This work was initiated due to the discovery that the Newton-Coulomb atom predicts the relationship between electrical resistance and temperature.

The purpose of this study is to answer the following questions:

*1) Can the Newton-Coulomb atomic model be used to predict the electrical resistivity of elemental matter using mathematics*?

*2) Can the Newton-Coulomb atomic model be used to predict the properties of an electrical conductor using mathematics*?

Note: The mathematical symbols are explained at the bottom of this page.

The answer to both the above questions is yes!

Not only is it now possible to describe the mechanical (or magnetic) characteristics of elemental matter using the mathematics of the Newton-Coulomb atom, but it is also possible to describe its electrical properties mathematically using the same model ...

... providing yet more evidence that the Newton-Coulomb version is the correct atomic model.

Due to the coupling ratio, the electrical (not the magnetic) charges unite all proton-electron pairs.

The electrical particle charges - positive (protons) and negative (electrons) - repel and attract to maintain balance within atoms. Contrary to popular belief, proton-neutron partners do not sit together in atomic nuclei, but are forced apart and oriented (within the atom's innermost shell) in a structural pattern that ensures all proton charges are neutralised (protected) by their neutrons (Fig 1). This pattern is called a lattice structure, and is replicated in atomic collections as elemental matter in both gaseous and viscous conditions.

The magnetic field generated by each proton-electron pair holds adjacent atoms together as viscous matter, and the electrical charges held by the proton partners push them apart. The magnetic field is therefore responsible for the density of elemental matter and the respective inter-atomic forces (Fₑ & Fₘ) define their viscous-gas condition; transition occurs when the repulsive electrical charge and attractive magnetic field forces are equal; Ṯg: Fₑ = Fₘ.

The density of an atom is that of all the proton-electron pairs - and their neutron partners - within its outermost shell.

The measured gas transition temperature - that of the proton-electron pairs in shell-1 - of elemental matter is that above which, the magnetic field forces are no longer able to resist the inter-atomic [proton] electrical repulsion forces.

All atomic proton-electron pairs, the potential energy of which is less than that induced (per electron) by the applied Voltage, will release their electrons for transmission through a conductor (electrical current).

Fig 1. A Typical Atomic Structure

Another term for **Voltage** is *potential difference*, which represents the potential energy (PE) required to pull electrons from their orbits.

Because Voltage is a *relative* value (Joules per Coulomb), it applies equally to any part of the conductor. The applied potential energy is distributed according to the requirements of each electron shell#.

Voltage is a measurement of potential energy per electron; 1 Volt = 1 Joule per Coulomb. One Coulomb represents 1/e electrons.

In other words; the [potential] energy applied by an electrical supply may be defined thus: PEₑ = V.e {J} per electron.

All the atomic electrons (in a conductor) with a potential energy# <PEₑ will be mobilised. Their number may be established thus: Nc = Zꞌ.m/mₐ; in which Zꞌ represents the number of mobilised electrons per atom (see **The Electrical Atom** below).

Conductor temperature rises with increasing Voltage (J/C) due to the consequential increase in PE, making it harder to pull electrons from their orbits (proton-electron pair; Fig 2).

# based upon the measured temperature of the elemental matter in the electrical circuit

**DC Current** (I) is the rate of flow of electrons along a conductor. It is measured in Coulombs per second and may be calculated thus:

**I = V/R = P/V** {C/s}.

Because Current is a *relative* value (Coulombs per second), it applies equally to any part of the conductor. The rate of flow of electrons is distributed according to the requirements of each electron shell#.

Once an electron has been released from its proton partner, it will travel to an adjacent atom at a velocity commensurate with the applied voltage, which may be calculated thus: v = k.I / V.Z = k / R.Z

*(refer to The Electrical Atom and Alternative Calculation Method below for the relevance of Z in the above calculation)*

Current intensity (Iꞌ) is the current per unit distance; in this case, it is the current divided by the orbital radius of the transmitted electron; 'Rₛ' ...

