Maxwell’s equations

$\displaystyle \begin{aligned} \nabla \cdot \mathbf{E} &=\frac{\rho}{\varepsilon_{0}} \\ \nabla \cdot \mathbf{B} &=0 \\ ~~\nabla \times \mathbf{E} &=-\frac{\partial \mathbf{B}}{\partial t} \\ ~ ~ \nabla \times \mathbf{B} &=\mu_{0} \mathbf{j}+\frac{1}{c^{2}} \frac{\partial \mathbf{E}}{\partial t} \end{aligned}$

$\displaystyle \mathbf{E}$ is the electric field, $\displaystyle \mathbf{B}$ is the magnetic field, $\displaystyle \rho$ is the charge density, $\displaystyle \mathbf{j}$ is the current density, $\displaystyle \varepsilon_{0}$ is the electric constant, $\displaystyle \mu_{0}$ is the magnetic constant, and $\displaystyle c$ is the speed of light.

They say:

Electric charge is the source/sink of electric field.
There are no isolated magnetic charges.
Changing magnetic field creates circulating electric field.
Electric current and changing electric field create circulating magnetic field.

Maxwell’s equations are not four unrelated facts. At first, they look like four separate laws: one about electric charge, one about magnetic fields, one about induction, and one about currents. They are a single geometric structure seen from four different angles. They describe how electric and magnetic fields are allowed to begin, end, curl, propagate, and exchange energy.

I. Gauss’s Law

The integral form of Gauss’s law is

$\displaystyle \oint_{\partial V} \mathbf{E}\cdot d\mathbf{A}=\frac{Q_{\text{inside}}}{\varepsilon_{0}}$

This says the following. Take any closed surface. Add up the total electric flux leaving that surface. The result is exactly the total charge enclosed by the surface, divided by $\displaystyle \varepsilon_{0}$ . Here $\displaystyle d\mathbf{A}$ is the outward-pointing area element. The quantity $\displaystyle \mathbf{E}\cdot d\mathbf{A}$ measures how much electric field passes through a small piece of the surface. If $\displaystyle \mathbf{E}$ points outward, the flux is positive. If $\displaystyle \mathbf{E}$ points inward, the flux is negative. If $\displaystyle \mathbf{E}$ is tangent to the surface, the flux is zero. Thus Gauss’s law says, in geometric language: Electric field lines begin on positive charge and end on negative charge.

Historically, Gauss’s law is the field-theoretic form of Coulomb’s inverse-square law. To see this, consider a point charge $\displaystyle q$ at the origin. By spherical symmetry, the electric field must point radially outward or inward, so

$\displaystyle \mathbf{E}(\mathbf{r})=E(r)\,\hat{\mathbf{r}}$

Choose a sphere of radius $\displaystyle r$ centered on the charge. On this sphere, $\displaystyle E(r)$ has the same magnitude everywhere, and the field is perpendicular to the surface. Therefore

$\displaystyle \oint \mathbf{E}\cdot d\mathbf{A}=E(r)\oint dA=E(r)\,4\pi r^{2}$

Gauss’s law gives $\displaystyle E(r)\,4\pi r^{2}=\frac{q}{\varepsilon_{0}}$ and hence

$\displaystyle E(r)=\frac{1}{4\pi\varepsilon_{0}}\frac{q}{r^{2}}$

So Coulomb’s law is not an independent mystery. It is Gauss’s law, together with spherical symmetry.

The differential form of Gauss’s law is obtained by writing the enclosed charge as an integral of the charge density:

$\displaystyle Q_{\text{inside}}=\int_{V}\rho\,dV$

Thus the integral law becomes

$\displaystyle \oint_{\partial V}\mathbf{E}\cdot d\mathbf{A}=\frac{1}{\varepsilon_{0}}\int_{V}\rho\,dV$

By the divergence theorem,

$\displaystyle \oint_{\partial V}\mathbf{E}\cdot d\mathbf{A}=\int_{V}\nabla\cdot\mathbf{E}\,dV$

Therefore

$\displaystyle \int_{V}\nabla\cdot\mathbf{E}\,dV=\int_{V}\frac{\rho}{\varepsilon_{0}}\,dV$

Since this holds for every region $\displaystyle V$ , the integrands must agree pointwise:

$\displaystyle \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}$

This is the local form of Gauss’s law.

It says that the divergence of the electric field at a point measures the amount of electric charge present at that point. Positive charge gives $\displaystyle \nabla\cdot\mathbf{E}>0$ so field lines locally spread outward. Negative charge gives $\displaystyle \nabla\cdot\mathbf{E}<0$ so field lines locally converge inward. In a region with no charge, $\displaystyle \nabla\cdot\mathbf{E}=0$ so electric field lines neither begin nor end there.

