By the 1860s physics had a drawer full of separate laws — one for the force between charges, one for the field of a magnet, Faraday's rule for induced voltages, Ampère's rule for the field around a wire. James Clerk Maxwell did something audacious: he wrote them all as statements about what the electric and magnetic fields are doing at a single point in space, added one missing term that consistency demanded, and discovered that the four equations together predict light. Radio, X-rays, the glow of your screen — all of it falls out of four lines.

This page is about those four lines in their differential (local) form — the way they are written with the two operators of vector calculus, the divergence \nabla\cdot and the curl \nabla\times. The whole trick is that each equation is really a plain-language sentence about a field, dressed in symbols. Once you can read \nabla\cdot as "how much this field spreads out from here" and \nabla\times as "how much this field swirls around here", the four equations stop being intimidating and start telling a story. That translation — from symbol to physical meaning — is the one idea of this page.

Two verbs: divergence and curl

A vector field assigns an arrow to every point of space. There are exactly two independent things a field can be doing at a point, and divergence and curl measure them.

Divergence \nabla\cdot\mathbf{F} is a number at each point: the source/sink density. Picture a tiny box around the point. If more field-flux flows out of the box than in, the divergence is positive — there is a source inside. If field flows in and vanishes, it is negative — a sink. If exactly as much enters as leaves, the divergence is zero. Think of it as "is stuff being created here?"

Curl \nabla\times\mathbf{F} is a vector at each point: the circulation density. Drop a tiny paddle-wheel at the point; if the field pushes it round, the field swirls there, and the curl vector points along the wheel's axis (right-hand rule), with a length telling you how hard it spins. Think of it as "is the field going around here?"

Maxwell's four equations are simply: two divergences (what sources the fields spread out from) and two curls (what makes the fields swirl). Reveal the figure to see a field with pure divergence and no swirl, and then a field with pure swirl and no divergence.

The four equations, one at a time

Here they are. Read each as an English sentence first, symbols second. Throughout, \mathbf{E} is the electric field, \mathbf{B} the magnetic field, \rho the charge density (charge per volume), \mathbf{J} the current density (current per area), and \varepsilon_0,\ \mu_0 the electric and magnetic constants of the vacuum.

1. Gauss's law — charge is the source of \mathbf{E}

The divergence of the electric field at a point equals the charge density there (over \varepsilon_0). Where there is positive charge, \mathbf{E} spreads outward — a source. Where there is negative charge, it converges inward — a sink. In empty space, \rho = 0, so \nabla\cdot\mathbf{E} = 0: electric field lines only ever begin and end on charges.

2. Gauss's law for magnetism — no magnetic charge

The magnetic field has zero divergence everywhere. There is no magnetic source or sink — no magnetic monopole has ever been found. Break a bar magnet in half and you get two smaller magnets, each with its own north and south, never an isolated pole. Consequently magnetic field lines never start or stop: they always close into loops.

3. Faraday's law — a changing \mathbf{B} curls up an \mathbf{E}

The curl of the electric field equals minus the rate of change of the magnetic field. If the magnetic field at a point is changing in time, it wraps a swirling electric field around itself. That circulating \mathbf{E} is exactly what drives current in a nearby loop of wire — this is electromagnetic induction, the principle behind every generator. The minus sign is Lenz's law: the induced effect always opposes the change that made it.

4. Ampère–Maxwell law — currents and changing \mathbf{E} curl up a \mathbf{B}

The curl of the magnetic field is fed by two things: electric current, and a changing electric field. A current \mathbf{J} wraps a swirling \mathbf{B} around itself (the original Ampère's law). Maxwell's own addition is the second term — the displacement current \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t — which says a changing electric field also produces a magnetic swirl, even with no wire present. That extra term is what lets the two curl equations feed each other and send a wave off through empty space.

State them together

Read the pattern down the list: divergence, divergence, curl, curl. The two divergence equations name the sources (charge for \mathbf{E}, nothing for \mathbf{B}); the two curl equations name the swirls (a changing partner field for each, plus current for \mathbf{B}).

Where the local form comes from

You have probably met these laws first in their integral form — as statements about whole surfaces and loops. The differential form is the same physics, squeezed down to a point by two theorems of vector calculus.

Divergence theorem turns the two Gauss laws local. The integral Gauss's law says the flux of \mathbf{E} out of any closed surface equals the enclosed charge over \varepsilon_0:

The divergence theorem rewrites the left side as a volume integral of \nabla\cdot\mathbf{E}, and the enclosed charge as \int_V \rho\,dV:

Since this holds for every volume V, however small, the integrands must match at every point — giving \nabla\cdot\mathbf{E} = \rho/\varepsilon_0. The same move on \oint\mathbf{B}\cdot d\mathbf{A} = 0 gives \nabla\cdot\mathbf{B} = 0.

Stokes' theorem turns the two circulation laws local. Faraday's integral law says the circulation of \mathbf{E} around a loop equals minus the rate of change of magnetic flux through it:

Stokes' theorem rewrites the left side as \int_S (\nabla\times\mathbf{E})\cdot d\mathbf{A}, and moving the time-derivative inside the integral on the right gives -\int_S (\partial\mathbf{B}/\partial t)\cdot d\mathbf{A}. Equal for every surface S, so the integrands match: \nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t. The identical argument on the Ampère–Maxwell loop law gives the fourth equation.

Worked examples: reading off the physics

Example 1 — empty space. Far from any charge or current, set \rho = 0 and \mathbf{J} = 0. The four equations collapse to

Notice the two curl equations now feed each other with no sources at all: a changing \mathbf{B} makes a curling \mathbf{E}, whose change makes a curling \mathbf{B}, and so on. Take the curl of Faraday's law and substitute; out drops a wave equation with speed c = 1/\sqrt{\mu_0\varepsilon_0}. Plug in the constants and you get 3\times10^8\ \text{m/s} — light is an electromagnetic wave. This is the payoff Maxwell found.

Example 2 — a static point charge. A lone positive charge sits at the origin. Its field is radial, \mathbf{E} \propto \hat{\mathbf{r}}/r^2. What do the equations say? Nothing is changing in time, so \partial\mathbf{B}/\partial t = 0 and \partial\mathbf{E}/\partial t = 0; there is no magnetic field, and no current. That leaves only \nabla\cdot\mathbf{E} = \rho/\varepsilon_0 doing any work: the divergence is zero everywhere except at the charge itself, where \rho is concentrated. The field spreads out from a single source point — exactly the radial picture in the figure above.

Example 3 — a steady current in a wire. A constant current \mathbf{J} flows and nothing changes in time. Then \partial\mathbf{E}/\partial t = 0, so the Ampère–Maxwell law loses its second term and becomes just \nabla\times\mathbf{B} = \mu_0\mathbf{J}. Reading it: the magnetic field curls around the current — non-zero curl right where the current flows, encircling the wire. Meanwhile \nabla\cdot\mathbf{B} = 0 guarantees those field lines close into loops. That is precisely the circulating field in the second half of the figure.