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  "title": "EM Fields & Plane Wave — EMFT",
  "category": "electromagnetics",
  "category_label": "Electromagnetics",
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  "description": "The EM Fields & Plane Wave workbench is the foundation tool for electromagnetics. It animates Maxwells equations in free space: the time-varying E and B fields, the Poynting vector for power flow, the boundary conditions at dielectric interfaces, and the wave polarization. Use it to see why a plane wave has E, B, and S mutually perpendicular, and how the wave reflects and refracts at a material boundary.",
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    "emft",
    "electromagnetics",
    "maxwell equations",
    "plane wave",
    "electric field",
    "magnetic field",
    "point charge",
    "dipole",
    "infinite wire",
    "smith chart",
    "poynting vector",
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  "content": {
    "formulas": [
      "Maxwells equations (free space, no sources): ∇·E = 0, ∇·B = 0, ∇×E = −∂B/∂t, ∇×B = μ₀ε₀ ∂E/∂t",
      "Wave equation: ∇²E = μ₀ε₀ ∂²E/∂t² (c = 1/√(μ₀ε₀))",
      "Plane wave: E = E₀·cos(k·r − ωt), B = B₀·cos(k·r − ωt), |E|/|B| = c",
      "Poynting vector: S = E × H (W/m²), time-averaged: ⟨S⟩ = |E|²/(2·Z₀) where Z₀ = √(μ₀/ε₀) = 377 Ω",
      "Boundary conditions (dielectric-dielectric): tangential E continuous, normal D jumps by σ_s, tangential H jumps by J_s, normal B continuous"
    ],
    "citations": [
      "Sadiku, M. N. O., \"Elements of Electromagnetics,\" 7th ed., Oxford, 2018.",
      "Jackson, J. D., \"Classical Electrodynamics,\" 3rd ed., Wiley, 1998."
    ],
    "faq": [
      {
        "q": "Why are E and B perpendicular in a plane wave?",
        "a": "Maxwells equations: ∇×E = −∂B/∂t means ∂E/∂x = −∂B/∂t, so a wave travelling in +z with E in +x has B in +y (90° rotation in space + 90° in time)."
      },
      {
        "q": "What is the difference between polarization and phase?",
        "a": "Polarization is the spatial orientation of E (linear, circular, elliptical). Phase is the relative timing of the wave at two points. They are independent: a wave can be circularly polarized and have any phase."
      }
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  "last_updated": "2026-07-05"
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