II. Gauss’s Law for Magnetism

The integral form of Gauss’s law for magnetism is

$\displaystyle \oint_{\partial V}\mathbf{B}\cdot d\mathbf{A}=0$

This says that the net magnetic flux through any closed surface is zero. In words: Magnetic field lines never begin or end. They form closed loops. This is the mathematical statement that isolated magnetic charges have not been observed. There are no observed magnetic monopoles. If you cut a bar magnet in half, you do not get a north pole by itself and a south pole by itself. You get two smaller magnets, each with its own north and south pole.

The differential form follows from the divergence theorem:

$\displaystyle \oint_{\partial V}\mathbf{B}\cdot d\mathbf{A}=\int_{V}\nabla\cdot\mathbf{B}\,dV$

But the integral law says that the left-hand side is always zero. Hence

$\displaystyle \int_{V}\nabla\cdot\mathbf{B}\,dV=0$

Since this holds for every region $\displaystyle V$ , the integrand must vanish pointwise:

$\displaystyle \nabla\cdot\mathbf{B}=0$

This is one of the deepest structural constraints in electromagnetism. It says that the magnetic field has no sources and no sinks. Unlike the electric field, which can begin on positive charge and end on negative charge, the magnetic field cannot begin or end anywhere.

This condition also explains why the vector potential is so natural. At least locally, any magnetic field satisfying $\displaystyle \nabla\cdot\mathbf{B}=0$ can be written as a curl:

$\displaystyle \mathbf{B}=\nabla\times\mathbf{A}$

Why is this a good form? Because the divergence of any curl is automatically zero:

$\displaystyle \nabla\cdot(\nabla\times\mathbf{A})=0$

So the vector potential $\displaystyle \mathbf{A}$ is not introduced as an artificial trick. It is naturally suggested by Gauss’s law for magnetism itself.

III. Faraday’s Law of Induction

The integral form of Faraday’s law is

$\displaystyle \oint_{\partial S}\mathbf{E}\cdot d\boldsymbol{\ell}=-\frac{d}{dt}\int_{S}\mathbf{B}\cdot d\mathbf{A}$

This says that a changing magnetic flux through a surface creates a circulating electric field around the boundary of that surface. This was Michael Faraday’s great experimental discovery. Take a loop of wire. Move a magnet near it. A current flows. Why? Because the magnetic flux through the loop is changing. That changing magnetic flux creates an electric field tangent to the wire, and that electric field pushes charges around the loop.

The left-hand side, $\displaystyle \oint_{\partial S}\mathbf{E}\cdot d\boldsymbol{\ell}$ is the circulation of the electric field around the boundary curve. In circuit language, this is the electromotive force: $\displaystyle \mathcal{E}=\oint_{\partial S}\mathbf{E}\cdot d\boldsymbol{\ell}$

The right-hand side is the negative rate of change of magnetic flux. The magnetic flux through the surface is $\displaystyle \Phi_{B}=\int_{S}\mathbf{B}\cdot d\mathbf{A}$ so Faraday’s law may be written as

$\displaystyle \mathcal{E}=-\frac{d\Phi_{B}}{dt}$

The minus sign is Lenz’s law. It says that the induced effect opposes the change that produced it. If the magnetic flux through a loop increases upward, the induced current creates a magnetic field downward to oppose that increase. If the upward flux decreases, the induced current creates a magnetic field upward to oppose the decrease.

The differential form follows from Stokes’s theorem:

$\displaystyle \oint_{\partial S}\mathbf{E}\cdot d\boldsymbol{\ell}=\int_{S}(\nabla\times\mathbf{E})\cdot d\mathbf{A}$

Substituting this into Faraday’s law gives

$\displaystyle \int_{S}(\nabla\times\mathbf{E})\cdot d\mathbf{A}=-\frac{d}{dt}\int_{S}\mathbf{B}\cdot d\mathbf{A}$

For a fixed surface, the time derivative can be moved inside the integral:

$\displaystyle \int_{S}(\nabla\times\mathbf{E})\cdot d\mathbf{A}=\int_{S}\left(-\frac{\partial\mathbf{B}}{\partial t}\right)\cdot d\mathbf{A}$

Since this holds for every surface $\displaystyle S$ , the integrands must agree pointwise:

$\displaystyle \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$

This is the local form of Faraday’s law. It says that wherever the magnetic field changes in time, the electric field curls around it.

A static electric field produced by stationary charges is conservative: $\displaystyle \nabla\times\mathbf{E}=0$ But an induced electric field is not conservative: $\displaystyle \nabla\times\mathbf{E}\neq 0$ That is the profound shift introduced by Faraday. Electric fields are not merely gradients of potentials produced by charges. They can also be generated by changing magnetic fields.

