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5. System-Independent Electromagnetic Equations

Electric Field, Polarization and Displacement

Let's introduce a constant ε0 and a coefficient λ in Coulomb's law and Gauss's law:

which hold in all unit systems, then we have different ε0 for different unit systems:

  • SI units:
  • ESU, Gaussian, Heaviside-Lorentz and natural units:
  • EMU:

and different λ:

  • SI, Heaviside-Lorentz and natural units:
  • ESU, EMU and Gaussian units:

The electric polarization is defined as: [1]

and Gauss's law for P is: [1]

where ρb is the bound charge density and the total charge density is: [1]

Let's define the electric displacement field as:

which also holds in all unit systems.

After applying (5.1), (5.2) and (5.3) on (5.4), we get the macroscopic form of Gauss's law:

In linear, homogeneous, isotropic dielectric with instantaneous response to changes in electric field: [2]

where ε is the absolute permittivity, εr is the relative permittivity (≥ 1) and χe is the electric susceptibility (≥ 0).

Although χe is dimensionless, it does not have the same value in different systems, according to its definition: [3]

Magnetic B, M and H Fields

Let's introduce a constant µ0 and a coefficient αL in Ampère's force law:

which holds in all unit systems, then we have different µ0:

  • SI units:
  • ESU:
  • EMU, Gaussian, Heaviside-Lorentz and natural units:

and different αL:

  • SI units, ESU and EMU: αL = 1
  • Gaussian and Heaviside-Lorentz units: αL = 1/c
  • natural units: αL = 1/c = 1

According to special relativity, , which also holds in all unit systems.

Note that αL also exists in Lorentz force, Biot-Savart law, Maxwell-Faraday equation, Ampère-Maxwell equation and the magnetization equation, which also hold in all unit systems:

where the magnetization is defined as: [1]

and Jm is the magnetization current density and the total current density is: [1]

and Jp is the polarization current density: [1]

Let's define the magnetic H field as:

which also holds in all unit systems.

After applying (5.4), (5.6), (5.7), (5.8) and (5.9) on (5.10), we get the macroscopic form of Ampère-Maxwell equation:

In diamagnets and paramagnets, the relation between B and H is usually linear: [4]

where µ is the absolute permeability, µr is the relative permeability and χm is the magnetic susceptibility (negative for diamagnets and positive for paramagnets).

Although χm is dimensionless, it does not have the same value in different systems, according to its definition: [5]

Summary

We introduced coefficients λ and αL in different unit systems.

λ depends on whether the unit system is rationalized or non-rationalized:

  • λ = 1 (rationalized) prefers simple forms for Maxwell's equations.
  • λ = 4π (non-rationalized) prefers simple forms for Coulomb's law, Ampère's force law and Biot-Savart law.

αL depends on the definition of magnetic B field:

  • αL = 1 defines B by Lorentz force.
  • αL = c makes B the same dimension as E.
  • αL = c = 1 takes both above advantages.
αL = 1 αL = c = 1 αL = c
λ = 1 (rationalized) SI Natural Heaviside-Lorentz
λ = 4π (non-rationalized) ESU, EMU - Gaussian

The differences between ESU and EMU are ε0 and µ0:

  • ESU: ε0 = 1 and µ0 = 1/c2, prefer simple Coulomb's law.
  • EMU: ε0 = 1/c2 and µ0 = 1, prefer simple Ampère's force law and Biot-Savart law.
System ε0 µ0 λ αL kC kA
SI 8.8541878128(13)×10-12 F/m [6] 1.25663706212(19)×10-6 N/A2 [6] 1 1 1/4πε0 µ0/4π
ESU 1 1/c2 4π 1 1 1/c2
EMU 1/c2 1 4π 1 1/c2 1
Gaussian 1 1 4π c 1 1/c2
Heaviside-Lorentz 1 1 1 c 1/4π 1/4πc2
Natural 1 1 1 c = 1 1/4π 1/4π

Here kC is the constant in Coulomb's law: [7]

and kA is the constant in Ampère's force law and Biot-Savart law: [8]

where ε0, µ0 and αL follow the special relativity:

The system-independent Maxwell equations are:

microscopic macroscopic
Gauss's law
Gauss's law for magnetism
Maxwell-Faraday equation
Ampère-Maxwell equation

Additional equations are:

charge or electric current or magnetic
total density
polarization and magnetization
bound charge and magnetization current
displacement field and H field
permittivity [2] and permeability [4]
susceptibility [2,3,4,5]
polarization current density
continuity equation
Poynting vector [9]
Lorentz force or

Notes

  1. These always hold in all unit systems.
  2. In anisotropic dielectric, the permittivity and electric susceptibility are second rank tensors.
  3. Refer to here and here (30,31).
  4. In anisotropic materials, the permeability and magnetic susceptibility are second rank tensors. This does not hold for ferromagnets, ferrimagnets and antiferromagnets.
  5. Refer to here and here (36,37).
  6. and , where e, h and c are exact values by SI definition. Before 2019, exactly.
  7. Coulomb's law is the combination of Gauss's law and Lorentz force.
  8. Ampère's force law is the combination of Biot-Savart law and Lorentz force, and Biot-Savart law is the magnetostatic situation of Ampère-Maxwell equation.
  9. Here is the explanation.