Chapter 2 — The Maxwell Equations

Quote

If you wake up a physicist in the middle of the night and say “Maxwell”, he is sure to say “electromagnetic field”.

— Rudolf Peierls (1962)

Motivation

Electromagnetism concerns the origin and behavior of the vector fields $\mathbf{E}(\mathbf{r},t)$ and $\mathbf{B}(\mathbf{r},t)$, defined operationally by the force they exert on a test charge $q$ moving with velocity $\mathbf{v}$:

$$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$$

This Coulomb-Lorentz force together with the Maxwell equations forms the complete foundation of classical electrodynamics.

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Electric Charge

A brief history: Amber (Greek: ēlektron) was known to attract light objects when rubbed. In 1600, Gilbert systematized such experiments. In 1751, Franklin proposed that rubbing transfers a tangible “electric fluid.” Aepinus introduced the idea that a numerical quantity $Q$ — electric charge — can be assigned to a body.

Today, $Q$ is known to be an intrinsic property of matter. All free-particle charges are integer multiples of the elementary charge:

$$e=1.602\times 10^{-19}\text{C}$$

In general, charge in a volume $V$, on a surface $S$, or along a line $C$ (volume, surface and line densities $\rho$, $\sigma$ and $\lambda$, respectively):

$$Q={\int }_V{\mathrm{d}}^{3}r\rho (\mathbf{r})={\int }_S\mathrm{d}S\sigma ({\mathbf{r}}_S)={\int }_C\mathrm{d}l\lambda (l)$$

Point charge

The classical point charge representation is an useful abstraction, which must be used with care as it often produces unphysical results (such as diverging self-energies and other subtleties that require regularization/renormalization). For example, for a bunch of charges $q_k$ at positions ${\mathbf{r}}_k$:

$$\rho (\mathbf{r})=\sum _k{q}_k\delta (\mathbf{r}-{\mathbf{r}}_k)$$

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Electric Current

Electric charge in organized motion (Galvani with electrodes inducing frog leg contractions and Volta with the invention of battery in 1800).

The current through a surface $S$ is calculated from current density $\mathbf{j}$:

$$I=\frac{\mathrm{d}Q}{\mathrm{d}t}={\int }_S\mathrm{d}\mathbf{S}\cdot \mathbf{j}$$

When a velocity field $\mathbf{v}(\mathbf{r},t)$ characterizes the charge distribution $\rho (\mathbf{r},t)$:

$$\mathbf{j}=\rho \mathbf{v}$$

For charge confined to a surface with surface charge density $\sigma$ and velocity $\mathbf{v}$:

$$\mathbf{K}=\sigma \mathbf{v},I={\int }_C\mathbf{K}\cdot (\hat{\mathbf{n}}\times \mathrm{d}\mathbf{\ell })$$

Intuition

$\mathbf{K}\cdot (\hat{\mathbf{n}}\times \mathrm{d}\mathbf{\ell })$ projects out how much surface current passes through a line element $\mathrm{d}\mathbf{\ell }$ — the vector $\hat{\mathbf{n}}\times \mathrm{d}\mathbf{\ell }$ is perpendicular to the line element and lies in the surface, pointing in the direction that “catches” current flowing across the strip.

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Charge Conservation

Experimental Fact

If charge were not exactly conserved, an electron could spontaneously decay into lighter neutral particles (e.g., $e^{-}\to \nu +\gamma$). The experimental lower bound on the electron lifetime is $>10^{24}$ years (Belli et al., 1999) — roughly $10^{14}$ times the age of the universe. The fact that no such decay has ever been observed is some of our strong direct evidence that electric charge is exactly conserved.

