EM-22 · Time-varying synthesis

Maxwell's Equations

The four laws you already learned gather in one place. The two divergences say where the fields spring from (charge exists, the magnetic monopole does not); the two curls give the loop in which changing electric and magnetic fields beget each other. Out of this closed system comes the electromagnetic wave — light.

See the four equations one by one

Pick the four equations one by one with the buttons. Two use the divergence (∇·) to draw where a field springs from; two use the curl (∇×) to draw the swirl of a field. See how the four weave electricity and magnetism into one.

Maxwell's equation

Drag to orbit. Buttons select the equation.

The selected equation

Gauss's law

Integral form∮E·dA = Q/ε₀

Differential form∇·E = ρ/ε₀

The electric field springs from charge; its divergence is the charge density.

The four equations in one place

Everything we have built up separately for electricity and magnetism gathers into four equations. Gauss's law (∇·E = ρ/ε₀, EM-06) says the electric field springs from charge; 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 current and a changing electric field make a magnetic field. Nothing here is new — it only gathers the four you already learned into one place.

Observe∇·E = ρε₀

The electric field springs from charge.

Choose∇·B = ?

The magnetic field has no source; there is no monopole.

Two divergences and two curls

The four split into two kinds. The two divergences say where a field springs from; the two curls say how a field turns. By divergence, the electric field has a source — charge (∇·E = ρ/ε₀) — but the magnetic field has none (∇·B = 0), because no magnetic monopole has ever been found. So magnetic field lines, with neither start nor end, always form closed loops. The two curls are the pair that bind the electric and magnetic fields through time: when one changes, the other swirls.

Symmetry and its breaking

In the four, the electric and magnetic fields face each other almost like a mirror. A changing magnetic field makes an electric one (Faraday); a changing electric field makes a magnetic one (Ampère–Maxwell). But the symmetry is not perfect. Charges exist while magnetic monopoles do not, and the signs of the two curl equations differ — minus for Faraday, plus for Ampère–Maxwell. It is exactly this asymmetry that makes the two fields sustain each other rather than cancel. Were the signs the same and the symmetry perfect, the two would erase each other.

Fill in∇×E = ?

A changing magnetic field makes an electric field.

On your own∇×B = μ₀J + ?

Current and a changing electric field make a magnetic field.

The wave hidden in the equations

Consider empty space with no charge and no current (ρ = 0, J = 0). A changing electric field makes a magnetic field, and that changing magnetic field makes an electric field again. Mesh the two curl equations and they become a wave equation in which the electric and magnetic fields sustain each other as they spread. Its speed is c = 1/√(μ₀ε₀), arising from those two constants alone. When Maxwell computed this value it matched the measured speed of light: light is an electromagnetic wave. The next unit extracts this wave directly.

Back to the first screen

The four equations you toured with the buttons on the first screen were not separate laws. The two divergences drew where the electric and magnetic fields spring from — charge exists, the monopole does not — and the two curls drew the loop that binds them through time. A changing field begets the other, and that field begets the first again. Because the four form one closed system, a single small disturbance regenerates itself and runs through space. That is light.

Maxwell's equations are the four that underpin all of electromagnetism: Gauss ∇·E = ρ/ε₀, magnetic Gauss ∇·B = 0, Faraday ∇×E = -∂B/∂t, and Ampère–Maxwell ∇×B = μ₀J + μ₀ε₀∂E/∂t. The two divergences say where the fields spring from; the two curls give the loop in which changing electric and magnetic fields beget each other. Out of this closed system comes the electromagnetic wave — light. c = 1/√(μ₀ε₀).

What comes next

Next (EM-23) meshes the two curl equations in empty space to extract the wave equation directly. You will see a plane wave in which the electric and magnetic fields are perpendicular to each other and to the direction of travel, running at speed c = 1/√(μ₀ε₀). The light that Maxwell’s equations held within them comes into view.