Maxwell's equations as a system

Two fields and two integral geometries give four scalars — that is the whole inventory. Over a closed surface you can take the flux, ∮ E ⋅ dA and ∮ B ⋅ dA; around a closed loop you can take the circulation, ∮ E ⋅ ds and ∮ B ⋅ ds. Maxwell's equations are the four statements that fix those four numbers, and there is no fifth integral of this kind left to write. The geometry is what students confuse. A flux law uses a closed surface, which has no edge and encloses a volume, and q_enc means the charge inside that volume. A circulation law uses a closed loop, and the surface appearing on its right is an open surface hung on that loop like a soap film on a wire, so I_enc means the current piercing that film. The surface and the loop are geometry you invent: you may draw either one anywhere, and the drawing itself is not a physical object. Every source sits on the right.

Read the four right-hand sides as a list of what can make a field. Charge makes electric flux: enclose 1.0 nC and the outward flux is 1.0 × 10⁻⁹ C ÷ 8.854 × 10⁻¹² F m⁻¹ = 1.13 × 10² N m² C⁻¹, whatever shape the surface takes. Nothing makes magnetic flux: that right-hand side is 0, not a small number, and the magnetic charge that would change it has never been found. A changing magnetic flux makes electric circulation. Current makes magnetic circulation — 1.0 A through the spanning surface gives ∮ B ⋅ ds = μ₀ × 1.0 A = 1.26 × 10⁻⁶ T m — and so does a changing electric flux. Two asymmetries deserve names. E has sources you can hold in your hand and B has none, so E lines can end while B lines must close. And the magnetic circulation has two source terms where the electric circulation has one, which is where the coupling between the halves lives.

Hang two different surfaces on the same loop — a flat disc and a deep bag — and the Ampere-Maxwell law must return the same circulation for both, because its left side never mentions which surface you chose. Subtract the two versions. The disc and the reversed bag together make one closed surface, and what survives is Iₒᵤₜ + ε₀ dΦ_E/dt = 0 over that closed surface, with Iₒᵤₜ the net current leaving it. Now bring in Gauss's law: ε₀ Φ_E = q_enc, so ε₀ dΦ_E/dt = dq_enc/dt and the condition reads Iₒᵤₜ = − dq_enc/dt. Charge leaving a region must come out of the charge inside it. Nobody wrote that in as a separate postulate; it is forced by demanding that the system give one answer per loop. Numerically, a surface taking 2.0 A in along one wire while 1.4 A leaves along another accumulates charge at 0.6 C s⁻¹, and its ε₀ dΦ_E/dt term is exactly 0.6 A.

Set every time derivative to zero and look at what is left. Gauss's law and ∮ E ⋅ ds = 0 mention only E and charge; ∮ B ⋅ dA = 0 and ∮ B ⋅ ds = μ₀ I_enc mention only B and current. The two halves no longer refer to each other, so in the static limit electrostatics and magnetostatics are separate theories — which is why a course can spend weeks on each before the two are introduced to one another. The vanishing circulation of E is also what licenses the potential: if the work done on a charge round every closed path is zero, a single-valued V can be hung on every point. How static is static enough? The terms being dropped are smaller than the terms kept by roughly ωL/c, where L is the size of the region. A 0.30 m circuit at 50 Hz gives 2π × 50 s⁻¹ × 0.30 m ÷ (3.00 × 10⁸ m s⁻¹) = 3.1 × 10⁻⁷, so there the static rules are not merely usable, they are more exact than the components.

The time derivatives couple the two halves in both directions: a changing Φ_B drives E round a loop, and a changing Φ_E drives B round a loop. That is a feedback path. Set a magnetic field changing and it builds a circulating E; if that E is itself changing it builds a circulating B; the pair can pass the disturbance along with no charge and no current anywhere near, which is the ingredient the next topic turns into a wave. Two features of the coupled system are already visible. It cannot be instantaneous, because a field here is set by what the sources were doing earlier. And the constant tying the two coupling terms together, μ₀ε₀, has units of s² m⁻², so 1/√(μ₀ε₀) = 3.00 × 10⁸ m s⁻¹ is a speed hiding inside two constants that were measured with capacitors and coils on a bench.

The four integral statements are exactly true for every field, every geometry and every speed of charge, and that generality is what makes them poor calculating tools. To extract a field from one you must already know enough about its direction and magnitude to bring it outside the integral, which in practice means spherical, cylindrical or planar symmetry, or a long solenoid. Without symmetry the laws still hold; they simply constrain a total and say nothing about any single point. Zero net flux through a closed surface does not mean zero field on it, and zero circulation round a loop does not mean E vanishes anywhere along it. The general tool is the local form of these same four laws, solved numerically, which a later course develops. One more boundary: these four say what the sources do to the fields and nothing about what the fields do to the sources. The Lorentz force F = q(E + v × B) supplies that half, and only the two together make a theory you can test.