IV. Ampère’s Law Before Maxwell

Historically, Ampère discovered that electric currents create magnetic circulation. The old integral form of Ampère’s law was

$\displaystyle \oint_{\partial S}\mathbf{B}\cdot d\boldsymbol{\ell}=\mu_{0}I_{\text{through }S}$

where the current passing through the surface is

$\displaystyle I_{\text{through }S}=\int_{S}\mathbf{J}\cdot d\mathbf{A}$

Thus Ampère’s law can also be written as

$\displaystyle \oint_{\partial S}\mathbf{B}\cdot d\boldsymbol{\ell}=\mu_{0}\int_{S}\mathbf{J}\cdot d\mathbf{A}$

This says that the circulation of the magnetic field around a closed curve is produced by the electric current passing through any surface bounded by that curve.

By Stokes’s theorem, $\displaystyle \oint_{\partial S}\mathbf{B}\cdot d\boldsymbol{\ell}=\int_{S}(\nabla\times\mathbf{B})\cdot d\mathbf{A}$ so Ampère’s law becomes

$\displaystyle \int_{S}(\nabla\times\mathbf{B})\cdot d\mathbf{A}=\mu_{0}\int_{S}\mathbf{J}\cdot d\mathbf{A}$

Since this holds for every surface $\displaystyle S$ , the integrands must agree pointwise:

$\displaystyle \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}$

This was the old differential form of Ampère’s law. It works beautifully for steady currents.

For example, take a long straight wire carrying current $\displaystyle I$ . By cylindrical symmetry, the magnetic field circles around the wire. Its magnitude depends only on the distance $\displaystyle r$ from the wire. Choose a circle of radius $\displaystyle r$ centered on the wire. On this circle, $\displaystyle \mathbf{B}$ is tangent everywhere and has constant magnitude. Therefore

$\displaystyle \oint \mathbf{B}\cdot d\boldsymbol{\ell}=B(r)\oint d\ell=B(r)\,2\pi r$

Ampère’s law gives $\displaystyle B(r)\,2\pi r=\mu_{0}I$

and hence

$\displaystyle B(r)=\frac{\mu_{0}I}{2\pi r}$

This is the magnetic analogue of the Coulomb calculation. A symmetry assumption turns an integral law into an explicit field formula.

The Problem: Ampère’s Law Violates Charge Conservation

Now comes Maxwell’s great insight. Charge conservation says

$\displaystyle \frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf{J}=0$

This is called the continuity equation. It says that charge cannot simply disappear. If the charge density in a region decreases, then charge must be flowing out of that region. If the charge density increases, then charge must be flowing in. Now consider the old differential form of Ampère’s law:

$\displaystyle \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}$

Take the divergence of both sides:

$\displaystyle \nabla\cdot(\nabla\times\mathbf{B})=\mu_{0}\nabla\cdot\mathbf{J}$

But the divergence of a curl is always zero: $\displaystyle \nabla\cdot(\nabla\times\mathbf{B})=0$

Therefore old Ampère’s law implies $\displaystyle \mu_{0}\nabla\cdot\mathbf{J}=0$ and hence $\displaystyle \nabla\cdot\mathbf{J}=0$ This is the problem. Charge conservation says $\displaystyle \nabla\cdot\mathbf{J}=-\frac{\partial\rho}{\partial t}$ . So old Ampère’s law forces $\displaystyle \frac{\partial\rho}{\partial t}=0$ . In other words, the old law says that charge density cannot change anywhere. But this is false.

The cleanest example is a charging capacitor. Current flows through the wire into one plate of the capacitor. Charge accumulates on that plate. At the same time, charge of the opposite sign accumulates on the other plate. So the charge density is changing in time. Between the plates, however, there is no ordinary conduction current. The gap is empty space or an insulator. No charges cross the gap from one plate to the other. And yet the magnetic field around the wire does not simply stop existing in the region between the plates. This creates an ambiguity in the old integral form of Ampère’s law. Take a closed loop around the wire. The same loop can bound many different surfaces. One surface cuts through the wire. That surface sees the conduction current $\displaystyle I$ . Old Ampère’s law gives

$\displaystyle \oint_{\partial S}\mathbf{B}\cdot d\boldsymbol{\ell}=\mu_{0}I$

But another surface can bulge outward and pass between the capacitor plates. That surface cuts through the gap, where no conduction current passes. Old Ampère’s law then gives

$\displaystyle \oint_{\partial S}\mathbf{B}\cdot d\boldsymbol{\ell}=0$

The same boundary loop cannot give two different answers. That is the contradiction. So the old law is incomplete. It works for steady currents, but it fails when charge densities and electric fields change in time. Maxwell’s correction will repair this by adding one missing source of magnetic circulation: the changing electric field.

