Maxwell Equation

Maxwell's equations are four short equations that describe every electric and magnetic field, from the charge on a capacitor to the wave leaving an antenna. Each equation says one thing about the sources of a field or about how a field changes. Together they predict that light and radio are the same kind of wave. I'll start with the four equations in their differential form. Then we'll read them in integral form, see why the displacement current is needed, and derive the wave equation from them.

What are the four Maxwell equations ?

Each Maxwell equation answers a different question about a field. Two of them say where field lines start. The other two say how a changing field of one kind creates a field of the other kind. The table below writes them in differential form, in SI units and for fields in vacuum.

Two vector operators appear in the table. The divergence, div F or ∇ ⋅ F, measures how much a field spreads out from a point. The curl, curl F, measures how much the field turns around a point. The Divergence and Curl pages explain both with examples. E is the electric field in V/m, B is the magnetic flux density in T, ρ is the charge density in C/m³ and J is the current density in A/m².

Name	Differential form	What it says
Gauss's law	div E = ρ / ε₀	Electric field lines start on positive charge and end on negative charge.
Gauss's law for magnetism	div B = 0	Magnetic field lines never start or end. There is no magnetic charge.
Faraday's law	curl E = -∂B/∂t	A changing magnetic field creates a circulating electric field.
Ampere-Maxwell law	curl B = μ₀J + μ₀ε₀ ∂E/∂t	A current, or a changing electric field, creates a circulating magnetic field.

The two constants set the strength of each effect. ε₀ = 8.854 x 10^-12 F/m is the permittivity of free space. μ₀ = 1.2566 x 10^-6 H/m is the permeability of free space, and its value is very close to 4π x 10^-7 H/m. Inside a material, ε₀ and μ₀ become ε and μ of that material.

Two equations are about sources : the divergence equations say that electric charge is the source of E, and that B has no source of its own.
Two equations couple the fields : the curl equations link a change in time of one field to a turning of the other field. This coupling makes a wave possible.
The minus sign in Faraday's law is Lenz's law : the induced electric field drives a current whose magnetic field opposes the change that created it.
Static fields decouple : if nothing changes in time, both time derivatives are 0. Then E depends only on charges and B depends only on currents.

What does each equation say in integral form ?

The differential form describes a field at a single point. An engineer usually measures something over a region instead, such as the charge inside a box or the voltage around a wire loop. The integral form states the same four laws in those terms.

Two theorems connect the two forms. The divergence theorem turns the volume integral of div F into the flux of F through the closed surface around the volume. Stokes' theorem turns the surface integral of curl F into the circulation of F around the edge of the surface. Applying them to the table above gives the four integral laws below. In each law, S is a closed surface or an open surface with edge C, and dA and dl are the small surface and line elements.

Gauss's law : the flux of E through a closed surface equals Q_enc / ε₀, where Q_enc is the charge inside. ∯ E ⋅ dA = Q_enc / ε₀.
Gauss's law for magnetism : the flux of B through any closed surface is 0. Every field line that enters the surface also leaves it.
Faraday's law : the voltage around a loop equals minus the rate of change of the magnetic flux Φ_B through the loop. ∮ E ⋅ dl = -dΦ_B/dt. This is the law behind a transformer and a generator.
Ampere-Maxwell law : the circulation of B around a loop has two parts. The first is μ₀ times the current through the loop. The second is μ₀ε₀ times the rate of change of the electric flux Φ_E. ∮ B ⋅ dl = μ₀I_enc + μ₀ε₀ dΦ_E/dt.

Let's try Gauss's law on one charge. Put a point charge Q at the centre of a sphere of radius r. By symmetry, E points outward and has the same size everywhere on the sphere. So the flux is E x 4πr², and Gauss's law gives E = Q / (4πε₀r²). This is Coulomb's law, obtained here without doing any integral by hand.

Choose the surface to match the symmetry : Gauss's law gives a field quickly only when the field is constant over the surface, as for a sphere, a long cylinder or a large plane.
The integral form works across a boundary : it still holds where the field jumps, for example at the surface of a conductor. The derivatives in the differential form do not exist there.

Why did Maxwell add the displacement current ?

Before Maxwell, the last equation was Ampere's law, curl B = μ₀J. That law works for steady currents, but it contradicts charge conservation when the current changes. Maxwell fixed the contradiction by adding one term, and that term is what makes radio possible.

The contradiction shows up when you take the divergence of both sides. The divergence of any curl is 0, so Ampere's law forces div J = 0. But charge conservation says div J = -∂ρ/∂t. A capacitor that is charging makes the problem concrete. Current flows in the wire, but no charge crosses the gap between the plates. A loop around the wire sees the current I, while a surface that passes through the gap sees no current at all.

The fix is the term ε₀ ∂E/∂t, which Maxwell called the displacement current density. Take the divergence of the full Ampere-Maxwell law. Gauss's law replaces div E with ρ / ε₀, and the result is div J + ∂ρ/∂t = 0. That is exactly charge conservation. In the capacitor gap, the electric field grows while the capacitor charges. The displacement current ε₀A dE/dt through a plate of area A equals the wire current I, so both surfaces now see the same current.

The divergence of a curl is always 0 : this identity is why the old Ampere's law could not hold for a changing current.
The displacement current is not a flow of charge : it is a changing electric field that acts as a source of B in the same way as a real current.
Charge conservation is built in : the continuity equation div J + ∂ρ/∂t = 0 follows from the Ampere-Maxwell law and Gauss's law together.
The new term closes the loop : a changing E now creates B, just as a changing B creates E in Faraday's law.

How do the equations predict electromagnetic waves ?

With the displacement current in place, a changing E creates B, and a changing B creates E. So the two fields can sustain each other in empty space, with no charge and no current. Let's derive how fast such a disturbance travels.

In vacuum, ρ = 0 and J = 0. Take the curl of Faraday's law, and replace curl B using the Ampere-Maxwell law. This gives curl curl E = -μ₀ε₀ ∂²E/∂t². The vector identity curl curl E = grad(div E) - ∇²E applies next, and div E = 0 in vacuum. So the result is the wave equation ∇²E = μ₀ε₀ ∂²E/∂t². B satisfies the same equation.

A wave equation of this form describes a wave with speed c = 1/√(μ₀ε₀). With the constants above, c = 2.998 x 10⁸ m/s, which is the measured speed of light. The simplest solution is a plane wave that travels along z. For example, E = E₀ cos(kz - ωt) in the x direction and B = (E₀/c) cos(kz - ωt) in the y direction, with k = ω/c. This pair satisfies all four equations.

Light and radio are the same wave : only the frequency differs. At 3.5 GHz the wavelength c/f is 85.7 mm, and at 2.4 GHz it is 125 mm.
E and B are perpendicular : both are also perpendicular to the direction of travel. So an electromagnetic wave in free space is a transverse wave.
The ratio E/B is fixed : in free space E/B = c. The ratio E/H is the wave impedance √(μ₀/ε₀) = 376.7 ohm, which appears in antenna and link budget calculations.
A material slows the wave : with ε and μ of the material, the speed becomes 1/√(με). This is why a signal travels more slowly in a coaxial cable or a PCB trace than in air.

YouTube

The four lectures below cover the same topics in video form. They run from the intuition behind each equation to the wave solutions and polarization.