Maxwell's equations

Gauss law identification

Record how electric flux is related to charge and how the standard system expresses vanishing magnetic divergence.

1. Which expressions represent electric Gauss law and the absence of magnetic monopole sources? definition
2. Does the electric source denote total charge or free charge, and is the selected electric field variable consistent with that choice? boundary

Dynamic coupling completeness

Record Faraday induction and the Ampère-Maxwell law, including whether displacement current or other time-dependent terms have been removed.

1. How are magnetic induction, electric current and the displacement-current term represented? definition
2. If a time-dependent term is absent, what approximation or restricted regime justifies its removal? boundary

Form conversion conditions

Record the regularity, domain and surface-motion assumptions needed when converting differential and integral forms.

1. Are the fields smooth, piecewise smooth or distributional, and how does the formulation handle interfaces or singular sources? definition
2. Are integration surfaces and contours fixed or moving, and what transport terms are required when differentiating their fluxes? action

Convention-preserving translation

Make the chosen unit system and any tensor, dual, coordinate or harmonic-time conventions explicit before comparing formulations.

1. Which unit system, field definitions and orientation conventions determine the constants and signs? definition
2. For a relativistic or frequency-domain presentation, which metric, tensor-component or time-dependence conventions must a translation preserve? action

Source partition consistency

Record whether charge and current are total or partitioned into free and bound contributions, and how that partition enters D and H.

1. What averaging and free-versus-bound source definitions support the selected macroscopic description? definition
2. Are polarization or magnetization effects counted both in explicit sources and in constitutive relations? boundary

Closure requirements

Record how material response relates the field variables and whether source motion or transport must be solved alongside the fields.

1. Which constitutive relations apply, and do they include anisotropy, dispersion, nonlinearity or spatially nonlocal response? definition
2. Are charge and current prescribed, or which additional transport, motion or continuity equations are required to determine them? action

Constraint-compatible initialization

Assess whether initial data satisfy the relevant Gauss laws and whether source evolution respects charge continuity.

1. Do the initial electric and magnetic fields satisfy the divergence constraints for the supplied sources? measurement
2. Does the prescribed or computed source evolution satisfy charge continuity consistently with the field equations? measurement

Boundary condition adequacy

Record interface jumps, surface sources, conductor idealizations and radiation or incoming-field conditions relevant to the problem.

1. Which field components are constrained at each interface, and which surface charges or currents support the specified jumps? boundary
2. What external boundary, radiation or incoming-field conditions are required, and are the supplied conditions compatible with the chosen problem formulation? action

Approximation justification

Distinguish full time-dependent behavior from static or quasistatic reductions and document where the selected classical description becomes insufficient.

1. Which length, time, frequency and material-response scales justify the retained and neglected terms? measurement
2. What features of the intended application require a different material description, quantum treatment or other extension beyond this formulation? boundary

Solution consistency evidence

Assess field-equation residuals, constraints, boundary satisfaction and the applicable energy balance; retain assumptions when deriving wave or potential formulations.

1. What residuals, boundary checks and applicable energy-balance checks demonstrate that the proposed solution satisfies the specified problem? measurement
2. If solving through wave equations or potentials, which source assumptions, divergence constraints and gauge conditions must be retained or checked? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

• This entry covers the physical laws in their modern sense, not a historical publication; the supplied domain classification should be checked.
• The modern four-equation vector presentation should not be identified without qualification with Maxwell's original notation and equation count.
• These are recalled distinctions, not researched findings; no sources or identifiers have been verified.

1. Which of these check these first hold for the sense of Maxwell's equations this model covers, and on what evidence? provenance

Kinds and varieties

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• Microscopic formulation in terms of electric and magnetic fields and total charge and current
• Macroscopic formulation using electric displacement, magnetic field strength, and free charge and current
• Differential formulation
• Integral formulation
• Relativistically covariant tensor formulation

1. Which of these kinds and varieties hold for the sense of Maxwell's equations this model covers, and on what evidence? provenance

Real-world use

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• Designing antennas, waveguides, and radio communication systems
• Calculating electromagnetic fields in motors, generators, and transformers
• Modelling light propagation, diffraction, and optical devices
• Simulating electromagnetic compatibility and interference
• Modelling electromagnetic fields in medical imaging equipment

1. Which of these real-world use hold for the sense of Maxwell's equations this model covers, and on what evidence? provenance

Failure modes and hazards

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• Macroscopic problems require appropriate material constitutive relations; the equations alone do not specify a material's response.
• Incorrect boundary conditions or inconsistent initial fields can produce invalid solutions.
• Mixing SI and Gaussian unit conventions introduces incorrect factors and physical dimensions.
• Numerical approximations can introduce instability, artificial dispersion, or violations of the divergence constraints.
• Classical Maxwell theory alone does not describe quantum phenomena such as discrete photon detection or spontaneous emission.

1. Which of these failure modes and hazards hold for the sense of Maxwell's equations this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

• Lorentz force law - It specifies the force electromagnetic fields exert on a charge; Maxwell's equations specify the fields' relationships to sources and their evolution.
• Electromagnetic wave equation - It is derived from Maxwell's equations under specified assumptions and does not independently express all their source and constraint relations.
• Quantum electrodynamics - It is a quantum theory of electromagnetic interactions, whereas Maxwell's equations describe classical electromagnetic fields.
• Constitutive relations - They describe material responses, such as polarization and magnetization, needed to close macroscopic electromagnetic models.
• A Dynamical Theory of the Electromagnetic Field - It is a historical publication by Maxwell; Maxwell's equations denotes the physical laws rather than a particular paper, edition, or copy.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of Maxwell's equations this model covers, and on what evidence? provenance