Maxwell’s Correction: The Displacement Current

Maxwell repaired Ampère’s law by adding one new term:

$\displaystyle \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}$

The extra term is $\displaystyle \mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}$ . This is Maxwell’s correction. In integral form, the corrected law is

$\displaystyle \oint_{\partial S}\mathbf{B}\cdot d\boldsymbol{\ell}=\mu_{0}\int_{S}\mathbf{J}\cdot d\mathbf{A}+\mu_{0}\varepsilon_{0}\frac{d}{dt}\int_{S}\mathbf{E}\cdot d\mathbf{A}$

The electric flux through the surface is $\displaystyle \Phi_{E}=\int_{S}\mathbf{E}\cdot d\mathbf{A}$ so the new contribution can be written as

$\displaystyle \mu_{0}\varepsilon_{0}\frac{d\Phi_{E}}{dt}$

The quantity $\displaystyle \varepsilon_{0}\frac{d\Phi_{E}}{dt}$ is called the displacement current. This name is slightly misleading. It is not necessarily a current made of charges physically crossing the surface. In the empty gap of a capacitor, no conduction charge crosses from one plate to the other. But the electric field between the plates changes in time, and that changing electric field contributes to the magnetic circulation exactly as a current would. Maxwell’s claim was therefore simple and profound: A changing electric field produces magnetic circulation, just as a conduction current does.

Now check charge conservation. Start with the corrected Ampère-Maxwell law:

$\displaystyle \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}$

Take the divergence of both sides:

$\displaystyle \nabla\cdot(\nabla\times\mathbf{B})=\nabla\cdot\left(\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}\right)$

The left-hand side is zero, because the divergence of a curl is always zero: $\displaystyle \nabla\cdot(\nabla\times\mathbf{B})=0$ . Therefore $\displaystyle 0=\mu_{0}\nabla\cdot\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial}{\partial t}(\nabla\cdot\mathbf{E})$ . Now use Gauss’s law for electricity: $\displaystyle \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}$ Substitute this into the previous equation: $\displaystyle 0=\mu_{0}\nabla\cdot\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial}{\partial t}\left(\frac{\rho}{\varepsilon_{0}}\right)$ . This becomes: $\displaystyle \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0$

This is exactly the continuity equation. So Maxwell’s extra term is not decorative. It is forced by charge conservation. That is the first miracle. The second miracle is that it predicts light.

Electromagnetic Waves

Now take Maxwell’s equations in vacuum: $\displaystyle \rho=0,\qquad \mathbf{J}=0$

Then Maxwell’s equations become $\displaystyle \nabla\cdot\mathbf{E}=0$ , $\displaystyle \nabla\cdot\mathbf{B}=0$ , $\displaystyle \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$ , $\displaystyle \nabla\times\mathbf{B}=\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}$

Now we derive the wave equation for the electric field. Start with Faraday’s law:

$\displaystyle \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$

Take the curl of both sides:

$\displaystyle \nabla\times(\nabla\times\mathbf{E})=-\frac{\partial}{\partial t}(\nabla\times\mathbf{B})$

Now use the Ampère-Maxwell law:

$\displaystyle \nabla\times\mathbf{B}=\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}$

Substituting gives

$\displaystyle \nabla\times(\nabla\times\mathbf{E})=-\mu_{0}\varepsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}$

Use the vector identity

$\displaystyle \nabla\times(\nabla\times\mathbf{E})=\nabla(\nabla\cdot\mathbf{E})-\nabla^{2}\mathbf{E}$

In vacuum, $\displaystyle \nabla\cdot\mathbf{E}=0$

so the identity becomes $\displaystyle \nabla\times(\nabla\times\mathbf{E})=-\nabla^{2}\mathbf{E}$

Therefore

$\displaystyle -\nabla^{2}\mathbf{E}=-\mu_{0}\varepsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}$

and hence

$\displaystyle \nabla^{2}\mathbf{E}=\mu_{0}\varepsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}$

This is a wave equation. The standard wave equation has the form

$\displaystyle \nabla^{2}f=\frac{1}{v^{2}}\frac{\partial^{2}f}{\partial t^{2}}$

So here $\displaystyle \frac{1}{v^{2}}=\mu_{0}\varepsilon_{0}$ and therefore $\displaystyle v=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}$

But this number is exactly the speed of light: $\displaystyle c=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}$

That was Maxwell’s astonishing conclusion: Light is an electromagnetic wave.

A changing magnetic field creates a curling electric field. A changing electric field creates a curling magnetic field. The two fields bootstrap each other forward through space. The same calculation gives a wave equation for the magnetic field:

$\displaystyle \nabla^{2}\mathbf{B}=\mu_{0}\varepsilon_{0}\frac{\partial^{2}\mathbf{B}}{\partial t^{2}}$

So both $\displaystyle \mathbf{E}$ and $\displaystyle \mathbf{B}$ propagate as waves.

