Reading the Four Laws: Flux, Circulation, Divergence, Curl

Two languages, one content

Maxwell's equations wear two costumes. The integral form talks about flux through a surface and circulation around a loop — global statements you evaluate by exploiting symmetry. The differential form talks about a single point — the local law, valid even where no symmetry helps. The bridge between them is two theorems of vector calculus.

\oint_{S}\mathbf{F}\cdot d\mathbf{A} = \int_{V}(\nabla\cdot\mathbf{F})\,dV \qquad\text{and}\qquad \oint_{C}\mathbf{F}\cdot d\boldsymbol{\ell} = \int_{S}(\nabla\times\mathbf{F})\cdot d\mathbf{A}

The divergence (Gauss) theorem and the curl (Stokes) theorem — the machinery that turns a surface/loop statement into a point statement.

Gauss's law: charge is the source of E

The electric flux out of any closed surface equals the enclosed charge over \varepsilon_0. Apply the divergence theorem and the same statement becomes local: the divergence of \mathbf{E} at a point is the charge density there. This is where the inverse-square law of a point charge actually comes from — it is the unique spherically symmetric field with the right flux.

\oint_{S}\mathbf{E}\cdot d\mathbf{A} = \dfrac{Q_{\text{enc}}}{\varepsilon_0} \quad\Longleftrightarrow\quad \nabla\cdot\mathbf{E} = \dfrac{\rho}{\varepsilon_0}

Gauss's law: integral form (flux) on the left, differential form (divergence) on the right — the same law.

No magnetic monopoles

The magnetic flux through any closed surface is exactly zero: whatever field lines enter must leave. Locally, \mathbf{B} has no divergence anywhere. This is Gauss's law for magnetism, and it is the mathematical way of saying there are no isolated magnetic charges — no north pole without its south. Every magnetic field line is a closed loop.

\oint_{S}\mathbf{B}\cdot d\mathbf{A} = 0 \quad\Longleftrightarrow\quad \nabla\cdot\mathbf{B} = 0

No net magnetic flux out of any closed surface — no magnetic monopoles have ever been observed.

Faraday and Ampère–Maxwell: the dynamic duo

The two curl equations are where the fields come alive. Faraday's law: a changing magnetic flux drives a circulating electric field — the principle behind every generator. Ampère's law with Maxwell's correction: an electric current, and a changing electric flux, drive a circulating magnetic field. Each field, when it changes, reaches out and creates a loop of the other — the mechanism that lets them fly off together as a wave.

\begin{aligned} \oint_{C}\mathbf{E}\cdot d\boldsymbol{\ell} &= -\dfrac{d\Phi_B}{dt} & &\Longleftrightarrow & \nabla\times\mathbf{E} &= -\dfrac{\partial\mathbf{B}}{\partial t}, \\[4pt] \oint_{C}\mathbf{B}\cdot d\boldsymbol{\ell} &= \mu_0 I_{\text{enc}} + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt} & &\Longleftrightarrow & \nabla\times\mathbf{B} &= \mu_0\mathbf{J} + \mu_0\varepsilon_0\dfrac{\partial\mathbf{E}}{\partial t} \end{aligned}

Faraday's law and the Ampère–Maxwell law in integral and differential form. The last term on the bottom right is the displacement current.

Matching across a boundary

Real problems have interfaces — a conductor's surface, the face of a lens. Right at a surface the fields can jump, but not arbitrarily: the four equations, applied to a thin pillbox and a narrow loop straddling the surface, pin down exactly how. The parallel part of \mathbf{E} and the perpendicular part of \mathbf{B} pass through smoothly; the perpendicular part of \mathbf{E} jumps by any surface charge, and the parallel part of \mathbf{B} jumps by any surface current.

\begin{aligned} E^{\parallel}_{\text{above}} &= E^{\parallel}_{\text{below}}, & B^{\perp}_{\text{above}} &= B^{\perp}_{\text{below}}, \\[4pt] E^{\perp}_{\text{above}} - E^{\perp}_{\text{below}} &= \dfrac{\sigma}{\varepsilon_0}, & B^{\parallel}_{\text{above}} - B^{\parallel}_{\text{below}} &= \mu_0\,K \end{aligned}

Boundary conditions from Maxwell's equations: σ is the surface charge density, K the surface current density.