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 question:

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

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

Not only is it now possible to describe the (mechanical (or magnetic) behaviour (and properties)) 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.

It is important to understand the limitations of documented values for resistivity.

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. 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.

The temperature of the proton-electron pairs transferring their electrons in an electrical circuit is that of the outermost proton-electron pair(s).

Fig 1. A Typical Atomic Structure

**Voltage** (V) is the potential energy (per Coulomb) required to pull electrons from an atom's outermost electron shell. The reason *these* electrons are active in an electrical circuit is because they have the lowest kinetic energy (KE) and therefore the lowest potential energy (PE); they are the easiest to transmit.

This potential energy is that required to hold the electron in orbit about its proton partner, and it may be calculated thus#:

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

PE = mₑ.vₛ² {J}

vₛ = √[Ṯₛ/X] {m/s}

Ṯₛ = Xᴿ/Rₛ {K}

Rₛ = R₁.Nₛ {m}

R₁ = Xᴿ/Ṯ {m}

PEₛ = mₑ.Ṯ / X.Nₛ {J}

**V = PEₛ/e** {J/C}

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

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

Voltage magnitude (PE) confirms that increasing the temperature (Ṯ) of the elemental matter will increase both its electrical resistance and its resistivity; because rising temperature increases PE making it harder to pull an electron from its orbit (proton-electron pair; Fig 2).

In other words, as the measured temperature of elemental matter (in an electrical circuit) increases, you need to raise the voltage to generate an operational current. But because the orbital velocity of the outermost electrons increases with rising temperature, you will also need to increase the applied current.

Or alternatively; electrical resistance increases with increasing temperature due to the increased potential energy required to pull electrons from their orbits (J/C) at a greater velocity (C/s).

vₛ = 2π.Rₛ.ƒ {m/s}

h/g = vₛ/(2πƒ)² = Rₛ/(2πƒ) {m.s}

Ω = V/I {J.s / C²}

ρ = V/I . Rₛ {Ω.m = J.s.m / C²}

ρ = PEₛ/e² . vₛ/(2πƒ)² = PEₛ/e² . 2π.Rₛ.ƒ/(2πƒ)²

The resistivity of a conductor may be calculated thus:

**ρ = PEₛ/e² . Rₛ/2πƒ**

**DC Current** (I) is the flow rate of electrons in an electrical circuit; Coulombs per second:

From resistance above: I = V/Ω = V . Rₛ/ρ {C/s}

I = PEₛ/e . Rₛ / (PEₛ/e² . Rₛ / 2πƒ)

**I = e.2πƒ** {C/s}

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' ...

Iꞌ = e . 2πƒ/Rₛ {C/s / m}

... providing an additional method for calculating resistivity as follows:

ρ = V/I' {J.s.m / C² = Ω.m}

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

P = V.I {J/s}

The above is true of theoretical [viscous] matter comprising collections of proton-electron pairs. However, real-life atoms comprise collections of deuterium and tritium atoms. The surplus tritium neutrons affect the internal forces, and therefore, the structure of elemental matter; such as density and gas-point temperature.

Therefore, we must consider these when predicting the resistance and resistivity of elemental matter, for example:

Kₜ = Ṯg/Ṯₛ

Kᵨ = ρₛ/ρₘ

But we also need to consider the effect of excess tritium neutrons

ψ' = ψ-1

Electrical resistance; Ω = V/I . Kₜ . Kᵨ . ψ'

Electrical resistivity of elemental matter: ρ = V/I' . Kₜ . Kᵨ . ψ'

Fig 2. Documented resistivity converted to operational temperature

As can be seen in the above Table, and in Fig 3, there are some differences between the calculated resistivity values and those extracted from various 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).

Units

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 | |

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] | 2883.922725 | 2883.842614 | K |

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

Pₚ | PEₛ.ƒₛ | 7.66506E-09 | J | |

Rₐ | ³√[mₐ/ρₘ] | 2.51592E-10 | m | |

Aₐ | π.Rₐ² | 1.98858E-19 | m² | |

Ø | √[P.ρ.ℓ.4/π]/V | 4.45185E-05 | 4.45200E-05 | m |

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

Nₐ | A/Aₐ | 7.82758E+09 | 7.82812E+09 | m² |

Pₐ | Pₚ.Nₐ | 59.99881803 | 60.00298491 | 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] | 3136.332741 | 3136.245619 | K |

PEₛ | mₑ.Ṯ / X.Nₛ | 1.11327E-20 | J | |

Pₚ | PEₛ.ƒₛ | 9.45385E-09 | J | |

Rₐ | ³√[mₐ/ρₘ] | 2.51592E-10 | m | |

Aₐ | π.Rₐ² | 1.98858E-19 | m² | |

Ø | √[P.ρ.ℓ.4/π]/V | 4.00860E-05 | 4.00874E-05 | m |

A | π.Ø²/4 | 1.26204E-09 | 1.26213E-09 | m² |

Nₐ | A/Aₐ | 6.34645E+09 | 6.34689E+09 | m² |

Pₐ | Pₚ.Nₐ | 59.99843412 | 60.00260097 | J/s (Watts) |

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

The majority of sources appear to indicate an operational temperature of 2884K rather than 3136K⁽⁴⁾, and the calculated resistivity gives a closer match for the actual filament diameter.