Suppose an electromagnetic wave travels in the $\displaystyle x$ -direction. Let

$\displaystyle \mathbf{E}=E_{0}\cos(kx-\omega t)\,\hat{\mathbf{y}}$

Then Maxwell’s equations imply

$\displaystyle \mathbf{B}=B_{0}\cos(kx-\omega t)\,\hat{\mathbf{z}}$

The electric field points in the $\displaystyle y$ -direction. The magnetic field points in the $\displaystyle z$ -direction. The wave travels in the $\displaystyle x$ -direction.

Thus $\displaystyle \mathbf{E}\perp\mathbf{B}\perp\text{direction of propagation}$

The speed relation is $\displaystyle \omega=ck$ and the magnitudes satisfy $\displaystyle E_{0}=cB_{0}$ Energy flows in the direction of the Poynting vector:

$\displaystyle \mathbf{S}=\frac{1}{\mu_{0}}\mathbf{E}\times\mathbf{B}$

So the geometry is simple: $\displaystyle \mathbf{E}\times\mathbf{B}$ points in the direction the wave travels.

Potentials

Because $\displaystyle \nabla\cdot\mathbf{B}=0$ we can write the magnetic field as a curl:

$\displaystyle \mathbf{B}=\nabla\times\mathbf{A}$

Now use Faraday’s law: $\displaystyle \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$ . Substitute $\displaystyle \mathbf{B}=\nabla\times\mathbf{A}$ :

$\displaystyle \nabla\times\mathbf{E}=-\frac{\partial}{\partial t}(\nabla\times\mathbf{A})$

Since the time derivative commutes with the curl,

$\displaystyle \nabla\times\mathbf{E}=-\nabla\times\frac{\partial\mathbf{A}}{\partial t}$

Thus $\displaystyle \nabla\times\left(\mathbf{E}+\frac{\partial\mathbf{A}}{\partial t}\right)=0$

A curl-free field is locally a gradient. Therefore there exists a scalar potential $\displaystyle \Phi$ such that

$\displaystyle \mathbf{E}+\frac{\partial\mathbf{A}}{\partial t}=-\nabla\Phi$

So $\displaystyle \mathbf{E}=-\nabla\Phi-\frac{\partial\mathbf{A}}{\partial t}$ and $\displaystyle \mathbf{B}=\nabla\times\mathbf{A}$ . These two definitions automatically satisfy

$\displaystyle \nabla\cdot\mathbf{B}=0$

and

$\displaystyle \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$

So two of Maxwell’s equations become identities. The remaining two equations determine the potentials.

Gauge Freedom

The potentials are not unique. For any smooth scalar function $\displaystyle \psi(\mathbf{x},t)$ , define new potentials by

$\displaystyle \mathbf{A}'=\mathbf{A}+\nabla\psi$

and

$\displaystyle \Phi'=\Phi-\frac{\partial\psi}{\partial t}$

These look like different potentials. But they produce the same physical fields.

First compute the transformed magnetic field: $\displaystyle \mathbf{B}'=\nabla\times\mathbf{A}'$ . Substitute $\displaystyle \mathbf{A}'=\mathbf{A}+\nabla\psi$ :

$\displaystyle \mathbf{B}'=\nabla\times(\mathbf{A}+\nabla\psi)$

so $\displaystyle \mathbf{B}'=\nabla\times\mathbf{A}+\nabla\times\nabla\psi$

But the curl of a gradient is always zero: $\displaystyle \nabla\times\nabla\psi=0$

Therefore

$\displaystyle \mathbf{B}'=\nabla\times\mathbf{A}=\mathbf{B}$

So the magnetic field is unchanged.

Now compute the transformed electric field: $\displaystyle \mathbf{E}'=-\nabla\Phi'-\frac{\partial\mathbf{A}'}{\partial t}$ . Substitute the transformed potentials:

$\displaystyle \mathbf{E}'=-\nabla\left(\Phi-\frac{\partial\psi}{\partial t}\right)-\frac{\partial}{\partial t}(\mathbf{A}+\nabla\psi)$

Expand both terms:

$\displaystyle \mathbf{E}'=-\nabla\Phi+\nabla\frac{\partial\psi}{\partial t}-\frac{\partial\mathbf{A}}{\partial t}-\frac{\partial}{\partial t}(\nabla\psi)$

Since space and time derivatives commute,

$\displaystyle \nabla\frac{\partial\psi}{\partial t}=\frac{\partial}{\partial t}(\nabla\psi)$

These two terms cancel. Hence

$\displaystyle \mathbf{E}'=-\nabla\Phi-\frac{\partial\mathbf{A}}{\partial t}$

Therefore $\displaystyle \mathbf{E}'=\mathbf{E}$

So both physical fields are unchanged:

$\displaystyle \mathbf{E}'=\mathbf{E},\qquad \mathbf{B}'=\mathbf{B}$

This is gauge freedom. The transformation

$\displaystyle \mathbf{A}\mapsto \mathbf{A}+\nabla\psi$

$\displaystyle \Phi\mapsto \Phi-\frac{\partial\psi}{\partial t}$

changes the potentials, but it does not change the electromagnetic fields. In compact form,

$\displaystyle (\Phi,\mathbf{A})\sim\left(\Phi-\frac{\partial\psi}{\partial t},\mathbf{A}+\nabla\psi\right)$

This is gauge invariance. The meaning is particularly important: the potentials contain redundancy. Many different pairs $\displaystyle (\Phi,\mathbf{A})$ describe the same physical electromagnetic field. The fields $\displaystyle \mathbf{E}$ and $\displaystyle \mathbf{B}$ are observable. The particular choice of potentials is partly a matter of description.