The even stronger local statement of charge conservation is the continuity equation. The change of charge in a volume can happen only through the surface(s) bounding that volume:

$$I={\oint }_S\mathrm{d}\mathbf{S}\cdot \mathbf{j}={\int }_V{\mathrm{d}}^{3}r\nabla \cdot \mathbf{j}$$

But also (minus sign because the charge leaves the volume through $S$):

$$-\frac{\mathrm{d}Q}{\mathrm{d}t}=-{\int }_V{\mathrm{d}}^{3}r\frac{\partial \rho }{\partial t}$$

(partial, because it’s evaluated at each fixed point $\mathbf{r}$ in the fixed integration volume $V$). Equating yields the continuity equation:

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

Eulerian vs. Lagrangian & the Convective Derivative

The continuity equation $\partial \rho /\partial t+\nabla \cdot (\rho \mathbf{v})=0$ is the Eulerian (fixed-point) statement: the rate of change at a grid point equals the net flux convergence. Expanding the divergence:

$$\frac{\partial \rho }{\partial t}+\mathbf{v}\cdot \nabla \rho +\rho \nabla \cdot \mathbf{v}=0⟹\frac{\mathrm{d}\rho }{\mathrm{d}t}+\rho \nabla \cdot \mathbf{v}=0$$

This is the Lagrangian form: the density experienced by a co-moving fluid parcel changes only if the flow is compressible ($\nabla \cdot \mathbf{v}\ne 0$). For incompressible flow, $\mathrm{d}\rho /\mathrm{d}t=0$ — a moving observer riding the fluid sees constant $\rho$.

The $\partial \rho /\partial t$ description talks about what happens at fixed grid points. The $\mathrm{d}\rho /\mathrm{d}t$ description talks about what happens along worldlines.

Application: Moving Point Charges and the Chain Rule Subtlety

For $\rho (\mathbf{r},t)={\sum }_k{q}_k\delta (\mathbf{r}-{\mathbf{r}}_k(t))$, the field is defined at every fixed point $\mathbf{r}$ (Eulerian), but the time dependence enters through the worldlines ${\mathbf{r}}_k(t)$ (Lagrangian information).

Taking $\partial /\partial t$ at fixed $\mathbf{r}$, the only thing that moves is the argument ${\mathbf{r}}_k(t)$. Let $\mathbf{s}=\mathbf{r}-{\mathbf{r}}_k(t)$, so $\partial s_i/\partial t=-v_{k,i}$:

$$\frac{\partial }{\partial t}\delta (\mathbf{r}-{\mathbf{r}}_k(t))=-{\mathbf{v}}_k\cdot \nabla \delta (\mathbf{r}-{\mathbf{r}}_k)$$

This Is NOT the Convective Term $-{\mathbf{v}}_k\cdot \nabla \delta$ looks like a convective derivative, but it's the Eulerian $\partial /\partial t$ — it arises from the chain rule because the source is moving, not because the observer is moving.

The

Since ${\mathbf{v}}_k(t)$ is a property of particle $k$ and doesn’t depend on $\mathbf{r}$, we have $\nabla \cdot {\mathbf{v}}_k=0$, so:

$$\frac{\partial \rho }{\partial t}=-\nabla \cdot \sum _k{q}_k{\mathbf{v}}_k\delta (\mathbf{r}-{\mathbf{r}}_k)=-\nabla \cdot \mathbf{j}$$

The continuity equation falls out, and we’ve identified $\mathbf{j}(\mathbf{r},t)={\sum }_k{q}_k{\mathbf{v}}_k\delta (\mathbf{r}-{\mathbf{r}}_k)$ without ever postulating it — it’s forced by charge conservation plus the microscopic worldlines.

The full convective derivative for an observer moving at velocity $\mathbf{u}$:

$$\frac{\mathrm{d}}{\mathrm{d}t}\delta (\mathbf{r}-{\mathbf{r}}_k)=\underset{\partial /\partial t\text{ (chain rule)}}{\underbrace{-{\mathbf{v}}_k\cdot \nabla \delta }}+\underset{\text{observer motion}}{\underbrace{\mathbf{u}\cdot \nabla \delta }}=(\mathbf{u}-{\mathbf{v}}_k)\cdot \nabla \delta$$

Setting $\mathbf{u}={\mathbf{v}}_k$ (ride on the charge): $\mathrm{d}\delta /\mathrm{d}t=0$ — if you’re sitting on the charge, the delta function is always centered on you.