As an atom's temperature rises, so too does the

# based upon the measured temperature of the elemental matter in the electrical circuit

**Electrical power** is the rate at which potential energy is consumed or expended, e.g. Joules per second;

**P = V.I {J/s}**

**Electrical resistance**; R = V/I {Ω = J.s / C²}

**Electrical resistivity**⁽²⁾; ρ = R.A/ℓ {Ω.m = J.s.m / C²}

Electrical resistance is simply a mathematical description of the difficulty in; a) pulling electrons from their atomic orbits, and b) transmitting them between adjacent atoms, which is why it is not only dependent upon the applied voltage and resultant current, but also on the resistivity of the elemental matter.:

**R = V/I**

whilst frequency varies linearly with orbital velocity (ƒ = v/2πR), voltage varies with its square (PE = mₑ.v²). It isn't difficult, therefore, to see that electrical resistance will rise linearly with increasing temperature.

**ρ = R.d**

electron transit distance (d) in viscous matter = orbital radius (Rₛ)

electron transit distance (d) in gaseous matter = distance between atoms (≈³√[mₐ/ρ])

The resistance and resistivity of elemental matter is as defined above for the atom, but also dependent upon the following factors.

relative temperature: Kₜ = Ṯg/Ṯₛ

relative density: Kᵨ = ρₐ/ρₘ

But we also need to consider the neutronic ratio of the [two] adjacent atoms between which the electrons travel:

**R = Rₐ . Kₜ . Kᵨ . 2.ψ** {J.s.m/C²}

**ρ₁ = Rₐ' . Kₜ . Kᵨ . 2.ψ** {J.s.m/C²}

Rₐ' is the atomic resistance (V/I) multiplied by the atomic radius (V/I') see **Current** above

Fig 2. Resistivity @ 300K converted to operational temperature

The electrical properties within an atom are based upon shell temperature (Ṯₛ);

Vₛ = PEₛ/e = Ṯₛ/Ṯₙ . mₑ.c²/e: Vₐ = Vₘ.Zꞌ

Iₛ = e.2πƒₛ = e.vₛ/Rₛ = e.gₛ/vₛ: Iₐ = Iₘ.Zꞌ

Rₐ = Vₐ/Iₐ {J.s/C²}

Pₐ = Vₐ.Iₐ {J/s}

As temperature rises, the potential energy holding electrons to their proton partners increases. But as the temperature of proton-electron shells reduces linearly from shell-1 to shell-Z/2, only the outer electron shells - those with a PE less than that applied by the battery - will be released.

Zꞌ refers to the number of electrons that will be released by the applied Voltage.

subscript 'ₘ' refers to mean shell number; e.g. ₘ for tungsten is 18.5, the mean value between shells 18 & 19

The electricity generated in a proton-electron pair at the time of the creation of a neutron may be calculated as follows:

Neutronic power: Pₙ = PEₙ.ƒₙ = 8.71003636610805E+09 {J/s}

Neutronic Voltage: Vₙ = PEₙ/e = 5.10999336611600E+05 {J/C}

Neutronic current: Iₙ = e.ƒₙ = 1.70451030795141E+04 {C/s}

Neutronic resistance: Rₙ = PEₙ / ƒₙ.e² = 2.99792459E+01 {J.s/C²}

note the relationship between neutronic resistance; 29.9792459 {**kg.m² / s.C²**}, EME velocity: c = 299792459 {**m/s**} and the magnetic constant: *μ* = 1.000E-07 {**kg.m/C²**}:

electrical resistance at the neutronic condition: Rₙ = *μ*.c

Non-neutronic electrical properties per electron shell may therefore be calculated as follows:

if magnetic [field] constant: *μ* = mₑ.Rₙ/e²

then magnetic [field] variable per electron shell: *μₛ* = mₑ.Rₛ/e²

where; Rₛ = electron shell radius

Non-neutronic Voltage: Vₛ = PEₛ/e

Non-neutronic current: Iₛ = e.ƒₛ

Non-neutronic power: Pₛ = Vₛ.Iₛ

Non-neutronic resistance: Rₛ = *μₛ*.vₛ = *μₛ*.gₛ/ƒₛ = Vₛ/Iₛ

where; ƒₛ = 2π/tₛ (tₛ = orbital period)

