Charge continuity

Take a FIXED volume V with outward normal n̂ and total enclosed charge Q_V(t)=∫_Vρ(r,t)d³r. By convention a positive J·n̂ means electric charge flows OUT of V. Conservation therefore states dQ_V/dt=−∮_∂V J·n̂dS. An incoming positive charge has J·n̂<0 on its entry face and raises Q_V. Electron number flow and positive conventional electric-current direction need not coincide.

This is a balance of physical charge transport across the boundary. The electromagnetic displacement term ε₀∂E/∂t is not an extra material carrier flux to be inserted beside J in this charge-particle balance; its role in Maxwell–Ampère was established in EM-03.02.

For a FIXED V the time derivative passes under the volume integral if ρ is sufficiently regular. The divergence theorem gives ∮J·n̂dS=∫_V∇·Jd³r. Since the volume can be chosen arbitrarily in a smooth region, the integrand must vanish locally: ∂ρ/∂t+∇·J=0.

For ideal point charges, delta-function sheets or discontinuous J, use the same law in a distributional or integral sense. A discontinuity is not a reason to drop the surface contribution, nor does ∇·J=0 follow unless ∂ρ/∂t=0 locally.

Now V=V(t) has a boundary moving at local velocity u_b(r,t). The Reynolds transport identity adds the charge swept by the moving surface: d/dt∫_V(t)ρd³r=∫_V(t)∂ρ/∂t d³r+∮_∂V(t)ρu_b·n̂dS. Insert local continuity to express the actual outward charge flow RELATIVE to the boundary.

Thus dQ_V/dt=−∮(J−ρu_b)·n̂dS. A moving boundary can enclose additional charge even if physical carriers are stationary (J=0). For a comoving material boundary with carrier velocity v, J=ρv and u_b=v, its relative charge flux vanishes if no charges cross that boundary. This formula is for a sufficiently smooth material/control boundary; it is not a replacement for the fixed-volume law at u_b=0.

Consider a STATIONARY planar interface z=0 with region 1 below and region 2 above, and n̂=+z from 1 to 2. Let σ_s(x,y,t) be charge per area on the sheet (C/m²), and K_s(x,y,t) the in-plane surface electric-current density (A/m). Integrate fixed-volume continuity over a thin pillbox and let its thickness tend to zero while retaining the sheet charge and in-plane edge current.

The interface law is ∂tσ_s+∇_s·K_s+n̂·(J₂−J₁)=0, with J₁ and J₂ limiting BULK current densities on each side. For J₁=+j ẑ and J₂=0 with no lateral sheet current, ∂tσ_s=+j: positive incoming charge accumulates. If instead K_s diverges laterally (∇_s·K_s>0), the sheet loses charge unless supplied by a normal influx.

For an isolated total-charge balance there is no source term: charge is conserved, including when it changes species or moves between material layers. If modeling ONLY one selected subsystem (such as electron carriers in one quantum well), a signed source S_s can represent net transfer from other subsystems, chemical recombination or contact injection. This bookkeeping is not the spontaneous creation of net electric charge.

Write ∂tσ_s+∇_s·K_s=S_s only after specifying which normal-transfer terms were omitted. If the normal currents J₁,J₂ already describe contact or tunneling injection, adding the same injected current to S_s would count it twice. For an electron density n_s, σ_s=−e n_s and K_s=σ_s v_d if every electron shares a signed drift velocity v_d.

Taking the divergence of the VACUUM Maxwell–Ampère equation gives 0=μ₀∇·J+μ₀ε₀∂t(∇·E). Gauss's law identifies ε₀∇·E=ρ and restores physical carrier continuity. The total source combination J+ε₀∂tE has zero divergence, explaining why different spanning surfaces yield the same magnetic circulation.

Nevertheless J and ε₀∂tE have different meanings: J counts transport of physical electric charge, while the changing-field term can exist between vacuum capacitor plates without charges crossing the gap. Do not insert both into −∮J·n̂dS when tracking Q_V of MATERIAL charge.

For one stationary electrode of a simple capacitor, define Q(t) as its excess physical charge and I_feed=dQ/dt as positive electric current entering that electrode from a metal lead, with no leakage or other contacts. No conduction carriers traverse an ideal vacuum gap, so the electrode charge increases by its feed-current influx. For ideal full-area plates, E_gap≈Q/(ε₀A), hence ε₀d(E_gap A)/dt≈dQ/dt=I_feed.

The equality of feed current and displacement current is a MAXWELL source-balance identity, not an additional path for charge particles. For a lossy dielectric or multiple leads, account for physical leakage in dQ/dt=I_feed−I_leak−… and do not assert that the full ideal gap displacement flux always equals the incoming feed current.

Treat a 2D sheet with electron number density n_s(x,y,t) (m⁻²), signed areal charge density σ_e=−e n_s, and in-plane electron drift v_d. The signed electric surface current is K_e=−e n_s v_d, not e n_s v_d: negatively charged carriers moving +x carry conventional electric current toward −x.

Without vertical injection, recombination or traps exchanging electrons, sheet continuity ∂tσ_e+∇_s·K_e=0 reduces, after dividing by −e, to ∂t n_s+∇_s·(n_s v_d)=0. With net number injection G_s−R_s in m⁻²s⁻¹, write ∂t n_s+∇_s·(n_s v_d)=G_s−R_s, making sure vertical normal currents are NOT also counted separately. This continuity law is independent of which Drude/RPA/nonlocal constitutive model later relates K_e to E.

A uniform sheet strip width w=10.0 μm carries in-plane surface current K_x=0.500 A/m across its width: its terminal current is I=K_xw=5.00 μA. If its electron density is n_s=1.00×10¹² cm⁻²=1.00×10¹⁶ m⁻², its sheet charge is σ_e≈−1.60×10⁻³ C/m². Units agree: [K_s w]=A and [e n_s]=C/m².

For a separate moving-volume thought experiment, stationary uniform positive ρ with J=0 and an outward-expanding boundary u_b·n̂>0 gives dQ_V/dt=+∮ρu_b·n̂dS, because the enlarged volume captures previously external charge. That is NOT transport of carriers into a fixed volume.

1. A fixed box contains Q=2.00 nC. A total outward conventional electric-current flux of +5.00 μA crosses its boundary. Find dQ/dt and the charge change over 0.100 ms if the flux remains constant.
2. For smooth ρ,J and a fixed arbitrary V, derive local continuity from the integral charge balance.
3. A moving box expands through stationary uniform positive ρ while J=0. Derive the sign of dQ/dt and explain how it differs from charge entering a fixed box.
4. At a stationary z=0 sheet, n̂=+z, J₁=+0.400 ẑ A/m², J₂=0 and ∇_s·K_s=+0.100 A/m². Calculate ∂tσ_s with sign and units.
5. For an isolated ideal capacitor electrode fed by I=+2.00 mA, find dQ/dt, and relate it to full-gap displacement-current flux without claiming electrons cross vacuum.
6. A 2D electron strip of width 10.0 μm has K_x=0.500 A/m and n_s=1.00×10¹² cm⁻². Find terminal conventional-current magnitude, signed electron sheet charge and the direction of conventional current relative to electron drift.

EM-03.05 will derive Poynting's local/integral theorem from Maxwell's equations and separate electromagnetic storage, boundary power flux, work on charges and dissipative or dispersive material response.