{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\varepsilon_0 \u003e 0$ and $\\mu_0 \u003e 0$ be constants, the vacuum permittivity and permeability, and put $c = 1 / \\sqrt{\\varepsilon_0 \\mu_0}$.\nLet $\\rho \\colon \\R^3 \\times \\R \\to \\R$ be a given charge density and $J \\colon \\R^3 \\times \\R \\to \\R^3$ a given current density.\n*Maxwell's equations* in vacuum are the system\n$$\\nabla \\cdot E = \\frac{\\rho}{\\varepsilon_0}, \\qquad \\nabla \\cdot B = 0, \\qquad \\nabla \\times E = - \\partial_t B, \\qquad \\nabla \\times B = \\mu_0 J + \\mu_0 \\varepsilon_0 \\, \\partial_t E,$$\nfor unknown [[vector fields|0299VCW1TH]] $E, B \\colon \\R^3 \\times \\R \\to \\R^3$, the electric and the magnetic field.\nHere $\\nabla \\cdot$ is the [[divergence|0MWV6ZTK6S]] and $\\nabla \\times$ is the [[curl|0NF6GQNXMY]] in the space variable.\n\n## Remarks\n\nExample: with $\\rho = 0$ and $J = 0$, for $E_0 \\in \\R$, $k \u003e 0$ and $\\omega = ck$, the plane wave\n$$E = \\big(0, E_0 \\cos(kx - \\omega t), 0\\big), \\qquad B = \\big(0, 0, \\tfrac{E_0}{c} \\cos(kx - \\omega t)\\big)$$\nis a solution.\nBoth fields depend only on $x$ and point across the $x$ direction, so both divergences vanish.\nOne has $(\\nabla \\times E)_z = \\partial_x E_y = -k E_0 \\sin(kx - \\omega t) = -\\partial_t B_z$, and $(\\nabla \\times B)_y = -\\partial_x B_z = \\tfrac{k E_0}{c} \\sin(kx - \\omega t) = \\tfrac{1}{c^2} \\partial_t E_y$.\n\nNon-example: $E = (-y, x, 0)$ and $B = 0$ with $\\rho = 0$ and $J = 0$ is not a solution, because $\\nabla \\times E = (0, 0, 2)$ while $-\\partial_t B = 0$.\n\nThe first two equations are Gauss's laws, the third is Faraday's law, and the fourth is the Ampère–Maxwell law.\nTaking the divergence of the fourth equation gives the continuity equation $\\partial_t \\rho + \\nabla \\cdot J = 0$ as a condition on the sources.\nIn matter one also uses the fields $D$ and $H$, related to $E$ and $B$ by constitutive relations.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9BWWD4QK8Y",
    "kind": "definition",
    "msc": [
      "35Q61",
      "35Q60"
    ],
    "name": "Maxwell's equations",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Maxwell's_equations"
      }
    ],
    "tag": "M665JWCEXY",
    "uses": [
      "GC5BDZ6QBC",
      "0299VCW1TH",
      "0MWV6ZTK6S",
      "0NF6GQNXMY"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/M665JWCEXY/",
  "links": [
    {
      "field": "defines",
      "target": "9BWWD4QK8Y"
    },
    {
      "field": "uses",
      "target": "GC5BDZ6QBC"
    },
    {
      "field": "uses",
      "target": "0299VCW1TH"
    },
    {
      "field": "uses",
      "target": "0MWV6ZTK6S"
    },
    {
      "field": "uses",
      "target": "0NF6GQNXMY"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/M665JWCEXY.md",
  "msc": [
    "35Q61",
    "35Q60"
  ],
  "name": "Maxwell's equations",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M66/M665JWCEXY.md",
  "status": null,
  "tag": "M665JWCEXY",
  "uri": "m2://M665JWCEXY",
  "url": "https://tamnd.github.io/math2/wiki/maxwells-equations-definition/",
  "writing": {
    "tag": "M665JWCEXY",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}