Maxwell's Equations

See the four equations one by one

Each of the four buttons brings up one of Maxwell's equations as a picture. Two of them show, with the divergence (∇·), where a field springs from; the other two show, with the curl (∇×), how a field circles. It is worth comparing the sky-blue arrows streaming out of the red point charge with the gold loops around the bar magnet, which always close on themselves. The indicator shows the integral and differential forms of the chosen equation together. How the four equations bind electricity and magnetism into one is the question of this lesson.

The four equations in one place

Everything built up separately for electricity and magnetism comes together in four equations. Gauss's law (∇·E = ρ/ε₀, EM-06) says the electric field springs from charges, and Gauss's law for magnetism (∇·B = 0) says the magnetic field has no source. Faraday's law (∇×E = -∂B/∂t, EM-20) says a changing magnetic field makes an electric field, and Ampère's law completed by the displacement current (∇×B = μ₀J + μ₀ε₀∂E/∂t, EM-21) says currents and changing electric fields make a magnetic field. No new law is added; the four already learned are simply gathered together. The four-line form used today is not quite what Maxwell first wrote, though: Heaviside later bundled his twenty-odd component equations into vector notation. Add the Lorentz force a charge feels in these fields, and you have all of classical electromagnetism.

Two divergences and two curls

The four equations fall into two kinds. The two divergences say where fields spring from, and the two curls say how fields circle. From the divergences, the electric field has a source in charge (∇·E = ρ/ε₀) but the magnetic field has none (∇·B = 0). That is because no magnetic monopole has ever been found, so magnetic field lines always form closed loops with no start and no end. It is the same fact that gives every piece of a bar magnet, however finely you cut it, both a north and a south pole. The two curls pair the electric and magnetic fields through time: when one changes, the other circles around it.

Symmetry and its breaking

In the four equations, the electric and magnetic fields face each other almost like mirror images. A changing magnetic field makes an electric field (Faraday), and a changing electric field makes a magnetic field (Ampère–Maxwell). The symmetry is not perfect, though: there are charges but no magnetic monopoles, and the two curl equations differ in sign, negative for Faraday and positive for Ampère–Maxwell. Thanks to this sign difference, combining the two curl equations gives the equation of a wave in which the two fields trade back and forth; with the same sign there would be no oscillation, only fields that blow up or die away in one direction. In empty space with no charges, swapping the electric and magnetic fields (with matching units) while putting a minus sign on one of them leaves all four equations unchanged, and this swap works only because the signs differ. Take the divergence of the Ampère–Maxwell equation and combine it with Gauss's law, and charge conservation, the fact that charge is never created or destroyed, falls out on its own: another sign that the four equations lock together.

The wave hidden in the equations

In empty space with no charges and no currents (ρ = 0, J = 0), a changing electric field makes a magnetic field, and that field, as it changes, makes an electric field again. Interlocking the two curl equations gives a wave equation in which the two fields carry each other outward, and its speed, c = 1/√(μ₀ε₀), is set by the two constants alone. When Maxwell worked out this value, it matched the measured speed of light, and the conclusion that light is an electromagnetic wave came from there. Experiment confirmed it only after Maxwell's death, when Hertz made and detected radio waves with a spark gap. The next lesson pulls this wave straight out of the equations.

Back to the first screen

The equations you saw with the four buttons on the first screen were not separate laws. The two divergences showed where electric and magnetic fields spring from (there are charges but no monopoles), and the two curls drew the loop that binds the two fields through time. One changing field gives rise to the other, and that one gives rise to the first again, so the four equations form a single closed system. That is why a small disturbance in one place can keep itself going and race through space, and that is light.

The next lesson (EM-23) interlocks the two curl equations in empty space and pulls out the wave equation directly. You will see a plane wave, with electric and magnetic fields perpendicular to each other and to the direction of travel, running at the speed c = 1/√(μ₀ε₀): the light that Maxwell's equations held comes into view.