From the following Table, it would appear, that in its pure-crystalline form (calculated), copper may be a better electrical conductor than silver.

Elemental Matter | Calculated Value | Documented Value | Units | |
---|---|---|---|---|

copper | 8.71743E-08 | 1.70E-08 | Ω.m | |

silver | 9.37502E-08 | 1.63E-08 | Ω.m | |

gold | 1.02838E-07 | 2.44E-08 | Ω.m | |

Relative Resistivity Values of Copper, Silver & Gold |

But in its practical (drawn) condition, copper wire may be slighly less conductive than silver in the same condition. Albeit some of the documented values for resistivity declare these materials to be equally conductive (1.63E-08 Ω.m).

An additional consideration must be the orientation of the atomic crystals. The calculated resistivity reflects a theoretical optimum value, whereas the documented values refer to both deformed crystals (drawn) and/or variable orientation.

Considering that;

a) it makes perfect sense that voltage (J/C) is the energy required to overcome the potential energy (PEₛ) holding the electron to its proton partner in the atom's outermost proton-electron pairs;

b) it makes perfect sense that current, or electron transit rate, (C/s) is related to the velocity of the electron(s) in the atom's outermost orbiting electrons (ƒₛ);

c) the calculations are very simple and the units are correct;

d) the variable nature of the documented values for resistivity calls some of them into question;

e) respective filament diameters in the above Example Calculation favour the calculated resistivity value;

f) the average discrepancy of 30% between the calculated and documented values for the resistivity of all 92 elements, is less than the discrepancy between sources for many documented values;

g) ten of the calculated values are exactly the same as the documented values;

h) the above example calculation verifies itself; calculated power rating is identical to the design power rating;

i) no fudged formula could generate the pattern correlation shown in Fig 3, together with a) to h) above;

CalQlata concludes that the above is a compelling description of the electrical atom, and how it applies to elemental matter.

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 apply to imperfect matter.

Fig 3. Resistivity: calculated values vs documented values (see above Table)

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

A = filament cross-sectional area {m²}

Aₐ = atom cross-sectional area {m²}

ƒₛ = the orbital frequency of the electron in shell 'Nₛ' {/s}

g = potential acceleration induced on the orbiting electron (Nₛ) by its proton partner {m/s²}

h = Newton's constant of motion for the electron orbiting in shell 'Nₛ' {m²/s}

I = current {C/s}

ℓ = filament length {m}

mₐ = atomic mass {kg}

Nₐ = number of atoms across the filament's cross-sectional area

N₁ = an atom's innermost shell [number]

Nₛ = an atom's outermost shell [number; s = INT((Z-1)/2)+1]

P = power {J/s}

Pₚ = proton-electron power; shell 'Nₛ' {J/s}

Pₐ = filament power {J/s}

PEₛ = the potential energy uniting the proton-electron pair in shell 'Nₛ' {J}

R₁ = orbital radius of an electron in shell 'N₁' {m}

Rₛ = orbital radius of an electron in shell 'Nₛ' {m}

Rₐ = inter-atomic spacing (in elemental matter) {m}

Ṯ & Ṯ₁ = measured temperature of the matter; that of the proton-electron pair whose electron is orbiting shell 'N₁' {K}

Ṯₛ = the temperature of a proton-electron pair whose electron is orbiting in shell 'Nₛ' {K}

Ṯg = the gas-point temperature of elemental matter {K}

vₛ = orbital velocity of an electron in shell 'Nₛ' {m/s}

V = voltage {J/C}

ρ = resistivity {J.s.m / C²}

ρₛ = atomic density (mₐ/Rₛ³) {kg/m³}

ρₘ = mass density of elemental matter {kg/m³}

ψ = RAM/Z (neutronic ratio)

Ω = resistance {J.s / C²}

Ø = filament diameter {m}

- a) phosphorus; b) sulphur; c) erbium; d) boron;
- ρ = Ω.A/ℓ = Ω.A/Rₛ = Ω.Rₛ²/Rₛ = Ω.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)}