Lorenz Gauge and Wave Equations for Potentials

Choose the Lorenz gauge:

$\displaystyle \nabla\cdot\mathbf{A}+\frac{1}{c^{2}}\frac{\partial\Phi}{\partial t}=0$

Then Maxwell’s equations become clean wave equations for the potentials:

$\displaystyle \nabla^{2}\Phi-\frac{1}{c^{2}}\frac{\partial^{2}\Phi}{\partial t^{2}}=-\frac{\rho}{\varepsilon_{0}}$

and

$\displaystyle \nabla^{2}\mathbf{A}-\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial t^{2}}=-\mu_{0}\mathbf{J}$

This is conceptually beautiful.

Charges and currents do not create fields instantaneously. They create potentials that propagate at speed $\displaystyle c$ . The electric and magnetic fields are then derived from those potentials.

The fields are given by

$\displaystyle \mathbf{E}=-\nabla\Phi-\frac{\partial\mathbf{A}}{\partial t}$

and

$\displaystyle \mathbf{B}=\nabla\times\mathbf{A}$

So the potentials are the deeper objects from which the fields are built.

In relativistic notation, the scalar potential and vector potential combine into one four-potential:

$\displaystyle A^{\mu}=\left(\frac{\Phi}{c},\mathbf{A}\right)$

Charge density and current density combine into the four-current:

$\displaystyle J^{\mu}=(c\rho,\mathbf{J})$

Then Maxwell’s equations become, schematically,

$\displaystyle \Box A^{\mu}=\mu_{0}J^{\mu}$

in Lorenz gauge, where

$\displaystyle \Box=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}$

is the wave operator. So the true relativistic structure was already hidden inside Maxwell’s equations before Einstein.

Coulomb Gauge

Another useful gauge is the Coulomb gauge:

$\displaystyle \nabla\cdot\mathbf{A}=0$

Now start with Gauss’s law: $\displaystyle \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}$

Using $\displaystyle \mathbf{E}=-\nabla\Phi-\frac{\partial\mathbf{A}}{\partial t}$

we get

$\displaystyle \nabla\cdot\mathbf{E}=-\nabla^{2}\Phi-\frac{\partial}{\partial t}(\nabla\cdot\mathbf{A})$

But Coulomb gauge says $\displaystyle \nabla\cdot\mathbf{A}=0$

so this reduces to

$\displaystyle -\nabla^{2}\Phi=\frac{\rho}{\varepsilon_{0}}$

Therefore

$\displaystyle \nabla^{2}\Phi=-\frac{\rho}{\varepsilon_{0}}$

This looks instantaneous, like electrostatics.

But physical causality is not violated. The reason is that Coulomb gauge separates the electromagnetic field into two parts. One part is a constraint part, determined by the charge distribution. The other part is a dynamical radiation part, carried by the transverse part of the vector potential $\displaystyle \mathbf{A}$ . The radiation still propagates at the speed of light. The apparent instantaneous feature belongs to the gauge description, not to a violation of physical causality.

Energy in the Electromagnetic Field

Maxwell’s equations imply that fields carry energy. The electromagnetic energy density is

$\displaystyle u=\frac{\varepsilon_{0}}{2}|\mathbf{E}|^{2}+\frac{1}{2\mu_{0}}|\mathbf{B}|^{2}$

The energy flux is the Poynting vector:

$\displaystyle \mathbf{S}=\frac{1}{\mu_{0}}\mathbf{E}\times\mathbf{B}$

There is a local conservation law:

$\displaystyle \frac{\partial u}{\partial t}+\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$

This is Poynting’s theorem. The term $\displaystyle \frac{\partial u}{\partial t}$ is the rate of change of electromagnetic energy density. The term $\displaystyle \nabla\cdot\mathbf{S}$ measures electromagnetic energy flowing out of a region.

The term $\displaystyle \mathbf{J}\cdot\mathbf{E}$ is the rate at which the field does work on matter. If there are no currents, so that $\displaystyle \mathbf{J}=0$ , then the conservation law becomes

$\displaystyle \frac{\partial u}{\partial t}+\nabla\cdot\mathbf{S}=0$

So electromagnetic energy is locally conserved. This makes fields physically real. They are not just bookkeeping devices. They carry energy and momentum.

