In classical electromagnetism, the line integral of the magnetic field B around any closed curve C equals the permeability of free space μ0 times the net conduction current I_enc passing through any surface S bounded by C, extended in Maxwell–Ampère form by adding the displacement current term μ0ε0 d/dt of the electric flux through S: ∮_C B·dl = μ0 I_enc + μ0ε0 d/dt ∫_S E·dA. The law relates magnetic-field circulation to sources (conduction and time-varying electric flux) in the continuum field description.

Principle

Magnetic-field circulation around a closed loop is determined by currents crossing the loop and by changing electric flux through the loop; inclusion of the displacement-current term is required to preserve charge conservation and make the relation valid for time-varying fields.

Demonstration

Illustrative scenario — Charging capacitor: Situation: Two large plates form a capacitor connected to a circuit carrying a time-varying conduction current I(t). Recognition: Choose a circular loop that encircles the wire and spans between the plates; the conduction current passes through a surface bounded by the loop but not through a surface deformed to lie in the capacitor gap. Action: Evaluate ∮ B·dl around the loop and include μ0ε0 d/dt ∫_S E·dA across the gap. Consequence: The computed circulation is consistent for either surface choice only when the displacement-current term μ0ε0 d/dt ∫_S E·dA is included, producing a continuous magnetic field in the gap and preserving continuity of Ampère's relation.

Misapplication

Omitting the displacement-current term for time-varying circuits and assuming ∮_C B·dl = μ0 I_enc always holds leads to inconsistency (e.g., different surfaces bounded by the same loop give different right-hand sides), which is a semantic error of applying the steady-state form outside its domain of validity.

Consequence

Provides a local integral relation to compute magnetic fields produced by conduction currents and time-varying electric fields; Maxwell's correction (displacement current) restores local charge conservation and is necessary for deriving electromagnetic wave propagation in the classical continuum model.

Reversal

In materials with bound charges and magnetization the macroscopic relation is expressed using H and D fields: ∮ H·dl = I_free_enc + d/dt ∫_S D·dA; at atomic or quantum scales the classical field description requires replacement by quantum electrodynamics, so the continuum Ampère form no longer applies without reinterpretation.

Boundary

Clearly within: Classical continuum electrodynamics for macroscopic fields in vacuum or homogeneous media. Boundary case: At interfaces include surface currents, magnetization, and dielectric displacement; use H and D for free/bound separation. Clearly outside: Lumped-circuit descriptions that treat inductance empirically without resolving B-field circulation at wavelengths comparable to circuit dimensions, and microscopic regimes where quantum field theory is required.

Semantic Tension

Tension between distributed field descriptions (Ampère's law demands evaluation of B around continuous loops and surfaces) and lumped-circuit models (which summarize electromagnetic behavior by discrete circuit elements); resolving practical problems requires choosing the appropriate model scale.

Synthesis

Ampère's circuital law, with Maxwell's displacement-current term, ties magnetic circulation to both conduction currents and changing electric flux, thereby unifying steady and time-varying magnetostatics into a single local field law whose domain is classical continuum electrodynamics.