si-gaussian-conversion

name	si-gaussian-conversion
description	Complete SI ↔ Gaussian unit conversion for electrodynamics. Replace ε₀, μ₀, c factors when translating between Jackson (SI), Landau (Gaussian), and other sources. Essential when using landau-graph (Gaussian) alongside electrodynamics skill (SI).

SI ↔ Gaussian Unit Conversion for Electrodynamics

When using landau-graph (Gaussian units, following Landau-Lifshitz) alongside the electrodynamics skill (SI units, following Jackson, Griffiths), formulas differ by factors of 4πε₀, c, and μ₀/4π. This reference provides the systematic translation.

Fundamental Constants

Quantity	SI	Gaussian
Coulomb constant	k = 1/(4πε₀)	1
Vacuum permittivity	ε₀	1/(4π)
Vacuum permeability	μ₀ = 4π×10⁻⁷ H/m	4π/c²
Speed of light	c = 1/√(ε₀μ₀)	c
Impedance of vacuum	Z₀ = √(μ₀/ε₀) ≈ 377 Ω	4π/c

How to Convert

Replace ALL occurrences of ε₀ and μ₀ according to the table below, then set the remaining c factors. The conversion is EXACT — no approximations.

SI → Gaussian

SI expression	Gaussian equivalent
ε₀	1/(4π)
μ₀	4π/c²
1/(4πε₀)	1
μ₀/(4π)	1/c²
√(μ₀/ε₀)	4π/c

Gaussian → SI

Gaussian expression	SI equivalent
q₁q₂/r² (force)	q₁q₂/(4πε₀r²)
E	√(4πε₀) E_SI
B	√(4π/μ₀) B_SI
1 (Coulomb const)	1/(4πε₀)

Core Equations: Both Systems

Maxwell's Equations

	SI	Gaussian
Coulomb	∇·E = ρ/ε₀	∇·E = 4πρ
No monopoles	∇·B = 0	∇·B = 0
Faraday	∇×E = −∂B/∂t	∇×E = −(1/c)∂B/∂t
Ampère	∇×B = μ₀J + μ₀ε₀∂E/∂t	∇×B = (4π/c)J + (1/c)∂E/∂t
Lorentz force	F = q(E + v×B)	F = q(E + (v/c)×B)

Potentials

	SI	Gaussian
E from potentials	E = −∇φ − ∂A/∂t	E = −∇φ − (1/c)∂A/∂t
B from potentials	B = ∇×A	B = ∇×A
Wave equation □A	□A = −μ₀J	□A = −(4π/c)J
Lorenz gauge	∇·A + μ₀ε₀∂φ/∂t = 0	∇·A + (1/c)∂φ/∂t = 0

Radiation

	SI	Gaussian
Poynting vector	S = E×H	S = (c/4π) E×B
Dipole radiation P	P = μ₀p̈²/(6πc)	P = 2p̈²/(3c³)
Larmor formula (non-rel)	P = μ₀q²a²/(6πc)	P = 2q²a²/(3c³)
Thomson cross section	σ_T = (8π/3)(μ₀q²/(4πm))²	σ_T = (8π/3)(q²/mc²)²

Dielectric/Magnetic Response

	SI	Gaussian
D definition	D = ε₀E + P	D = E + 4πP
H definition	H = B/μ₀ − M	H = B − 4πM
Polarization P	P = ε₀χ_e E	P = χ_e E
Magnetization M	M = χ_m H	M = χ_m H
ε = ε₀(1+χ_e)	ε = ε₀ε_r	ε = 1 + 4πχ_e
Kramers-Kronig (same form!)	ε'(ω)−1 = (2/π)P∫...	ε'(ω)−1 = (2/π)P∫...

Moving Media (Minkowski)

	SI	Gaussian
D + v×H/c² =	ε(E + v×B)	D + (v/c)×H = ε(E + (v/c)×B)
B − v×E/c² =	μ(H − v×D)	B − (v/c)×E = μ(H − (v/c)×D)

Common Pitfalls

1. 4π factors: The most common error — Gaussian has explicit 4π in Maxwell but not in Coulomb's law; SI has 1/(4πε₀) in Coulomb but no 4π in Maxwell.

2. c in Lorentz force: Gaussian F = q(E + v×B/c). SI F = q(E + v×B). Missing the 1/c leads to errors in relativistic velocity regimes.

3. Kramers-Kronig is INVARIANT: The dispersion relations have the SAME form in both systems because ε(ω) is dimensionless in Gaussian and ε(ω)/ε₀ is dimensionless in SI. The conversion cancels out.

4. Impedance: SI has Z₀≈377Ω explicitly. Gaussian has Z₀=4π/c which in practical units is also 377Ω, but the form obscures the impedance concept.

5. The electrodynamics skill uses SI (following Jackson, Griffiths). The landau-graph skill uses Gaussian (following Landau). When agents use BOTH skills, apply the conversion table above.

Reference

• Wikipedia: https://en.wikipedia.org/wiki/Gaussian_units
• Jackson, Appendix §1 (unit conversion tables)
• Landau Vol.2 §27 (Gaussian Maxwell equations)