Lagrangian Formulation

The electromagnetic field can be derived from an action principle. One form of the Lagrangian density is

$\displaystyle \mathcal{L}=\frac{\varepsilon_{0}}{2}|\mathbf{E}|^{2}-\frac{1}{2\mu_{0}}|\mathbf{B}|^{2}-\rho\Phi+\mathbf{J}\cdot\mathbf{A}$

The signs matter. The Hamiltonian energy density is

$\displaystyle \mathcal{H}=\frac{\varepsilon_{0}}{2}|\mathbf{E}|^{2}+\frac{1}{2\mu_{0}}|\mathbf{B}|^{2}$

The Hamiltonian contains electric energy plus magnetic energy. The Lagrangian contains electric energy minus magnetic energy, analogous to kinetic energy minus potential energy. In relativistic notation, the field strength tensor $\displaystyle F_{\mu\nu}$ packages the electric and magnetic fields into one geometric object.

The Lagrangian becomes

$\displaystyle \mathcal{L}=-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}-J_{\mu}A^{\mu}$

Varying the four-potential $\displaystyle A^{\mu}$ gives Maxwell’s equations. This viewpoint is powerful because it shows that electromagnetism is governed by four principles: locality, Lorentz invariance, gauge invariance, and least action. Maxwell’s equations are not merely empirical laws. They are the Euler-Lagrange equations of the simplest gauge-invariant relativistic field theory.

Differential Forms: The Clean Geometric Version

Vector calculus is useful, but it is not the most natural language for Maxwell’s equations. In vector calculus, Maxwell’s equations are split into separate statements about divergence and curl. We write one equation for $\displaystyle \nabla\cdot\mathbf{E}$ , another for $\displaystyle \nabla\cdot\mathbf{B}$ , another for $\displaystyle \nabla\times\mathbf{E}$ , and another for $\displaystyle \nabla\times\mathbf{B}$ .

That works, but it hides the geometry. Differential forms give a cleaner language. They treat electromagnetism as geometry on spacetime, not as four separate-looking vector equations.

Use spacetime coordinates $\displaystyle (t,x,y,z)$

or, relativistically, $\displaystyle x^{\mu}=(ct,x,y,z)$

The electromagnetic potential is not only the vector potential $\displaystyle \mathbf{A}$ . It also includes the scalar potential $\displaystyle \Phi$ . In ordinary vector notation, the potentials are $\displaystyle \Phi$ and $\displaystyle \mathbf{A}=(A_x,A_y,A_z)$

The electric and magnetic fields are obtained from them by

$\displaystyle \mathbf{E}=-\nabla\Phi-\frac{\partial\mathbf{A}}{\partial t}$

and

$\displaystyle \mathbf{B}=\nabla\times\mathbf{A}$

In differential forms, these two formulas are packaged into one formula:

$\displaystyle F=dA$

Here $\displaystyle A$ is the electromagnetic potential one-form, and $\displaystyle F$ is the electromagnetic field two-form.

A useful coordinate way to write the potential one-form is

$\displaystyle A=-\Phi\,dt+A_x\,dx+A_y\,dy+A_z\,dz$

This says that $\displaystyle A$ has one time component, coming from $\displaystyle \Phi$ , and three space components, coming from $\displaystyle \mathbf{A}$ . The symbol $\displaystyle d$ is the exterior derivative. It is the geometric derivative that unifies gradient, curl, and divergence. Now apply $\displaystyle d$ to the potential:

$\displaystyle F=dA$

In coordinates, the electromagnetic field two-form has the schematic form

$\displaystyle F=E_x\,dx\wedge dt+E_y\,dy\wedge dt+E_z\,dz\wedge dt+B_x\,dy\wedge dz+B_y\,dz\wedge dx+B_z\,dx\wedge dy$

This is the key coordinate picture. The electric field components multiply space-time area elements:

$\displaystyle dx\wedge dt,\qquad dy\wedge dt,\qquad dz\wedge dt$

The magnetic field components multiply space-space area elements:

$\displaystyle dy\wedge dz,\qquad dz\wedge dx,\qquad dx\wedge dy$

So the electromagnetic field $\displaystyle F$ is a spacetime area-measuring object.

It measures electric flux through space-time planes and magnetic flux through space-space planes. This is why $\displaystyle F$ is called a two-form. A one-form measures directed line elements. A two-form measures directed area elements. The electromagnetic field naturally measures flux through small oriented areas in spacetime. In ordinary vector language, we usually speak about two fields: $\displaystyle \mathbf{E}$ and $\displaystyle \mathbf{B}$ .

But in spacetime, these are not truly separate objects. They are different parts of one electromagnetic field: $\displaystyle F$ . This is the first major simplification.