Observer velocity	$\mathrm{d}\delta /\mathrm{d}t$	Physical meaning
$\mathbf{u}=0$ (fixed)	$-{\mathbf{v}}_k\cdot \nabla \delta$	Charge rushes past you
$\mathbf{u}={\mathbf{v}}_k$ (co-moving)	$0$	Charge is always right here
General $\mathbf{u}$	$(\mathbf{u}-{\mathbf{v}}_k)\cdot \nabla \delta$	Relative motion determines the change

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The Maxwell Equations

Building Up: Electrostatics

Priestley, Cavendish, and Coulomb established the force between charged objects, with superposition (via an integral here) — the field produced by charges is the vector sum of constituent fields:

$$\mathbf{E}(\mathbf{r})=\frac{1}{4\pi {ϵ}_0}\int {\mathrm{d}}^3{r}^{\prime }\rho ({\mathbf{r}}^{\prime })\frac{\mathbf{r}-{\mathbf{r}}^{\prime }}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}$$

Recalling ${\nabla }^2\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}=-\nabla \cdot \frac{\mathbf{r}-{\mathbf{r}}^{\prime }}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}=-4\pi \delta (\mathbf{r}-{\mathbf{r}}^{\prime })$, this immediately gives:

$$\nabla \cdot \mathbf{E}=\frac{\rho }{{ϵ}_0}\text{and}\nabla \times \mathbf{E}=0\text{(electrostatics only)}$$

The Concept of Field

Introduced by Faraday. In the static case, $\mathbf{E}$ is a bookkeeping device for instantaneous action at a distance. But in the time-dependent case, the field becomes an entity of its own — endowed with energy, momentum, and angular momentum.

Magnetostatics

Gilbert’s De Magnete (1600); Michell (1750); Ørsted’s 1820 discovery that current-carrying wires produce effects similar to magnets; Ampère’s force law on a closed loop: $\mathbf{F}=\oint I\mathrm{d}\mathbf{\ell }\times \mathbf{B}(\mathbf{r})$, with the Biot–Savart law (1820):

$$\mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\sum _k\oint I_k\frac{\mathrm{d}{\mathbf{\ell }}_k\times (\mathbf{r}-{\mathbf{r}}_k)}{|\mathbf{r}-{\mathbf{r}}_k{|}^3}$$

Substituting $I\oint \mathrm{d}\mathbf{\ell }\to \int {\mathrm{d}}^3{r}^{\prime }\mathbf{j}$, we define the magnetic field of any time-independent current density:

$$\mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int {\mathrm{d}}^3{r}^{\prime }\frac{\mathbf{j}({\mathbf{r}}^{\prime })\times (\mathbf{r}-{\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}$$

In 1851, William Thomson (Lord Kelvin) showed:

$$\nabla \cdot \mathbf{B}=0\text{and}\nabla \times \mathbf{B}={\mu }_0\mathbf{j}\text{(magnetostatics only)}$$

Faraday’s Law

Faraday — the greatest experimentalist of the 19th century (his Diary and Experimental Researches in Electricity are worth reading). Together with Neumann, Helmholtz, Thomson, Weber, and Maxwell, the key observation was that a changing magnetic flux through a circuit drives a current:

$$-\frac{\mathrm{d}}{\mathrm{d}t}{\int }_S\mathrm{d}\mathbf{S}\cdot \mathbf{B}=IR$$

Combined with Ohm’s law $IR=\oint \mathrm{d}\mathbf{\ell }\cdot \mathbf{E}$:

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

The Displacement Current

Maxwell’s great contribution. Today we argue for it via the continuity equation: taking $\nabla \cdot$ of Ampère’s law $\nabla \times \mathbf{B}={\mu }_0\mathbf{j}$ gives $0={\mu }_0\nabla \cdot \mathbf{j}$, which contradicts $\partial \rho /\partial t+\nabla \cdot \mathbf{j}=0$ unless $\rho$ is static.

