Electromagnetic gauge and Lorenz wave equations
Vacuum fields E=−∇φ−∂_tA and B=∇×A are unchanged by A′=A+∇χ, φ′=φ−∂_tχ.
Key knowledge
Vacuum fields E=−∇φ−∂_tA and B=∇×A are unchanged by A′=A+∇χ, φ′=φ−∂_tχ. In Lorenz gauge ∇·A+c⁻²∂_tφ=0, the potentials obey □φ=−ρ/ε₀ and □A=−μ₀J, with □=∇²−c⁻²∂_t² and c²=1/(ε₀μ₀).
Values and equations
E=−∇φ−∂_tA; B=∇×A; ∇·A+c⁻²∂_tφ=0; (∇²−c⁻²∂_t²)φ=−ρ/ε₀; (∇²−c⁻²∂_t²)A=−μ₀J
- E
- Electric field V m⁻¹
- B
- Magnetic flux density T
- φ
- Scalar potential V
- A
- Vector potential T m
- χ
- Gauge function V s
- ρ
- Charge density C m⁻³
- J
- Conventional electric current density A m⁻²
- ε₀
- Vacuum permittivity F m⁻¹
- μ₀
- Vacuum permeability H m⁻¹
- c
- Vacuum light speed m s⁻¹
- t
- Time s
Conditions and scope
- Classical vacuum Maxwell equations, consistent differentiable conserved source ∂_tρ+∇·J=0 and adequate domains/boundary data.
- Gauge-invariant fields do not make potentials unique: Lorenz gauge retains transformations satisfying □χ=0, and wave solutions still require initial/boundary or radiation conditions. Material wave speeds and gauges need their stated constitutive models.
Sources · 1
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Evidence and review
Classical Electromagnetism (2014): Scalar and vector potentials: Equations 11–15 and the gauge-transformation paragraph: potentials, Lorenz constraint and source wave equations. Replace source gauge ψ by −χ and reverse its d'Alembertian sign to the stated □ convention. Excludes the printed light-speed numerical typo and any claim that the gauge alone uniquely fixes potentials.
2026-10-10 · Independent AI editorial review: All 136 unique authored Physics compact statements, all hypotheses and displayed equations/variable/units tables, all 192 Physics-origin semantic links, exact 165 author inspection matrix and mapped retained conceptual claims have been independently read. Three Physics-tagged Measurement realization records are delegated to root Measurement review. Scientific source verification combines exact author section inspection with independent whole compact-content/semantic review and targeted primary rechecks recorded below; it does not claim each full source body was independently reread. Independent AI editorial review is not human specialist certification or experimental replication. Field inventory is approved after independently reviewing the two added source-inspected band-construction methods. Receipt: physics-independent-content-inventory-review.1.
Entry details
Permanent link to this entry ↗- Record identity
- physics-electromagnetic-gauge-wave-equations
- Revision
- university-core.1
- Structured reference record
- University core expansion