A Field Guide to Maxwell’s Equations

Maxwell’s equations are statements about two vector fields defined over space and time: the electric field \(\vE(\vr,t)\), in volts per metre, and the magnetic field \(\vB(\vr,t)\), in teslas. Each is a map \(\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}^3\). The sources are a scalar charge density \(\rho\) and a vector current density \(\vJ\). A separate law, the Lorentz force, says what the fields do to a charge \(q\) moving with velocity \(\vv\):

\[\mathbf{F}=q\left(\vE+\vv\times\vB\right)\]

Maxwell’s equations say what the fields are, given the sources. They are built from two differential operators, divergence and curl, and both are easiest to understand through linear algebra.

Divergence and curl are pieces of the Jacobian→

At any point, a smooth field \(\vF\) has a Jacobian matrix \(J_{ij}=\partial F_i/\partial x_j\), the best linear approximation of how \(\vF\) changes nearby. Split it into symmetric and antisymmetric parts:

\[J=\underbrace{\tfrac12\left(J+J^{\mathsf T}\right)}_{\text{stretch}}+\underbrace{\tfrac12\left(J-J^{\mathsf T}\right)}_{\text{rotation}}\]

The divergence is the trace of the Jacobian, so it measures expansion:

\[\nabla\cdot\vF=\operatorname{tr}J=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}\]

The antisymmetric part has three independent entries, and they are the components of the curl. For any vector \(\mathbf{u}\), \(\tfrac12(J-J^{\mathsf T})\mathbf{u}=\tfrac12(\nabla\times\vF)\times\mathbf{u}\). The curl measures local rotation: a tiny paddle wheel placed in the flow spins with angular velocity \(\tfrac12\nabla\times\vF\).

The integral meaning

Divergence is outward flux per unit volume, and each component of the curl is circulation per unit area, in the limit of a shrinking region:

\[\nabla\cdot\vF=\lim_{V\to0}\frac{1}{V}\oint_{\partial V}\vF\cdot\dA,\qquad \vn\cdot(\nabla\times\vF)=\lim_{A\to0}\frac{1}{A}\oint_{\partial A}\vF\cdot\dl\]

Add up those local statements over a finite region and you get the two theorems that convert every Maxwell equation between its differential and integral forms:

\[\underbrace{\int_V\nabla\cdot\vF\,dV=\oint_{\partial V}\vF\cdot\dA}_{\text{divergence theorem}}\qquad\underbrace{\int_S(\nabla\times\vF)\cdot\dA=\oint_{\partial S}\vF\cdot\dl}_{\text{Stokes’ theorem}}\]

Three identities that shape the equations

\[\nabla\cdot(\nabla\times\vF)=0,\qquad \nabla\times(\nabla\varphi)=\mathbf{0},\qquad \nabla\times(\nabla\times\vF)=\nabla(\nabla\cdot\vF)-\nabla^2\vF\]

Curls have no divergence, and gradients have no curl. Helmholtz’s theorem adds the decisive fact: a field that falls off fast enough at infinity is determined uniquely by its divergence and its curl. In linear-algebra language, square-integrable fields split into two orthogonal subspaces, gradients (curl-free) and curls (divergence-free). That is why Maxwell’s equations come as two divergence equations and two curl equations. Together they pin down \(\vE\) and \(\vB\).

Background

Charles-Augustin de Coulomb measured the force between charged spheres with a torsion balance in 1785 and found the inverse-square law. Carl Friedrich Gauss recast it as a statement about flux; he formulated the law in 1835, and it was published in 1867. The divergence theorem it rests on goes back to Lagrange (1762), Gauss (1813) and Ostrogradsky (1826).

The picture: field lines start on positive charge and end on negative charge. The net number of lines leaving a closed surface counts the charge inside. The shape of the surface does not matter, and a charge outside the surface contributes as many lines going in as coming out.

From Coulomb to Gauss

A point charge \(q\) at the origin has field \(\vE=q\,\hat{\mathbf r}/(4\pi\varepsilon_0 r^2)\). Away from the origin its divergence is zero: in spherical coordinates, \(\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2E_r\right)=0\). Yet the flux through any sphere around the origin is \(q/\varepsilon_0\). All of it comes from the origin, so the divergence is a delta function:

\[\nabla\cdot\frac{\hat{\mathbf r}}{r^2}=4\pi\,\delta^3(\vr)\quad\Longrightarrow\quad\nabla\cdot\vE=\frac{q}{\varepsilon_0}\,\delta^3(\vr)\]

Write any charge distribution as a superposition of point charges, \(\rho(\vr)=\int\rho(\vr')\,\delta^3(\vr-\vr')\,d^3r'\), and linearity gives \(\nabla\cdot\vE=\rho/\varepsilon_0\).