Resistivity may also be calculated for a perfect crystal as follows:

**ρ₂ = Rₛ.d/Z . (ψ-1)²** {J.s.m/C²}

where:

'Rz' refers to the resistance in the proton-electron pair, the electron of which orbits in the atom's outermost shell

'd' refers to the nominal distance between two adjacent atoms in a perfect crysal of elemental matter; d = ³√[mₐ/ρ]

'ρ' represents the density of a perfect crystal of elemental matter

'mₐ' represents the mass of an atom.

'Z' represents the number of electrons released from the atom.

As can be seen in Table 2, and in Fig 3, there are some differences between the calculated resistivity values and those extracted from various reference sources. It would be helpful to see which are correct.

We can do this by comparing the performance of the calculated and documented resistivity values in a well-known application, such as the tungsten filament in an incandescent light-bulb.

We can make this comparison using two versions of the bulb; say a 60-Watt bulb used in the USA (120V) and the same power rating in the UK (220V).

Input Data | ||||
---|---|---|---|---|

P | check | 60 | Watts {J/s} | |

Ø | check | 4.6E-05 | m | |

ℓ | input | 0.595 | 2.0 | m |

Calculated Resistivity | ||||

Property | Formula | US | UK | Units |

ρ (293K) | input | (2.0E-07) 6.28E-07 | Ω.m | |

V | input | 120 | 220 | Volts |

Ṯ₁ | Nₛ.Ṯₙ . ²˙⁵√[π.V².R².tₙ / PEₙ.ρ.ℓ] | 2883.918179 | 2883.842614 | K |

d | ³√[mₐ/ρₘ] | 2.51592E-10 | m | |

Ø | √[P.ρ.ℓ.4/π]/V | 4.45186E-05 | 4.452014E-05 | m |

A | π.Ø²/4 | 1.55658E-09 | 1.55669E-09 | m² |

Nₐ | (Ø²/4)/d² | 7.82761E+09 | 7.82815E+09 | |

PEₛ | mₑ.Ṯ₁ / X.Nₛ | 1.02377E-20 | J | |

Pₛ | PEₛ.ƒₛ | 7.66506E-09 | J | |

Pƒ | Pₛ.Nₐ | 60.00023644 | 60.00440341 | J/s (Watts) |

Documented Resistivity | ||||

Property | Formula | US | UK | Units |

ρ (293K) | input | (5.65E-08) 5.09E-07 (Fig 2) | Ω.m | |

V | input | 120 | 220 | Volts |

Ṯ₁ | Nₛ.Ṯₙ . ²˙⁵√[π.V².R².tₙ / PEₙ.ρ.ℓ] | 3136.480618 | 3136.393492 | K |

d | ³√[mₐ/ρₘ] | 2.51592E-10 | m | |

Ø | √[P.ρ.ℓ.4/π]/V | 4.00860E-05 | 4.00850E-05 | m |

A | π.Ø²/4 | 1.26190E-09 | 1.26198E-09 | m² |

Nₐ | (Ø²/4)/d² | 6.34570E+09 | 6.34615E+09 | |

PEₛ | mₑ.Ṯ / X.Nₛ | 1.11343E-20 | J | |

Pₛ | PEₛ.ƒₛ | 9.45385E-09 | J | |

Pƒ | Pₚ.Nₐ | 60.0000000 | 60.00416696 | J/s (Watts) |

Table 1: Filament Performance Comparison: Calculated vs Documentedverifies the calculation method identifies the correct resistivity |

As can be seen in Table 1, the calculated resistivity value (6.28E-07 Ω.m) provides a more representative result for filament temperature and diameter (2884K & 4.6E-05m), than the documented resistivity value (5.09E-07 Ω.m) for filament temperature and diameter (3136K & 4.0E-05m)⁽⁴⁾.

