{
  "id": "PWM-L1-em-maxwell-vacuum",
  "name": "Maxwell's Equations in Vacuum",
  "domain": "electromagnetism",
  "statement": "In vacuum with no charges or currents, the four Maxwell equations reduce to a self-consistent wave equation for both E and B, predicting light as a transverse wave at speed c = 1/sqrt(mu_0 * epsilon_0).",
  "equations_latex": [
    "\\nabla \\cdot \\mathbf{E} = 0",
    "\\nabla \\cdot \\mathbf{B} = 0",
    "\\nabla \\times \\mathbf{E} = -\\partial \\mathbf{B}/\\partial t",
    "\\nabla \\times \\mathbf{B} = \\mu_0 \\varepsilon_0 \\, \\partial \\mathbf{E}/\\partial t",
    "\\nabla^2 \\mathbf{E} = (1/c^2) \\, \\partial^2 \\mathbf{E}/\\partial t^2"
  ],
  "variables": [
    {"symbol": "E", "name": "Electric field", "unit": "V/m"},
    {"symbol": "B", "name": "Magnetic field", "unit": "T"},
    {"symbol": "epsilon_0", "name": "Vacuum permittivity", "unit": "F/m"},
    {"symbol": "mu_0", "name": "Vacuum permeability", "unit": "H/m"},
    {"symbol": "c", "name": "Speed of light in vacuum", "unit": "m/s"}
  ],
  "conservation_laws": ["energy", "momentum", "charge"],
  "symmetries": ["lorentz_invariance", "gauge_invariance", "time_translation", "spatial_translation", "rotation", "parity", "time_reversal"],
  "validity_regime": "Classical fields at macroscopic length scales; photon energies much less than 511 keV; refractive index near 1. Breaks down in quantum-coherent regimes, fields above the Schwinger limit (~1e18 V/m), and curved spacetime.",
  "references": [
    {
      "title": "Classical Electrodynamics",
      "authors": ["John David Jackson"],
      "year": 1999,
      "venue": "Wiley (3rd edition)",
      "doi": null
    },
    {
      "title": "Introduction to Electrodynamics",
      "authors": ["David J. Griffiths"],
      "year": 2017,
      "venue": "Pearson (4th edition)",
      "doi": null
    },
    {
      "title": "A Dynamical Theory of the Electromagnetic Field",
      "authors": ["James Clerk Maxwell"],
      "year": 1865,
      "venue": "Philosophical Transactions of the Royal Society of London, vol. 155",
      "doi": "10.1098/rstl.1865.0008"
    }
  ]
}