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All Maxwell equations, point and integral

Short answer

Maxwell's equations: ∇·D=ρ_v, ∮D·dS=∫ρ_v dv; ∇·B=0, ∮B·dS=0; ∇×E=-∂B/∂t, ∮E·dl=-∫∂B/∂t·dS; ∇×H=J+∂D/∂t, ∮H·dl=∫(J+∂D/∂t)·dS.

Write all the Maxwell equations for the time varying field in point form as well as integral form.

The four Maxwell's equations for time-varying fields (with ρv the free volume charge density, J the conduction current density):

No.LawPoint (differential) formIntegral form
1Gauss's law for E∇·D = ρv∮ D·dS = ∫ ρv dv
2Gauss's law for B∇·B = 0∮ B·dS = 0
3Faraday's law∇×E = −∂B/∂t∮ E·dl = −∫ (∂B/∂t)·dS
4Ampere's law (Maxwell's form)∇×H = J + ∂D/∂t∮ H·dl = ∫ (J + ∂D/∂t)·dS

Meaning:

  • Equation 1: electric flux out of a closed surface = enclosed charge.
  • Equation 2: no isolated magnetic charge; B flux lines are closed.
  • Equation 3: a changing magnetic field produces an electric field.
  • Equation 4: a current or a changing electric field produces a magnetic field.

Constitutive relations: D = εE, B = μH, J = σE. The continuity equation ∇·J = −∂ρv/∂t follows from equations 1 and 4.

✓ Answer checked by Claude Opus 5.5 (high effort)

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