Displacement current

In vacuum, magnetostatic Ampère is ∇×B=μ₀J for a stationary current with ∂ρ/∂t=0. Taking its divergence gives 0=μ₀∇·J because the divergence of any curl is zero. But charge conservation in a time-dependent system says ∂ρ/∂t+∇·J=0. Applying the stationary law to a region where charge accumulates would incorrectly require ∂ρ/∂t=0.

The defect is not a violation of continuity: it is an incorrect extension of the steady-current source equation to a time-varying process. The field law needs another term whose divergence compensates the charge accumulation.

Vacuum Gauss states ∇·E=ρ/ε₀. Differentiate at a fixed position: ε₀∇·∂E/∂t=∂ρ/∂t. Combine this with continuity to find ∇·(J+ε₀∂E/∂t)=0. Therefore a natural corrected vacuum curl equation is ∇×B=μ₀[J+ε₀∂E/∂t]. The additional density is called displacement-current density.

Continuity fixes the divergence needed of the extra term; continuity BY ITSELF does not uniquely determine every divergence-free part of an electromagnetic field law. The complete Ampère–Maxwell relation is an empirical member of Maxwell's equations, supported by induction and field propagation as well as this consistency check.

Choose a FIXED smooth surface S bounded by an oriented closed contour C. Use the right-hand rule to relate the boundary traversal to surface normal n̂. Stokes then gives the vacuum integral law. The electric-flux derivative is taken through the same fixed oriented surface used to count conduction current; both terms carry the same orientation convention.

J_d has dimensions A/m²; its flux through S has units A. It does not imply that free electrons move through vacuum or that a dielectric dissipates power merely because an electric field varies.

Consider a wire feeding a capacitor with a vacuum gap. Take a small circular contour C around its lead, oriented positively for the feed current I=dQ/dt. A flat spanning surface S₁ cuts the lead and carries I_conduction=I. A deformed surface S₂ shares exactly the same oriented boundary but bulges between the plates; in the vacuum gap it cuts no conduction current.

The old static circulation law predicts μ₀I from S₁ but zero from S₂. The corrected law includes the electric-flux change across S₂. This is NOT physical conduction across the gap; it is the changing electric field. Currents, charging plate surfaces and fringing fields must be treated consistently if the surfaces cross nonideal regions.

For an ideal symmetric parallel-plate capacitor with plates area A and vacuum separation d much smaller than lateral dimensions, neglect fringe fields and assume approximately uniform gap field E(t)=Q(t)/(ε₀A) along the positive plate-to-negative plate direction. The oriented electric flux through the entire plate overlap is Φ_E≈E A=Q/ε₀.

Differentiate at fixed geometry: ε₀dΦ_E/dt≈dQ/dt=I. This equality uses the idealized full gap area. On a PARTIAL surface it is generally wrong to set the displacement-current flux equal to the complete wire current without first calculating how much changing electric flux the surface encloses.

Take ideal coaxial circular plates of radius a, vacuum gap and axial approximate E(t). Consider a concentric circular Ampèrian loop of radius r in the gap. Assume the complete SOURCE geometry makes B azimuthal and constant along that circle; choosing a circular contour by itself would not justify this step. For 0≤r<a, the electric-flux area is πr² and I_d(r)=I(r²/a²).

Use Ampère–Maxwell with the spanning disk wholly in the gap: Bφ(2πr)=μ₀I_d(r). For r>a the idealized enclosed varying flux saturates at I, yielding the external 1/r form. The result is a quasistatic symmetric approximate profile, not a universal exact field near finite plate edges, asymmetric leads or high-frequency radiative regimes.

Let S₁ and S₂ have the same oriented boundary C, so their union S₁∪(−S₂) is a closed surface. Gauss's theorem on the TOTAL source term gives the enclosed volume integral of ∇·[J+ε₀∂E/∂t]. By continuity and Gauss's law this divergence is zero, so the total current flux through the two surfaces is equal.

The equality does not assert that the individual conduction or electric-flux terms agree on every surface. It is the SUM of them that is independent of the choice of fixed spanning surface. This also explains why Ampère's magnetostatic expression becomes consistent when ∂ρ/∂t=0 and ∂E/∂t can be neglected in its proper regime.

In macroscopic matter, define D=ε₀E+P and H=B/μ₀−M. The material equation is ∇×H=J_free+∂D/∂t, and Gauss's material law is ∇·D=ρ_free. The polarization current ∂P/∂t is already included in ∂D/∂t; do not add it again to the macroscopic right-hand side. Bound magnetization currents are represented through H.

An ideal vacuum gap has P=0 and no conduction J_free across it, yet ε₀∂E/∂t may be nonzero. A real dielectric can support polarization, conduction leakage and loss separately. At THz, frequency dispersion and nonlinear material response prevent a naive identification of complex ε(ω) with a universal static capacitance or dissipated-power formula.

For ideal circular plates radius a=0.0500 m charging at I=0.0100 A, A=πa²≈7.85×10⁻³ m². The vacuum field-change rate is dE/dt≈I/(ε₀A)≈1.44×10¹¹ V/(m·s). At r=0.0250 m inside the gap, Bφ≈μ₀Ir/(2πa²)=2.00×10⁻⁸ T=20.0 nT; at r=0.100 m outside, Bφ≈μ₀I/(2πr)=20.0 nT.

These numbers illustrate the ideal field model, not a claim that a real finite capacitor has exactly this gap field at all radii. When I=0 and fields are stationary, J_d vanishes. Rapid variations that invalidate the uniform-gap/instantaneous-circuit approximations require solving full Maxwell boundary-value dynamics.

1. Take the divergence of ∇×B=μ₀J and combine with ∂ρ/∂t+∇·J=0. Which condition is wrongly imposed if the stationary law is used while charge accumulates?
2. From vacuum Gauss's law and charge continuity derive the divergence-free total source density J+ε₀∂E/∂t. Explain whether this mathematical check by itself proves every part of Maxwell's law.
3. For an ideal vacuum capacitor with full overlapping area A and charge Q(t), calculate ε₀dΦ_E/dt and identify where actual conduction carriers flow.
4. For circular plates a=0.0500 m charging at I=0.0100 A, calculate the fraction of full displacement current enclosed by a coaxial disk r=0.0250 m and its approximate Bφ.
5. At r=0.100 m for the same ideal plate model, compute Bφ. State why the formula is not exact for arbitrary finite leads and fields.
6. A dielectric has D=ε₀E+P. Explain why ∂P/∂t must not be added twice to ∇×H=J_free+∂D/∂t and state whether a vacuum displacement current transports electrons across the gap.

EM-03.03 will assemble the four vacuum Maxwell equations in local and integral forms, explicitly distinguish sources and boundary conditions, and connect their consistency to waves and finite propagation speed.