Gauss’s law does not determine \(\vE\) by itself, since a divergence-free field can still swirl. In electrostatics the missing piece is \(\nabla\times\vE=\mathbf 0\), so \(\vE=-\nabla\varphi\), and Gauss’s law becomes Poisson’s equation:

\[\nabla^2\varphi=-\frac{\rho}{\varepsilon_0}\]

Discretize the Laplacian on a grid and Poisson’s equation becomes a linear system \(A\boldsymbol{\varphi}=\mathbf b\), with \(-A\) sparse, symmetric and positive definite. Capacitance extractors, semiconductor device simulators and electrostatic field solvers are, at their core, fast solvers for that system.

Background

William Gilbert’s De Magnete (1600) recorded that a magnet broken in two gives two complete magnets, each with a north and a south pole. Nobody has isolated a single pole since. In 1931 Paul Dirac showed that if even one magnetic monopole existed anywhere, electric charge would have to be quantized, which made the search attractive. Dedicated searches in cosmic rays, in old rock, and at colliders, most recently the MoEDAL experiment at CERN’s Large Hadron Collider, have found none. In Maxwell’s equations, that absence is the zero on the right-hand side where Gauss’s law has \(\rho/\varepsilon_0\).

In the field-line picture, B lines have nowhere to start or stop. They close on themselves or run off to infinity, and every line that enters a closed surface also leaves it.

Three consequences

A vector potential exists. On all of \(\mathbb R^3\), a divergence-free field is the curl of something (a form of the Poincaré lemma). So we can write

\[\vB=\nabla\times\vA\]

\(\vA\) is not unique. Adding any gradient leaves \(\vB\) unchanged because \(\nabla\times\nabla\chi=\mathbf 0\). In linear-algebra terms, the curl operator has a null space (the gradients), and \(\vA\) is only defined modulo that null space. Choosing a representative is called fixing a gauge.

Flux through a loop is well defined. Take two different surfaces \(S_1\) and \(S_2\) spanning the same closed curve. Together they bound a volume, so the net flux out is zero and the flux through each is the same. That is what makes “the magnetic flux through a circuit” meaningful in Faraday’s law.

The normal component is continuous. A thin pillbox across an interface between two materials gives \(B_{1n}=B_{2n}\). Magnetic flux crosses boundaries without loss, which is the basis of magnetic-circuit analysis.

Background

On 29 August 1831, Michael Faraday wound two coils on opposite sides of an iron ring. Closing a switch on one coil made a galvanometer on the other flick and then return to zero; opening the switch made it flick the other way. Steady current did nothing. Only change produced current. Joseph Henry found the same effect independently in the United States.

In 1834 Heinrich Lenz gave the sign rule: the induced current opposes the change that produced it. That is the minus sign, and it is energy conservation. With the opposite sign, an induced current would reinforce the change that created it and grow without limit.

Field law and flux rule

For a stationary loop, Stokes’ theorem turns the differential form directly into the integral form: the EMF around the loop equals the rate of decrease of magnetic flux through it. For a loop that moves or deforms, \(\mathcal E=-d\Phi_B/dt\) still holds, but part of the EMF now comes from the magnetic force \(\vv\times\vB\) on charges in the moving wire. The local statement \(\nabla\times\vE=-\partial\vB/\partial t\) is the more fundamental one.

Voltage is no longer path-independent

When \(\vB\) changes, \(\vE\) has curl, so it is not the gradient of a scalar. In terms of potentials,

\[\vE=-\nabla\varphi-\frac{\partial\vA}{\partial t}\]

The line integral of \(\vE\) between two points depends on the path, so “the voltage between A and B” needs a path to mean anything. Route an oscilloscope probe’s ground lead through changing magnetic flux and the scope reads a voltage that exists only in the loop formed by the probe tip and its ground lead. This is the physics behind short ground springs on probes, twisted pairs, and minimal loop area in circuit layout.

Background

In 1820 Hans Christian Ørsted saw a compass needle swing when current flowed in a nearby wire, the first link between electricity and magnetism. Within months André-Marie Ampère had worked out the forces between current-carrying wires, and Biot and Savart had a formula for the field of a current element. The magnetostatic law \(\nabla\times\vB=\mu_0\vJ\) held in every steady experiment.

Maxwell found a flaw in it in 1861. The divergence of a curl is always zero, so taking the divergence of Ampère’s law forces \(\nabla\cdot\vJ=0\). But charge conservation says

\[\frac{\partial\rho}{\partial t}+\nabla\cdot\vJ=0\]

and \(\nabla\cdot\vJ\) is not zero wherever charge is piling up, such as on a capacitor plate. Gauss’s law gives \(\rho=\varepsilon_0\nabla\cdot\vE\), so \(\nabla\cdot\left(\vJ+\varepsilon_0\,\partial\vE/\partial t\right)=0\) always. Replacing \(\vJ\) with that combination makes the equation consistent:

\[\nabla\cdot(\nabla\times\vB)=\mu_0\left(\nabla\cdot\vJ+\frac{\partial\rho}{\partial t}\right)=0\]

Maxwell called \(\varepsilon_0\,\partial\vE/\partial t\) the displacement current. It is tiny in most circuits and could not be measured directly in his day, but it lets a changing electric field create a magnetic field. Paired with Faraday’s law, that closes the loop that produces waves (Chapter V). Maxwell published the complete theory in 1865, and Heinrich Hertz generated and detected radio waves in 1887–88.