This conclusion is reinforced by the fact that an atomic resistance calculation for the tungsten atom of 809.58 J.s/C² at operating temperature ('2883.918179 K') using the formulas in **The Electrical Atom** above is so close to that calculated for the filament using the calculated version of resistivity (6.28E-07 J.s.m/C² Fig 2 & Table 3); R = 806.67 J.s/C².

As can be seen in Table 2, the documented values for copper and silver appear to indicate a reversal of generally accepted convention, whilst the calculated values conform. A similar situation occurs with iron, which is generally considered to be a good conductor, and tungsten; the conductivity of which is generally considered to be poor.

Elemental Matter | ρᵈ | ρ₁ | ρ₂ | Units |
---|---|---|---|---|

copper | 1.70E-08 | 8.6449E-08 | 4.818227E-08 | Ω.m |

silver | 1.63E-08 | 9.2675E-08 | 1.01215E-07 | Ω.m |

gold | 2.44E-08 | 1.0557E-07 | 2.16681E-07 | Ω.m |

iron | 9.80E-08 | 1.0108E-07 | 2.99265E-08 | Ω.m |

tungsten | 5.65E-08 | 2.0251E-07 | 2.09747E-07 | Ω.m |

Table 2: resistivity values of representative metals @ 300K |

Fig 3. Resistivity: calculated values vs documented values @ 300K (see Table 2)

Documented values for the resistivity of elemental matter must therefore be treated with caution.

It stands to reason that the mobilisation of electrons along a conductor (electricity) requires them to be pulled from their atoms. In fact, the alternative term for voltage; 'potential difference' is that required to overcome the potential energy holding the electrons in their atomic orbits. moreover, its units are Joules per Coulomb; or energy per electron.

It also stands to reason that the current in an electrical circuit is the rate of flow of electrons; Coulombs per second.

Given that potential energy in circular orbits is calculated thus; PE = m.v²

and that electron mobilisation simply requires an energy slightly in excess of the orbital potential energy,

the flow rate of mobilised electrons will be the same as the orbital velocity immediately prior to release; 2πƒ.

At the outermost shell temperature (Ṯₛ);

voltage; V = PEₛ/e = mₑ.Ṯₛ / X.e

note: vₛ² = Ṯₛ/X

and

current; I = e.2πƒₛ

note: v = 2πRₛ.ƒₛ & Rₛ = XR/Ṯₛ

Given that the calculated values are close to - and in some cases the same as - the documented values, which are themselves at best variable (see 'Fig 1. Electrical Resistivity Variability Range'), together with the obvious logic behind the calculations, and given that all the units are correct, the above mathematical description may be considered legitimate. Moreover, the calculated resistivity values for the best known metals are more representative of convention than the documented values (Table 2).

It must be pointed out, however, that the calculated values for resistivity apply to perfect crystals of the elemental matter concerned, whereas the documented values generally apply to formed matter; imperfect crystals.

However, with regard to the resistivity of a tungsten filament, ...

from: R = ρ.ℓ/A = V/I {J.s/C²}

P = I.V = V².A / **ρ**.ℓ {J/s}

conductor length may be calculated thus: ℓ = V².A / **ρ**.P {m}

In a 120-volt, 60-Watt tungsten light bulb, the filament (Ø = 4.6E-05m; A = 1.662E-07m²) length is either;

documented: ℓ = 120²x1.662E-07 ÷ 60x**5.09E-07** = 783.61mm

calculated: ℓ = 120²x1.662E-07 ÷ 60x**6.28E-07** = 635.12mm

Wikipedia⁽⁴⁾ defines the filament length as 580mm

... it would appear therefore, that the calculated value is more representative than the documented value.

Here is a formula you most probably will not have seen before:

v = k.I / V.Z {J.m/C² . C/s . C/J = m/s}

where 'v' is the mean velocity of all the orbiting electrons in an atom to which a Voltage (V) has been applied.

