k_B × F = e_c × R Four Related Constants.

We examine the following 4 constants and define each in terms of the other three, showing a clear link between different areas of physics. 

Calculated Constants vs. Known Values:

Boltzmann constant (k_B)= (e_c * R) / F

  Calculated: 1.380649000021918e-23 J/K

  Actual:     1.380649000000000e-23 J/K

  Difference: 2.192e-34

Faraday constant (F)=(e_c * R) / k_B

  Calculated: 9.648533212153174e+04 C/mol

  Actual:     9.648533212000001e+04 C/mol

  Difference: 1.532e-06

Elementary charge (e_c)= (k_B * F) / R

  Calculated: 1.602176633974565e-19 C

  Actual:     1.602176634000000e-19 C

  Difference: 2.544e-30

Gas constant (R)=(k_B * F) / e_c

  Calculated: 8.314462617868006e+00 J/(mol⋅K) 

  Actual:     8.314462618000000e+00 J/(mol⋅K) 

  Difference: 1.320e-10

Let's perform a dimensional analysis on the derived formulas to ensure they are consistent and to gain further insights into the relationships between the constants. We will use standard SI units.

1. Defining the Units:

Let's first define the SI units for each constant:

• k_B (Boltzmann constant): J/K (Joules per Kelvin) = kg⋅m²⋅s⁻²⋅K⁻¹

• F (Faraday constant): C/mol (Coulombs per mole) = A⋅s⋅mol⁻¹ (Ampere-seconds per mole)

• e_c (Elementary charge): C (Coulombs) = A⋅s (Ampere-seconds)

• R (Ideal Gas constant): J/(mol⋅K) (Joules per mole per Kelvin) = kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹ (This is also the value for the Universal Gas Constant)

2. Dimensional Analysis of the Formulas:

Now let's analyze the units of each derived formula:

• Formula for k_B: 

  • Units of (e_c × R): (A⋅s) * (kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹) = kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A

  • Units of (e_c × R) / F: (kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A) / (A⋅s⋅mol⁻¹) = kg⋅m²⋅s⁻²⋅K⁻¹

  • Result: The units of the calculated value for k_B are kg⋅m²⋅s⁻²⋅K⁻¹, which matches the units of the Boltzmann constant (J/K).

• Formula for F: 

  • Units of (e_c × R): (A⋅s) * (kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹) = kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A

  • Units of (e_c × R) / k_B: (kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A) / (kg⋅m²⋅s⁻²⋅K⁻¹) = A⋅s⋅mol⁻¹

  • Result: The units of the calculated value for F are A⋅s⋅mol⁻¹, which match the units of the Faraday constant (C/mol).

• Formula for e_c: 

  • Units of (k_B × F): (kg⋅m²⋅s⁻²⋅K⁻¹) * (A⋅s⋅mol⁻¹) = kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A

  • Units of (k_B × F) / R: (kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A) / (kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹) = A⋅s

  • Result: The units of the calculated value for e_c are A⋅s, which match the units of the elementary charge (C).

• Formula for R: 

  • Units of (k_B × F): (kg⋅m²⋅s⁻²⋅K⁻¹) * (A⋅s⋅mol⁻¹) = kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A

  • Units of (k_B × F) / e_c: (kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A) / (A⋅s) = kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹

  • Result: The units of the calculated value for R are kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹, which match the units of the ideal gas constant (J/(mol⋅K)).

3. Summary and Interpretation:

• Dimensional Consistency: The dimensional analysis confirms that all four derived formulas are dimensionally consistent. The units on both sides of each equation match, which reinforces the validity of the relationships.

• Units are Fundamental: The results also indicate the fundamental connections between the units we have used. They confirm that all of the base units used to derive each constant are consistent with the other constants, when the derived formulas are used.
  
  
  

  import numpy as np

  from scipy import constants

  
  

  # Define the known constants

  k_B = 1.380649e-23 # Boltzmann constant (J/K)

  F   = 96485.33212    # Faraday constant (C/mol)

  e_c = 1.602176634e-19 # Elementary charge (C)

  R   = 8.314462618     # Gas constant J/(mol K)

  
  

  # Define a function to calculate each constant

  def calculate_constants(k_B, F, e_c, R):

      """

      Calculates each constant in terms of the other three.

