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Maxwell's equations, integral to differential

Short answer

Integral form: ∮E·dA = q/ε₀, ∮B·dA = 0, ∮E·dl = −dΦ_B/dt, ∮B·dl = μ₀I + μ₀ε₀ dΦ_E/dt. Differential form: ∇·E = ρ/ε₀, ∇·B = 0, ∇×E = −∂B/∂t, ∇×B = μ₀J + μ₀ε₀ ∂E/∂t. They describe electric and magnetic field sources, induction, and conservation.

State Maxwell's equations in integral form. Convert them into differential form. Explain each of these equations.

Maxwell's equations in integral form

  1. ∮E·dA = q/ε₀ (Gauss's law for electricity)
  2. ∮B·dA = 0 (Gauss's law for magnetism)
  3. ∮E·dl = −dΦ_B/dt (Faraday's law)
  4. ∮B·dl = μ₀I + μ₀ε₀ dΦ_E/dt (Ampere-Maxwell law)

Conversion to differential form

We use the divergence theorem (∮F·dA = ∫∇·F dV) and Stokes' theorem (∮F·dl = ∫(∇×F)·dA), with q = ∫ρ dV and I = ∫J·dA.

  1. ∫∇·E dV = (1/ε₀)∫ρ dV for any volume, so ∇·E = ρ/ε₀.
  2. ∫∇·B dV = 0 for any volume, so ∇·B = 0.
  3. ∫(∇×E)·dA = −∫(∂B/∂t)·dA for any surface, so ∇×E = −∂B/∂t.
  4. ∫(∇×B)·dA = μ₀∫J·dA + μ₀ε₀∫(∂E/∂t)·dA, so ∇×B = μ₀J + μ₀ε₀ ∂E/∂t.

Meaning of each equation

EquationMeaning
∇·E = ρ/ε₀Electric charges are the sources of E; field lines start on + and end on − charge.
∇·B = 0No magnetic monopoles; B lines are closed loops.
∇×E = −∂B/∂tA changing magnetic field produces a circulating electric field (induction).
∇×B = μ₀J + μ₀ε₀∂E/∂tCurrents and changing electric fields produce a circulating magnetic field.

The last term, introduced by Maxwell (displacement current), makes charge conservation consistent and predicts electromagnetic waves. In free space ρ = 0 and J = 0.

✓ Answer checked by Claude Opus 5.5 (high effort)

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