetic field.

According to equations (4) and (14):

ke² = k(mυ)² = (R_f /m)(mυ)²,

(16)

where R_f /m = k. This leads to an important conclusion.

3. New Formula for the Electrostatic Potential

In the MKS system, the electric charge has the dimension of momentum, [q] = kg∙m/s, and the coefficient k has the dimension [k] = m/kg; thus, the electrostatic potential has the dimension of speed, [ϕ] = [kq/r] = [U/q] = m/s.

On the contrary, the gravitational potential has the dimension of the square of speed, [Ф] = [U/m] = (m/s)².

Let us note that the gravitational potential

Ф = Gm/r = kυ_g²m/r = kg²/mr,

(30)

where g = mυ_g is the body’s gravitational charge (in the MKS system) [4].

Therefore, we can introduce a new formula for the electrostatic potential (its modulus):

φ = U/m = ke²/mr

(31)

or, in general,

φ = kq²/mr,

(32)

where this potential has the dimension of the square of speed, [φ] = (m/s)².

4. Conclusions

Currently, it is believed that the force field surrounding a body (particle) has no boundaries. This long-standing misconception led to the problem of infinite values in particle theory.

In reality, the force field has the external boundary (radius) R_f = km and the internal boundary (radius): R₀ = ke²/mc² for the charged elementary particle’s field, and R_g = Gm/c² = kg²/mc² for the ordinary body’s gravitational field.

Consequently, the force and energy of interaction between bodies (particles) can vary from the minimum to maximum values, depending on the distance. Thus, the electromagnetic and gravitational forces have the finite radius of action.

In addition, (in the MKS system) there are two fundamentally different electrostatic potentials:

a) ϕ = kq/r (with the dimension of speed);

b) φ = kq²/mr (with the dimension of the square of speed).

The potential of the first type, apparently, can be considered as a radiation potential and used to solve problems related to the process of energy emission by charged particles.

Potential of the second type is the motion potential that should be used to solve issues of mechanics, i.e., to study the movement of bodies (particles) and the interaction between them.