The Standard Textbook Capacitor Argument

Consider a charging capacitor. Current $\mathbf{j}$ flows in the wire but not between the plates. An Amperian loop around the wire gives $\oint \mathrm{d}\mathbf{\ell }\cdot \mathbf{B}={\mu }_{0}I$, but a surface through the gap encloses no current. To resolve the contradiction, add ${\mu }_0{ϵ}_0\partial \mathbf{E}/\partial t$ (the displacement current) — between the plates, ${ϵ}_0\partial \mathbf{E}/\partial t$ carries exactly the “missing” current.

Maxwell's Own Reasoning

Maxwell did not associate displacement current with charge in motion — he envisioned rotating hexagonal vortices with interposed ball bearings, and inferred the displacement current from the kinematics of that mechanical model. But since adding it led to electromagnetic waves propagating at speed $c$, he was presumably convinced, and so was the rest of the world upon Hertz’s 1888 detection of EM waves.

The Four Equations

$$\begin{array}{rlrl}\nabla \cdot \mathbf{E}& =\frac{\rho }{{ϵ}_0}& \nabla \times \mathbf{E}& =-\frac{\partial \mathbf{B}}{\partial t}\\ \nabla \cdot \mathbf{B}& =0& \nabla \times \mathbf{B}& ={\mu }_0\mathbf{j}+{\mu }_0{ϵ}_0\frac{\partial \mathbf{E}}{\partial t}\end{array}$$

Integral forms (from the divergence and Stokes’ theorems):

$${\oint }_S\mathrm{d}\mathbf{S}\cdot \mathbf{E}=\frac{Q_{\text{enc}}}{{ϵ}_0}{\oint }_C\mathrm{d}\mathbf{\ell }\cdot \mathbf{E}=-\frac{\mathrm{d}}{\mathrm{d}t}{\int }_S\mathrm{d}\mathbf{S}\cdot \mathbf{B}$$ $${\oint }_S\mathrm{d}\mathbf{S}\cdot \mathbf{B}=0{\oint }_C\mathrm{d}\mathbf{\ell }\cdot \mathbf{B}={\mu }_0{I}_{\text{enc}}+{\mu }_0{ϵ}_0\frac{\mathrm{d}}{\mathrm{d}t}{\int }_S\mathrm{d}\mathbf{S}\cdot \mathbf{E}$$

Why $\mathrm{d}/\mathrm{d}t$ in the Integral Forms, Not $\partial /\partial t$

The integral ${\int }_S\mathrm{d}\mathbf{S}\cdot \mathbf{B}(\mathbf{r},t)$ integrates out all spatial dependence — the result is a function of $t$ alone, and a function of one variable has only one kind of derivative. The surface $S$ is fixed, so we can pull $\partial /\partial t$ outside the integral freely; once outside, there’s nothing left to hold fixed, so $\partial /\partial t$ and $\mathrm{d}/\mathrm{d}t$ are identical. If the surface were moving (as in Faraday’s law with a moving loop), the Leibniz integral rule would produce extra boundary terms and the distinction would matter.

The force on a distribution ${\rho }^{\ast },{\mathbf{j}}^{\ast }$ due to fields produced by other sources:

$$\mathbf{F}(t)=\int {\mathrm{d}}^{3}r[{\rho }^{\ast }(\mathbf{r},t)\mathbf{E}(\mathbf{r},t)+{\mathbf{j}}^{\ast }(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t)]$$

Heuristic "Derivation" from Symmetry

Assume rotational/inversion symmetry and linearity (superposition). Since $\mathbf{F}$, $\mathbf{E}$, and $\mathbf{v}\times \mathbf{B}$ are all vectors under the Lorentz force, $\mathbf{E}$ is a polar vector and $\mathbf{B}$ is an axial vector.

