Electrodynamics

The Maxwell equations

Maxwell’s four equations connect charge and current to changing electric and magnetic fields.

The differential form gives local field-source relations; the integral form expresses flux and circulation through a surface or around a closed path.

∇·E = ρ/ε₀, ∇·B = 0, ∇×E = −∂B/∂t, ∇×B = μ₀J + μ₀ε₀∂E/∂t

Definition: Quantities and model

Electric Gauss: ∇·E = ρ/ε₀. Magnetic Gauss: ∇·B = 0. Faraday: ∇×E = −∂B/∂t. Ampère–Maxwell: ∇×B = μ₀J + μ₀ε₀∂E/∂t.

Adjust parameters and rotate the view to inspect field structure; this is illustrative, not a general Maxwell solver.

Interpretation and consequences

Taking the divergence of Ampère–Maxwell together with continuity, ∂ρ/∂t + ∇·J = 0, shows why the ∂E/∂t term is required for charge conservation.

Example: Quantitative example

In source-free vacuum, taking the curl of Faraday and using Ampère–Maxwell gives ∇²E − μ₀ε₀∂²E/∂t² = 0, with speed c = 1/√(μ₀ε₀).

Solution

Substitute into the stated relation, keep SI units consistent, and check the result dimensionally.

Taking the divergence of both curl equations gives $\partial_t\rho+\nabla\cdot J=0$ , provided Gauss's law holds initially: local charge conservation. Conversely, if charge conservation is fundamental, the displacement-current term is what makes Ampère's law compatible with it. Integral forms expose the physical meaning: electric flux through a closed surface counts charge, while field circulation around its boundary is linked to conduction current and changing electric flux.

Gauss's laws and the two curl equations form a constrained evolution system. Since $\nabla\cdot B=0$ , the divergence of Faraday's curl vanishes identically; Gauss's electric law is consistent in time when the continuity equation holds. An initial-value solution must satisfy both Gauss constraints initially, and the curl equations then preserve them. This structure matters in numerical work, where discretization errors can otherwise cause constraint drift.

In vacuum, $\varepsilon_0$ and $\mu_0$ set the finite propagation speed $c=(\mu_0\varepsilon_0)^{-1/2}$ . This is more than a wave-solution result: it connects electric and magnetic measurements to the causal structure of electrodynamics.

Quick check

Which relation is correct in the idealized situation described?

What should be checked first when applying a field formula?

References

James Clerk Maxwell (1873). A Treatise on Electricity and Magnetism
David J. Griffiths (2017). Introduction to Electrodynamics