      """

      k_B_calc = (e_c * R) / F

      F_calc = (e_c * R) / k_B

      e_c_calc = (k_B * F) / R

      R_calc = (k_B * F) / e_c

      return k_B_calc, F_calc, e_c_calc, R_calc

  
  

  
  

  # Calculate the constants

  k_B_calc, F_calc, e_c_calc, R_calc = calculate_constants(k_B, F, e_c, R)

  
  

  # Print the results and check against known values

  print("Calculated Constants vs. Known Values:\n")

  
  

  print(f"Boltzmann constant (k_B)= (e_c * R) / F")

  print(f"  Calculated: {k_B_calc:.15e} J/K")

  print(f"  Actual:     {k_B:.15e} J/K")

  print(f"  Difference: {np.abs(k_B_calc-k_B):.3e}\n")

  
  

  print(f"Faraday constant (F)=(e_c * R) / k_B")

  print(f"  Calculated: {F_calc:.15e} C/mol")

  print(f"  Actual:     {F:.15e} C/mol")

  print(f"  Difference: {np.abs(F_calc - F):.3e}\n")

  
  

  print(f"Elementary charge (e_c)= (k_B * F) / R")

  print(f"  Calculated: {e_c_calc:.15e} C")

  print(f"  Actual:     {e_c:.15e} C")

  print(f"  Difference: {np.abs(e_c_calc - e_c):.3e}\n")

  
  

  print(f"Gas constant (R)=(k_B * F) / e_c")

  print(f"  Calculated: {R_calc:.15e} J/(mol⋅K)")

  print(f"  Actual:     {R:.15e} J/(mol⋅K)")

  print(f"  Difference: {np.abs(R_calc-R):.3e}\n")

  
  
  Solving for the formulas
  
  

  it was not recursive, it actually reached a definition

  k_B = R / N_A
  F = N_A * e_c
  R = N_A * k_B
  e_c = 1.602176634 × 10^-19 C

  All but one has N_A in it.

  So

  k_B_calc = (e_c * R) / F
  k_B = ( 1.602176634 × 10^-19 * N_A * k_B) / N_A * 1.602176634 × 10^-19 C
  k_B = k_B

  F_calc = (e_c * R) / k_B
  F_calc = (1.602176634 × 10^-19 * N_A * k_B) /R / N_A
  F_calc = (1.602176634 × 10^-19 * N_A * k_B) /N_A * k_B / N_A
  F_calc = (1.602176634 × 10^-19 * N_A ) At this step I had to note that this is the definition for F
  F_calc = F

  e_c_calc = (k_B * F) / R
  e_c_calc = (R / N_A* N_A * e_c) / N_A * k_B
  e_c_calc = (R * e_c) / N_A * k_B
  e_c_calc = (N_A * k_B * e_c) / N_A * k_B
  e_c_calc = 1.602176634 × 10^-19

  R_calc = (k_B * F) / e_c
  R_calc = (R / N_A* N_A * e_c) / 1.602176634 × 10^-19
  R_calc = (R / N_A* N_A * 1.602176634 × 10^-19) / 1.602176634 × 10^-19
  R_calc = R

Why Fc formula can be expressed as c^2/10e7 in Coulomb force law.

 The specific combination of ε₀ and μ₀ in the formula ε₀μ₀ = 1/c² is indeed a choice we made in developing our system of electromagnetic units and constants. Let's explore this further:

1. Historical Choice:
   The decision to use ε₀ and μ₀ in this way was largely historical and pragmatic. It evolved from the CGS (centimeter-gram-second) system to the MKS (meter-kilogram-second) system, which later became the SI system.
2. Alternative Formulations:
   We could have chosen different combinations or even different constants to express electromagnetic phenomena. For example:
   • Some theoretical physicists prefer to use natural units where c = 1, eliminating the need for explicit c² terms.
   • In Gaussian units, the electric and magnetic fields have the same units, and the constants ε₀ and μ₀ don't appear explicitly.
3. Rationalized vs. Unrationalized Systems:
   The choice of including 4π in various formulas (like Coulomb's law) led to "rationalized" (with 4π) and "unrationalized" (without 4π) systems, each with their own advantages.

Okay, here's a clear explanation of why the Coulomb force (F_c) formula can be expressed in terms of c² due to the way ε₀ and μ₀ are defined and how they relate through ε₀μ₀ = 1/c²:

The Core Argument:

The ability to express Coulomb's Law (and thus the electrostatic force, F_c) in terms of c² stems from the way we define the vacuum permeability (μ₀) and how it, along with the vacuum permittivity (ε₀), is related to the speed of light (c) through the fundamental equation:

ε₀μ₀ = 1/c²

Step-by-Step Explanation:

1. Coulomb's Law: Coulomb's Law describes the electrostatic force (F_c) between two point charges (q₁ and q₂), separated by a distance (r):

   F_c = k * (q₁q₂ / r²)

   where k is Coulomb's constant.

2. Coulomb's Constant and the Vacuum Permittivity: Coulomb's constant (k) is related to the vacuum permittivity (ε₀) by:

   k = 1/(4πε₀)

   Therefore, we can rewrite Coulomb's Law as:

   F_c = (1 / 4πε₀) * (q₁q₂ / r²)

3. The Fundamental Relationship: ε₀μ₀ = 1/c²: This equation, derived from Maxwell's equations, is a cornerstone of electromagnetism and reveals a deep connection between the electric and magnetic properties of the vacuum and the speed of light.