For source-free, linear, first-order equations: $\partial \mathbf{E}/\partial t\sim \mathbf{E}$ leads to unphysical exponential growth. $\partial \mathbf{E}/\partial t\sim \mathbf{B}$ mixes polar and axial. $\partial \mathbf{E}/\partial t\sim \mathbf{r}\times \mathbf{B}$ breaks translational invariance. But $\nabla$ is a polar vector that’s translationally invariant, so:

$$\frac{\partial \mathbf{E}}{\partial t}\sim \nabla \times \mathbf{B}\text{and}\frac{\partial \mathbf{B}}{\partial t}\sim \nabla \times \mathbf{E}$$

An experiment with two boxes and an electric field (initially no charge in each): turning on the field produces equal and opposite static charges and a negatively directed field on the walls, with no magnetic field detected. By charge conservation, current must have flowed, and since charge and field appear simultaneously, we infer $\partial \mathbf{E}/\partial t\sim \mathbf{j}$. Taking divergences with vanishing initial $\nabla \cdot \mathbf{B}$ and $\nabla \cdot \mathbf{E}+k\rho$ gives $\nabla \cdot \mathbf{E}\sim \rho$ and $\nabla \cdot \mathbf{B}=0$.

A more profound route exploits the symmetries of special relativity — starting from Coulomb’s law in one frame and Lorentz-boosting recovers the full Maxwell equations, demonstrating that magnetism is a relativistic effect of electrostatics.

TODO: Add some details on this special relativity approach, and even better for a blog post.

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Range of Validity

The range of classical electromagnetism is extraordinary. But how far down in scale do the underlying microscopic Maxwell equations (which are basis for spatially averaged macroscopic ones) actually hold?

Microscopic validity: Compelling evidence comes from energy levels of one-electron atoms (G.W. Erickson, J. Phys. Chem. Ref. Data 6, 1977). Today, atomic and molecular spectra are routinely computed by combining Maxwell’s theory with non-relativistic or relativistic quantum mechanics. Experiments suggest validity of the Maxwell equations down to the Compton wavelength $\sim 10^{-12}$ m, which we take as the spatial resolution scale of the theory.

No time averaging needed microscopically: Electron motions occur on timescales of $10^{-15}$–$10^{-16}$ s. Unlike spatial averaging (which we do because we can’t resolve atomic-scale variations), there is no reason to average over time — the microscopic Maxwell equations with the instantaneous $\rho (\mathbf{r},t)$ and $\mathbf{j}(\mathbf{r},t)$ correctly predict the fields at each instant, and those rapidly varying fields are what nearby charges actually respond to.

So the microscopic equations are valid — but we rarely want to track $\sim 10^{23}$ charges individually. The passage to the macroscopic theory requires replacing the wildly oscillating microscopic fields with smooth, slowly varying averages.

Spatial Averaging and Macroscopic Fields

In practice, many averaging schemes exist depending on the physical system and scale, based on replacing the extremely rapidly varying microscopic fields with slowly varying macroscopic ones:

$$\mathbf{E}(\mathbf{r},t)=\frac{1}{\Omega }{\int }_{\Omega }{\mathrm{d}}^{3}s{\mathbf{E}}_{\text{micro}}(\mathbf{s}+\mathbf{r},t)$$

then replace $\mathbf{E}(\mathbf{r},t)\to \mathbf{E}(\mathbf{R},t)$ where $\mathbf{R}$ is a low-resolution spatial variable — adjacent points in $\mathbf{R}$-space are separated on average by the diameter of the $\Omega$-sphere as measured in $\mathbf{r}$-space. For gases, $\Omega$ can be the inverse density of atoms; for a crystal, the volume of a unit cell.

Key Requirement

It does not matter much which averaging scheme is used (e.g., a smoothly varying weight function instead of a sharp cutoff). What matters most is that spatial averaging be linear, so that space and time derivatives in the Maxwell equations commute with it. Then the macroscopic Maxwell equations are simply the averages of the microscopic ones.