The general formulas for orbital velocity (v) and radius (R) of an electron at any temperature (Ṯ) are:

v = √[Ṯ/X] {m/s}

R = Xᴿ/Ṯ {m}

The mean orbital shell number in a tungsten atom is:

Nᵒ = Z/4 = 18.5

the mean orbital radius in each atom at a 'measured' temperature (Ṯ₁) is therefore:

Rₘ = Nᵒ . Xᴿ/Ṯ₁ {m}

the temperature of the proton-electron pairs in the mean shell (@ Rₘ):

Ṯₘ = Xᴿ/Rₘ {K}

the orbiting velocity of the electrons in the mean shell (@ Rₘ):

vₘ = √[Ṯₘ/X] {m/s}

Therefore, the 'measured' temperature of an electrical conductor may be determined as follows:

v = k.I / V.Z

Ṯ₁ = vₘ² . Z/4 . X

Symbol | Formula | Value | Units | |
---|---|---|---|---|

V | input | 220 | J/C | |

Ø | input | 4.45E-05 | m | |

ℓ | input | 2 | m | |

A | π.Ø²/4 | 1.55669E-09 | m² | |

ρ | input | 6.27866E-07 | J.s.m/C² | |

R | ρ.ℓ/A | 806.6666667 | J.s/C² | |

I | V/R | 0.272727273 | C/s | |

P | I.R | 60 | J/s | |

Verification: | ||||

Nₘ | Z/4 | 18.5 | ||

v | k.I / V.Z | 1.5056207E+05 | m/s | |

vₘ | (v₁₈+v₁₉)/2 | 1.4996554E+05 | m/s | |

v₁ | √[Ṯ₁/X] | 6.4484945E+05 | m/s | |

Rₘ | Nₘ.XR/Ṯ₁ | 1.1267509E-08 | m | |

Ṯₘ | XR/Rₘ | 155.8877149 | K | |

vₘ | √[Ṯₘ/X] | 1.4992445E+05 | m/s | |

Ṯ₁ # | vₘ².Nₘ.X | 2908.505150 | K | |

Table 3: verification calculationssubscript 'ₘ' refers to the atomic mean value 'v₁₈' & 'v₁₉' are electron velocities in shells 18 and 19 respectively # refer to 'Ṯ₁' (Calculated Resistivity and Documented Resistivity) in Table 1 for comparison. |

The minor variation in temperatures 2883.918179 K (Table 1) and 2908.505150 K (Table 3);

and orbital velocities 1.5056207E+05 m/s and 1.4996554E+05 (Table 3);

are due to the gas-transition temperature of tungsten used in the calculations, which affects resistivity; the book-based value used for the calculation is 6200K, but it should be 6233K.

However, the above calculation yet again verifies the Newton-Coulomb atomic model.

A = cross-sectional area of the electrical conductor {m²}

ƒ = electron orbital frequency {/s}

g = potential acceleration (proton-electron pair) {m/s²}

ℓ = filament length {m}

m = mass {kg}

N = number

PE = potential energy (proton-electron pair) {J}

R = electron orbital radius {m}

R = electrical resistance {J.s/C²}

Ṯ = temperature {K}

v = electron orbital velocity {m/s}

ρ = resistivity {J.s.m/C²}

ψ = RAM/Z (neutronic ratio)

Ø = conductor diameter {m}

Subscripts:

ₐ atomic

ₑ electrical

ₘ magnetic

z mean shell number

ₛ shell number

₁ shell number 1 (innermost shell)

- a) phosphorus; b) sulphur; c) erbium; d) boron;
- ρ = R.A/ℓ = R.A/Rₛ = R.Rₛ²/Rₛ = R.Rₛ {Ω.m}
- h/g = vₛ / (2πƒ)² = Rₛ / (2πƒ) {m.s}
- Wikipedia suggests a filament of diameter 4.6E-05m, length of 0.595m and a temperature of 2000K and 3300K;

"how stuff works" suggests a filament of diameter 'one-hundredth of an inch', length of about 2m and a temperature of around 2477K;

the Dutton Institute suggests a filament temperature of around 2823K;

There are numerous other sources that suggest a diameter of 'one-two-thousandths of inch' and temperatures between 2000K and 3000K

You will find further reading on this subject in reference publications^{(69, 70, 71 & 73)}