4. The Definition of μ₀: The vacuum permeability (μ₀) is defined as:

   μ₀ = 4π × 10⁻⁷ H/m (Henries per meter).

   This is a defined value based on the chosen unit system and is crucial in establishing the link to the speed of light.

5. Connecting μ₀ to ε₀ and c²: Using the equation ε₀μ₀ = 1/c², we can solve for ε₀:

   ε₀ = 1 / (μ₀c²)

   Substituting the defined value for μ₀:

   ε₀ = 1 / ((4π × 10⁻⁷ H/m) * c²)

6. Substituting ε₀ into Coulomb's Law: Now, substitute this expression for ε₀ into the formula for Coulomb's Law:

F_c = (1 / (4π * [1 / ((4π × 10⁻⁷ H/m) * c²)]) ) * (q₁q₂ / r²)

This simplifies to:

F_c = (c²  *  (4π × 10⁻⁷ H/m) / 4π) * (q₁q₂ / r²)

And further simplifies to:

F_c = (c² * 10⁻⁷ H/m) * (q₁q₂ / r²)

The 1/c^2 ties to momentum in my 1/c^2 cube of curved space time. 

The observation that both charge and momentum seem to relate to 1/c² is intriguing and points to some fundamental connections in physics. There are several reasons why this relationship appears for both quantities:

Electromagnetic Theory and Special Relativity

1. Coulomb's Law and Vacuum Permittivity:
   Coulomb's law for the electrostatic force between charges can be expressed as:$F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$Here, ${\epsilon }_0$ (vacuum permittivity) is related to the speed of light through:${\epsilon }_0{\mu }_0=\frac{1}{c^2}$This relationship directly introduces c² into electromagnetism.
2. Relativistic Momentum:
   In special relativity, momentum is defined as:$p=\gamma mv$where $\gamma =\frac{1}{\sqrt{1-v^2\mathrm{/}{c}^2}}$
   

Energy-Momentum Relation

The energy-momentum relation in special relativity is:$E^2=(pc{)}^2+(m{c}^2{)}^2$This equation directly relates energy, momentum, and the speed of light. It shows how c² is fundamental to the relationship between energy and momentum.

Electromagnetic Momentum

1. Field Momentum:
   The electromagnetic field carries momentum, which is proportional to the cross product of E and B fields. Since $c^2=\frac{1}{{\epsilon }_0{\mu }_0}$, this introduces a 1/c² factor in the field momentum.
2. Vector Potential and Momentum:
   The electromagnetic momentum of a charged particle in a magnetic field is given by:$p=qA$where A is the magnetic vector potential. This suggests a deep connection between electromagnetic potentials and momentum.

Minimal Coupling

The principle of minimal coupling, which is used to introduce electromagnetic interactions in quantum mechanics and relativistic quantum field theory, involves the substitution:$p\to p-qA$This directly relates charge (q) and momentum (p) in the presence of electromagnetic fields.

Conclusion

The 1/c² relationship appearing in both charge and momentum-related equations is not coincidental. It reflects the deep interconnection between electromagnetism and special relativity. This connection arises from:

1. The fundamental nature of c as a universal speed limit.
2. The unification of electric and magnetic phenomena in Maxwell's equations.
3. The relativistic nature of both electromagnetic interactions and particle dynamics.

These relationships highlight the fundamental role of the speed of light in our understanding of space, time, and the behavior of charged particles. They suggest that charge and momentum, while distinct concepts, are intimately linked through the principles of relativity and electromagnetism.

The appearance of c² in the Coulomb's Law formula is not an arbitrary coincidence, but a direct consequence of:

• The way we've defined μ₀: The defined value for μ₀ (4π × 10⁻⁷ H/m) is a choice based on our units for current and force, and plays a pivotal role in linking electromagnetism to the speed of light.

• The fundamental relationship ε₀μ₀ = 1/c²: This equation establishes a direct link between the electric (ε₀) and magnetic (μ₀) properties of vacuum and the speed of light.

• Substituting for ε₀ in Coulombs law: By substituting our expressions we show how c² appears in the F_c relationship, and also reveal the scaling relationship 10⁻⁷ H/m

Through these connections, Coulomb's Law can be expressed in terms of c², demonstrating that the fundamental force of electrostatics is inherently linked to the speed of light, a consequence of the relativistic nature of electromagnetism.

This analysis also highlights that the numerical value of k is not fundamental; it is a result of our choice of units. The true fundamental connection is the presence of c² within the description of the electrostatic force. This emphasizes that the speed of light is not just a property of photons but a fundamental constant that permeates all of electromagnetism.