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Maxwell's equations and continuity equation

Short answer

Take divergence of ∇×B = μ₀J + μ₀ε₀ ∂E/∂t. Using ∇·(∇×B)=0 and Gauss's law ∇·E=ρ/ε₀ gives ∇·J = −∂ρ/∂t, the continuity equation expressing charge conservation.

State and explain Maxwell's equations (in free space and dielectric medium). With the help of Maxwell's equations, derive the continuity equation (charge conservation theorem) ∇·J = −∂ρ/∂t.

Maxwell's equations

In free space (ρ = 0, J = 0):

LawDifferential form
Gauss (electric)∇·E = 0
Gauss (magnetic)∇·B = 0
Faraday∇×E = −∂B/∂t
Ampere-Maxwell∇×B = μ₀ε₀ ∂E/∂t

In a dielectric medium with free charge density ρ and free current density J, using D = εE and H = B/μ:

LawDifferential form
Gauss (electric)∇·D = ρ
Gauss (magnetic)∇·B = 0
Faraday∇×E = −∂B/∂t
Ampere-Maxwell∇×H = J + ∂D/∂t

Meaning: ∇·D = ρ says charges are the source of D. ∇·B = 0 says there are no monopoles. Faraday's law says a changing B produces E. The last says currents and a changing D (displacement current) produce H.

Continuity equation

  1. Start with the Ampere-Maxwell law in vacuum with sources: ∇×B = μ₀J + μ₀ε₀ ∂E/∂t.

  2. Take the divergence of both sides. The divergence of a curl is always zero:

    0 = μ₀ ∇·J + μ₀ε₀ ∂(∇·E)/∂t

  3. Use Gauss's law ∇·E = ρ/ε₀:

    0 = μ₀ ∇·J + μ₀ε₀ ∂(ρ/ε₀)/∂t = μ₀ ∇·J + μ₀ ∂ρ/∂t

  4. Divide by μ₀:

∇·J = −∂ρ/∂t

This says the net current flowing out of a small volume equals the rate of decrease of charge inside it, which is conservation of charge. Without Maxwell's displacement term, step 2 would give ∇·J = 0, which is wrong for charging capacitors.

✓ Answer checked by Claude Opus 5.5 (high effort)

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