The wave equation

Go to empty space: \(\rho=0\), \(\vJ=\mathbf 0\). Take the curl of Faraday’s law and substitute Ampère–Maxwell:

\[\nabla\times(\nabla\times\vE)=-\frac{\partial}{\partial t}(\nabla\times\vB)=-\mu_0\varepsilon_0\frac{\partial^2\vE}{\partial t^2}\]

Expand the left side with \(\nabla\times(\nabla\times\vE)=\nabla(\nabla\cdot\vE)-\nabla^2\vE\) and use \(\nabla\cdot\vE=0\):

\[\nabla^2\vE=\mu_0\varepsilon_0\frac{\partial^2\vE}{\partial t^2},\qquad\nabla^2\vB=\mu_0\varepsilon_0\frac{\partial^2\vB}{\partial t^2}\]

That is the wave equation, and it fixes the speed:

\[c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}\]

With the CODATA values above, this page computes \(c=\) 299,792,458 m/s. In 1862 Maxwell made the same calculation using Weber and Kohlrausch’s 1856 electrical measurements, got roughly 310,000 km/s, and noticed it agreed with Fizeau’s measured speed of light, roughly 315,000 km/s. He concluded that light is an electromagnetic wave.

Plane waves are an eigenvalue problem

Look for solutions \(\vE=\vE_0\,e^{i(\vk\cdot\vr-\omega t)}\). Each \(\nabla\) becomes \(i\vk\) and each \(\partial/\partial t\) becomes \(-i\omega\), so the four equations become algebra:

\[\vk\cdot\vE_0=0,\qquad\vk\cdot\vB_0=0,\qquad\vk\times\vE_0=\omega\vB_0,\qquad\vk\times\vB_0=-\frac{\omega}{c^2}\vE_0\]

Write \(\vk\times\) as the skew-symmetric matrix

\[[\vk]_\times=\begin{pmatrix}0&-k_z&k_y\\k_z&0&-k_x\\-k_y&k_x&0\end{pmatrix},\qquad\text{eigenvalues }0,\ \pm i|\vk|\]

The zero eigenvector is \(\vk\) itself, a longitudinal wave, and the divergence equations rule it out. That leaves a two-dimensional space of transverse polarizations. Combining the curl equations gives \([\vk]_\times^2\vE_0=-(\omega^2/c^2)\vE_0\), and on the transverse subspace \([\vk]_\times^2=-|\vk|^2\), so

\[\omega=c\,|\vk|\]

This is the dispersion relation of light in vacuum: every frequency travels at the same speed. The relations also give \(|\vB_0|=|\vE_0|/c\), with \(\vE\), \(\vB\) and \(\vk\) forming a right-handed triad, which is the picture at the top of this page.

Why the waves neither grow nor decay

Stack the fields into one state vector \(u=(\vE,\vB)\). The curl equations read

\[\frac{\partial u}{\partial t}=\mathcal L u,\qquad\mathcal L=\begin{pmatrix}0&c^2\,\nabla\times\\-\nabla\times&0\end{pmatrix}\]

Use the energy inner product \(\langle u,u\rangle=\int\left(\varepsilon_0|\vE|^2+|\vB|^2/\mu_0\right)dV\). The curl is self-adjoint for fields that vanish at infinity, and \(\varepsilon_0c^2=1/\mu_0\), so \(\mathcal L\) is skew-adjoint. Its spectrum lies on the imaginary axis, exactly like a linear ODE \(\dot x=Ax\) with \(A=-A^{\mathsf T}\): solutions oscillate and propagate, and the norm, the field energy, is conserved.

Energy and the Poynting vector

Where there are currents, energy can leave the fields as work on charges. The local balance is Poynting’s theorem:

\[\frac{\partial u}{\partial t}+\nabla\cdot\mathbf S=-\vJ\cdot\vE,\qquad u=\tfrac12\left(\varepsilon_0|\vE|^2+\frac{|\vB|^2}{\mu_0}\right),\qquad\mathbf S=\frac{1}{\mu_0}\vE\times\vB\]

Sunlight above the atmosphere carries about 1361 W/m². Setting \(\langle S\rangle=c\varepsilon_0E_{\text{rms}}^2\) gives \(E_{\text{rms}}\approx716\) V/m.