#electromagnetism

Gravity is the weakest force in the universe.No empowered crack of the electricarcing back and forth,no aurora painted by the solar wind,no polarity of the magneticto           push          apartor pull-in-close-together.No elemental bindingin all the flavors of matter.No quarks or gluons or plasmas,no  e x p l o s i v e  r e l e a s e  of the nuclear.No | wave | particle | duality | particle |…

The original 20 Maxwell's equation were eventually reduced to 4 equations that served the purpose of their reduction. Even though some of the additional terms, such as the displacement current, was added to the equations since the original formulation, the question that remains to date is this: Were more non-linear behavior embedded in the original Maxwell treatise sacrificed in favor of more "linearization" to make the reduced equations accessible at the engineering level?

The question is a little deeper especially that Maxwell dedicated a content from an earlier volume for the treatment only of Ampere's Law and the electric current, both the conducting and displacement ones. In fact, both terms on the right of the Maxwell equation shown below are genuinely present physics, and both are state-dependent for the correct reason — not a hidden Maxwell nonlinearity, but in plasma physics for example, they represent the plasma medium's response function, which is exactly where real nonlinearity is still maintained.

∇ × H = J + ∂t/∂D​

Both J (conducting current) and ∂D/∂t (displacement current) ride on the same variable: ne(t) which is the plasma electron density as it changes with time, such that:

J=σ(ne​) E D=ε(ne​,ω) E

While both are standard, textbook plasma electrodynamics that survived the reduction of Maxwell equations, both terms are driven by the same ionization state in a plasma source.

To find the actual crossover density (nc) between the density at which the plasma flips from displacement-current-dominated to conduction-current-dominated, or for a given driving frequency ω, the critical density nc:

nc = ε0 me ω^2/e^2

such that:

Underdense plasma (ne < nc): displacement current dominates, the plasma behaves as a weak dielectric, electromagnetic fields penetrate freely, and current is effectively unconfined.

Overdense plasma (ne > nc​): conduction current dominates, the plasma behaves as a conductor, and current becomes confined to a skin layer.

In active discharges (ne ∼ 10^15 – 10^21 m^−3) at 1 MHz frequency, the system is deep in the overdense, skin-effect regime. Idle plasma tubes with only background ionization can remain underdense until triggered.

The crossover density nc​ defines the local electromagnetic regime of the plasma: underdense (displacement-current dominated, dielectric-like) vs overdense (conduction-current dominated, skin-effect regime). This local response is fully encoded in the standard, reduced Maxwell equations together with plasma constitutive relations; no hidden nonlinearity is lost in the reduction from Maxwell’s original formulation.

The braid word of a designed discharge tube, for example, specifies the topological class of the current and field-line paths (linking, twisting, and possible knotting). Within a fixed braid-word class, topology constrains the allowed global configurations of current and magnetic flux, while the plasma’s density-dependent response determines the local distribution of conduction vs displacement current and the skin-depth profile.

Knotting and linking contribute through conserved (or slowly decaying) topological invariants such as magnetic helicity, which can stabilize certain field configurations and inhibit magnetic energy dissipation. However, these topological features are not independent of Maxwell’s equations; they emerge as constraints on their solutions when coupled to the plasma’s material response.

Image: a pastry braid word that is representative of a geometry with constrained topological properties.

Image is due to ChatGPT AI search engine.