Now consider the equation $\displaystyle dF=0$ . Since $\displaystyle F=dA$ we have $\displaystyle dF=d(dA)$ . But the exterior derivative always satisfies $\displaystyle d^{2}=0$ . So $\displaystyle d(dA)=0$ and therefore $\displaystyle dF=0$ . This single equation contains two of Maxwell’s equations. To see this more concretely, expand $\displaystyle dF=0$ in coordinates. The purely spatial part gives $\displaystyle \frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}=0$ . That is $\displaystyle \nabla\cdot\mathbf{B}=0$ . So one part of $\displaystyle dF=0$ says there are no magnetic monopoles. Magnetic field lines do not begin or end.

The mixed space-time parts give

$\displaystyle \frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}=-\frac{\partial B_x}{\partial t}$

$\displaystyle \frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}=-\frac{\partial B_y}{\partial t}$

$\displaystyle \frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}=-\frac{\partial B_z}{\partial t}$

These three equations are exactly $\displaystyle \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$ . So the one compact equation $\displaystyle dF=0$ contains both $\displaystyle \nabla\cdot\mathbf{B}=0$ and $\displaystyle \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$ . In ordinary language, it says two things at once: there are no magnetic monopoles, and a changing magnetic field creates a curling electric field. So no magnetic monopoles and Faraday induction are two faces of the same geometric fact. That is the first great compression.

Now we need the other two Maxwell equations. These are the equations with sources: electric charge and electric current. To write them, we use the Hodge star operator: $\displaystyle \star$ . The Hodge star depends on the geometry of spacetime. It converts a differential form into its dual form. In this context, it takes the electromagnetic field $\displaystyle F$ and produces its dual: $\displaystyle \star F$ .

Roughly speaking, the Hodge star turns one kind of spacetime area into the complementary kind of spacetime area. For example, in four dimensions, the dual of the $\displaystyle dx\wedge dy$ plane is related to the $\displaystyle dz\wedge dt$ plane. So $\displaystyle \star F$ rearranges the electric and magnetic pieces of $\displaystyle F$ in a way determined by spacetime geometry.

The sourced Maxwell equations are written as

$\displaystyle d\star F=\mu_{0}J$

Here $\displaystyle J$ is the current three-form. It contains both charge density and current density. In ordinary vector notation, charge-current is described by $\displaystyle \rho$ and $\displaystyle \mathbf{J}=(J_x,J_y,J_z)$ . In differential-form notation, these combine into one current three-form. Schematically,

$\displaystyle J=\rho\,dx\wedge dy\wedge dz-J_x\,dt\wedge dy\wedge dz-J_y\,dt\wedge dz\wedge dx-J_z\,dt\wedge dx\wedge dy$

This looks complicated, but the meaning is simple. The term $\displaystyle \rho\,dx\wedge dy\wedge dz$ measures charge inside a small volume of space. The terms involving $\displaystyle J_x,J_y,J_z$ measure charge flowing through surfaces as time passes. So $\displaystyle J$ is the spacetime version of charge and current together. Now expand the sourced equation $\displaystyle d\star F=\mu_{0}J$ in coordinates. The spatial-volume part gives

$\displaystyle \frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{\partial E_z}{\partial z}=\frac{\rho}{\varepsilon_{0}}$

That is Gauss’s law for electricity:

$\displaystyle \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}$

The three space-time parts give

$\displaystyle \frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z}=\mu_{0}J_x+\mu_{0}\varepsilon_{0}\frac{\partial E_x}{\partial t}$

$\displaystyle \frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}=\mu_{0}J_y+\mu_{0}\varepsilon_{0}\frac{\partial E_y}{\partial t}$

$\displaystyle \frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y}=\mu_{0}J_z+\mu_{0}\varepsilon_{0}\frac{\partial E_z}{\partial t}$

These three equations are exactly $\displaystyle \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}$

So the second differential-form equation contains both Gauss’s law for electricity and the Ampère-Maxwell law.

Putting everything together, Maxwell’s four vector equations in differential forms, they become only two equations:

$\displaystyle dF=0$

and

$\displaystyle d\star F=\mu_{0}J$

The four equations have not disappeared. They have been organized into two spacetime equations. This is the practical meaning of differential-form notation. It is not magic. It is a compact bookkeeping system that respects the geometry of spacetime.

So the whole structure is

$\displaystyle dF=0,\qquad d\star F=\mu_{0}J$

Now comes a beautiful point. Charge conservation follows automatically. Start with the sourced equation:

$\displaystyle d\star F=\mu_{0}J$ . Apply $\displaystyle d$ to both sides:

$\displaystyle d(d\star F)=d(\mu_{0}J)$

Since $\displaystyle \mu_{0}$ is constant, this is $\displaystyle d(d\star F)=\mu_{0}dJ$ . But $\displaystyle d^{2}=0$ so the left-hand side vanishes: