Revision #1399 → #1772 · back to history
modifiedMaxwell's microscopic equations (SI)921b2725eed8
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| anchors | [{"section":"Microscopic version in SI units","snippet":"Maxwell's microscopic equations are written as"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\nabla \\cdot \\mathbf {E} \\,\\,\\,&={\\frac {\\rho }{\\varepsilon _{0}}}\\\\\\nabla \\cdot \\mathbf {B} \\,\\,\\,&=0\\\\\\nabla \\times \\mathbf {E} &=-{\\frac {\\partial \\mathbf {B} }{\\partial t}}\\\\\\nabla \\times \\mathbf {B} &=\\mu _{0}\\left(\\mathbf {J} +\\varepsilon _{0}{\\frac {\\partial \\mathbf {E} }{\\partial t}}\\right)\\end{aligned}}}"}] | — |
modifiedDifferentiation under the integral signcb4e9b6f3abb
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| anchors | [{"section":"Integral equations","snippet":"we can bring the differentiation under the integral sign in Faraday's law"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\mathrm {d} }{\\mathrm {d} t}}\\iint _{\\Sigma }\\mathbf {B} \\cdot \\mathrm {d} \\mathbf {S} =\\iint _{\\Sigma }{\\frac {\\partial \\mathbf {B} }{\\partial t}}\\cdot \\mathrm {d} \\mathbf {S} \\,,}"}] | — |
modifiedTotal enclosed electric chargef0a58f1a15be
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| anchors | [{"section":"Integral equations","snippet":"The total electric charge Q enclosed in Ω is the volume integral over Ω of the charge density ρ"},{"type":"math_alttext","value":"{\\displaystyle Q=\\iiint _{\\Omega }\\rho \\ \\mathrm {d} V,}"}] | — |
modifiedNet magnetic flux1db72f1ffef2
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| anchors | [{"section":"Integral equations","snippet":"The net magnetic flux Φ B is the surface integral of the magnetic field B passing through a fixed surface"},{"type":"math_alttext","value":"{\\displaystyle \\Phi _{B}=\\iint _{\\Sigma }\\mathbf {B} \\cdot \\mathrm {d} \\mathbf {S} ,}"}] | — |
modifiedNet electric fluxfd866d96b138
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| anchors | [{"section":"Integral equations","snippet":"The net electric flux Φ E is the surface integral of the electric field E passing through Σ"},{"type":"math_alttext","value":"{\\displaystyle \\Phi _{E}=\\iint _{\\Sigma }\\mathbf {E} \\cdot \\mathrm {d} \\mathbf {S} ,}"}] | — |
modifiedNet electric current1b8517f33002
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| anchors | [{"section":"Integral equations","snippet":"The net electric current I is the surface integral of the electric current density J passing through Σ"},{"type":"math_alttext","value":"{\\displaystyle I=\\iint _{\\Sigma }\\mathbf {J} \\cdot \\mathrm {d} \\mathbf {S} ,}"}] | — |
modifiedEquivalence of differential and integral formsec33fd04a163
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| mathlib.decl | integral_divergence_of_hasFDerivAt_off_countable | MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable |
modifiedDivergence theorem for electric flux1fc9f750a931
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| mathlib.decl | integral_divergence_of_hasFDerivAt_off_countable | MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable |
modifiedIntegral and differential Gauss's law equivalencec249a0bdaf6d
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| anchors | [{"section":"Flux and divergence","snippet":"this is satisfied if and only if the integrand is zero everywhere"},{"type":"math_alttext","value":"{\\displaystyle \\iiint _{\\Omega }\\left(\\nabla \\cdot \\mathbf {E} -{\\frac {\\rho }{\\varepsilon _{0}}}\\right)\\,\\mathrm {d} V=0}"}] | — |
modifiedKelvin–Stokes theorem for circulationde940f2fb5a9
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| anchors | [{"section":"Circulation and curl","snippet":"By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ"},{"type":"math_alttext","value":"{\\displaystyle \\oint _{\\partial \\Sigma }\\mathbf {B} \\cdot \\mathrm {d} {\\boldsymbol {\\ell }}=\\iint _{\\Sigma }(\\nabla \\times \\mathbf {B} )\\cdot \\mathrm {d} \\mathbf {S} ,}"},{"type":"math_alttext","value":"{\\displaystyle \\iint _{\\Sigma }\\left(\\nabla \\times \\mathbf {B} -\\mu _{0}\\left(\\mathbf {J} +\\varepsilon _{0}{\\frac {\\partial \\mathbf {E} }{\\partial t}}\\right)\\right)\\cdot \\mathrm {d} \\mathbf {S} =0.}"}] | — |
| mathlib.decl | integral2_divergence_prod_of_hasFDerivAt | MeasureTheory.integral2_divergence_prod_of_hasFDerivAt |
modifiedCharge conservation (continuity equation)0330f26c132b
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| anchors | [{"section":"Charge conservation","snippet":"The invariance of charge can be derived as a corollary of Maxwell's equations"},{"type":"math_alttext","value":"{\\displaystyle 0=\\nabla \\cdot (\\nabla \\times \\mathbf {B} )=\\nabla \\cdot \\left(\\mu _{0}\\left(\\mathbf {J} +\\varepsilon _{0}{\\frac {\\partial \\mathbf {E} }{\\partial t}}\\right)\\right)=\\mu _{0}\\left(\\nabla \\cdot \\mathbf {J} +\\varepsilon _{0}{\\frac {\\partial }{\\partial t}}\\nabla \\cdot \\mathbf {E} \\right)=\\mu _{0}\\left(\\nabla \\cdot \\mathbf {J} +{\\frac {\\partial \\rho }{\\partial t}}\\right)}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\partial \\rho }{\\partial t}}+\\nabla \\cdot \\mathbf {J} =0.}"}] | — |
addedDiv–curl identity (∇·(∇×F) = 0)1d534111e47b
modifiedMaxwell's equations in vacuumd4568b530279
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| anchors | [{"section":"Vacuum equations, electromagnetic waves and speed of light","snippet":"such as in vacuum, Maxwell's equations reduce to"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\nabla \\cdot \\mathbf {E} &=0,&\\nabla \\times \\mathbf {E} +{\\frac {\\partial \\mathbf {B} }{\\partial t}}=0,\\\\\\nabla \\cdot \\mathbf {B} &=0,&\\nabla \\times \\mathbf {B} -\\mu _{0}\\varepsilon _{0}{\\frac {\\partial \\mathbf {E} }{\\partial t}}=0.\\end{aligned}}}"}] | — |
addedCurl-of-curl identitybab3b18b36dd
modifiedElectromagnetic wave equation7b222ad1e7e2
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| anchors | [{"section":"Vacuum equations, electromagnetic waves and speed of light","snippet":"the equations above have the form of the standard wave equations"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\frac {1}{c^{2}}}{\\frac {\\partial ^{2}\\mathbf {E} }{\\partial t^{2}}}-\\nabla ^{2}\\mathbf {E} =0,\\\\{\\frac {1}{c^{2}}}{\\frac {\\partial ^{2}\\mathbf {B} }{\\partial t^{2}}}-\\nabla ^{2}\\mathbf {B} =0.\\end{aligned}}}"}] | — |
modifiedPhase velocity in a medium446248623a67
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| anchors | [{"section":"Vacuum equations, electromagnetic waves and speed of light","snippet":"In materials with relative permittivity , ε r , and relative permeability , μ r"},{"type":"math_alttext","value":"{\\displaystyle v_{\\text{p}}={\\frac {1}{\\sqrt {\\mu _{0}\\mu _{\\text{r}}\\varepsilon _{0}\\varepsilon _{\\text{r}}}}},}"}] | — |
modifiedSplitting into free and bound parts6f53e040eacd
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| anchors | [{"section":"Macroscopic formulation in terms of displacement and magnetizing fields (in matter version)","snippet":"This reflects a splitting of the total electric charge Q and current I"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}Q&=Q_{\\text{f}}+Q_{\\text{b}}=\\iiint _{\\Omega }\\left(\\rho _{\\text{f}}+\\rho _{\\text{b}}\\right)\\,\\mathrm {d} V=\\iiint _{\\Omega }\\rho \\,\\mathrm {d} V,\\\\I&=I_{\\text{f}}+I_{\\text{b}}=\\iint _{\\Sigma }\\left(\\mathbf {J} _{\\text{f}}+\\mathbf {J} _{\\text{b}}\\right)\\cdot \\mathrm {d} \\mathbf {S} =\\iint _{\\Sigma }\\mathbf {J} \\cdot \\mathrm {d} \\mathbf {S} .\\end{aligned}}}"}] | — |
modifiedAuxiliary fields D and H14a932b282fb
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| anchors | [{"section":"Auxiliary fields, polarization and magnetization","snippet":"The definitions of the auxiliary fields are"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {D} (\\mathbf {r} ,t)&=\\varepsilon _{0}\\mathbf {E} (\\mathbf {r} ,t)+\\mathbf {P} (\\mathbf {r} ,t),\\\\\\mathbf {H} (\\mathbf {r} ,t)&={\\frac {1}{\\mu _{0}}}\\mathbf {B} (\\mathbf {r} ,t)-\\mathbf {M} (\\mathbf {r} ,t),\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\rho _{\\text{b}}&=-\\nabla \\cdot \\mathbf {P} ,\\\\\\mathbf {J} _{\\text{b}}&=\\nabla \\times \\mathbf {M} +{\\frac {\\partial \\mathbf {P} }{\\partial t}}.\\end{aligned}}}"}] | — |
modifiedBound charge and current densities5a42c3eb4cee
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| anchors | [{"section":"Auxiliary fields, polarization and magnetization","snippet":"The macroscopic bound charge density ρ b and bound current density J b in terms of polarization P and magnetization M are then defined as"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {D} (\\mathbf {r} ,t)&=\\varepsilon _{0}\\mathbf {E} (\\mathbf {r} ,t)+\\mathbf {P} (\\mathbf {r} ,t),\\\\\\mathbf {H} (\\mathbf {r} ,t)&={\\frac {1}{\\mu _{0}}}\\mathbf {B} (\\mathbf {r} ,t)-\\mathbf {M} (\\mathbf {r} ,t),\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\rho _{\\text{b}}&=-\\nabla \\cdot \\mathbf {P} ,\\\\\\mathbf {J} _{\\text{b}}&=\\nabla \\times \\mathbf {M} +{\\frac {\\partial \\mathbf {P} }{\\partial t}}.\\end{aligned}}}"}] | — |
modifiedMacroscopic equations reproduce microscopic03341fdbc73d
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| anchors | [{"section":"Auxiliary fields, polarization and magnetization","snippet":"use the defining relations above to eliminate D , and H"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\rho &=\\rho _{\\text{b}}+\\rho _{\\text{f}},\\\\\\mathbf {J} &=\\mathbf {J} _{\\text{b}}+\\mathbf {J} _{\\text{f}},\\end{aligned}}}"}] | — |
modifiedConstitutive relations without polarization/magnetization253999624060
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| anchors | [{"section":"Constitutive relations","snippet":"For materials without polarization and magnetization, the constitutive relations are"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {D} =\\varepsilon _{0}\\mathbf {E} ,\\quad \\mathbf {H} ={\\frac {1}{\\mu _{0}}}\\mathbf {B} ,}"}] | — |
modifiedLinear material constitutive relations928939bf2c7f
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| anchors | [{"section":"Constitutive relations","snippet":"for linear materials the constitutive relations are"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {D} =\\varepsilon \\mathbf {E} ,\\quad \\mathbf {H} ={\\frac {1}{\\mu }}\\mathbf {B} ,}"}] | — |
modifiedOhm's law constitutive relation0e7844615762
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| anchors | [{"section":"Constitutive relations","snippet":"included Ohm's law in the form"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {J} _{\\text{f}}=\\sigma \\mathbf {E} .}"}] | — |
addedMinkowski metric tensor697b80796ceb
addedd'Alembert operatora5395bed9344