Bilinear Quantities

This simplicity breaks down for quantities that are bilinear in fields and sources, such as force density $\rho \mathbf{E}$. We simply assume $⟨\rho \mathbf{E}⟩=⟨\rho ⟩⟨\mathbf{E}⟩$ and $⟨\mathbf{j}\times \mathbf{B}⟩=⟨\mathbf{j}⟩\times ⟨\mathbf{B}⟩$. This is unambiguous as long as $\mathbf{F}$ is the total force on an isolated sample of macroscopic matter in vacuum, but subtleties arise when computing the force on a part of a dielectric volume.

Edge effects: At the surface of a conductor, the quantum-mechanical wavefunction spills slightly outside, giving the surface charge a finite thickness $\sim \lambda (z)$. But this width cannot be resolved macroscopically — it becomes $\sigma \delta (z)$. Consequently, the macroscopic field just outside a perfect conductor drops discontinuously to zero just inside.

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Matching Conditions

The non-analytic nature of macroscopic fields at interfaces requires explicit matching conditions. An intuitive way to derive them relies on using step functions for sharp region separations.

Derivation via Step Functions

Model the field across an interface at $z=0$:

$$\mathbf{E}(\mathbf{r})={\mathbf{E}}_R(\mathbf{r})\Theta (z)+{\mathbf{E}}_L(\mathbf{r})\Theta (-z)$$

Allow localized charge: $\rho (\mathbf{r})={\rho }_R\Theta (z)+{\rho }_L\Theta (-z)+\sigma \delta (z)$.

Apply Gauss’s law $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$. The $\Theta$-function terms give the bulk equations in each region. The $\delta$-function terms from $\partial \Theta /\partial z=\pm \delta (z)$ give:

$$\hat{\mathbf{n}}\cdot ({\mathbf{E}}_R-{\mathbf{E}}_L)=\frac{\sigma }{{ϵ}_0}$$

This procedure also highlighting the non-analytic nature of macroscopic fields at surfaces and interfaces.

With ${\hat{\mathbf{n}}}_2$ pointing outward from region 2:

$$\begin{array}{rl}{\hat{\mathbf{n}}}_2\cdot ({\mathbf{E}}_1-{\mathbf{E}}_2)& =\sigma /{ϵ}_0\\ {\hat{\mathbf{n}}}_2\cdot ({\mathbf{B}}_1-{\mathbf{B}}_2)& =0\\ {\hat{\mathbf{n}}}_2\times ({\mathbf{E}}_1-{\mathbf{E}}_2)& =0\\ {\hat{\mathbf{n}}}_2\times ({\mathbf{B}}_1-{\mathbf{B}}_2)& ={\mu }_0\mathbf{K}\end{array}$$

Warning

These apply only to an interface at rest in the frame where the fields are measured. Moving interfaces require the modified conditions of Namias (1988).

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Maxwell Equations in Matter

Kelvin and Maxwell appreciated that fields responsible for forces differ in some essential way from fields induced in matter by external charges. Today we distinguish charge and current densities intrinsic to the matter from fields produced by sources extrinsic to it, but retain the 19th century language due to literature convenience and the amount of intuition built around quantities they introduced.

Neutral matter: The macroscopic charge density (in the sense of Lorentz averaging) is zero everywhere inside — though the microscopic charge density varies rapidly in space. Same goes for macroscopic current density (except in ferromagnetic matter).

Assuming no sources in an isolated sample of matter, no fields are produced. But introducing external ${\rho }_f$ and ${\mathbf{j}}_f$ (“free” in the sense of extrinsic to matter) induces charge reorganization and current flow:

$$\rho ={\rho }_f(\mathbf{r},t)-\nabla \cdot \mathbf{P}(\mathbf{r},t)$$ $$\mathbf{j}={\mathbf{j}}_f+\nabla \times \mathbf{M}(\mathbf{r},t)+\frac{\partial \mathbf{P}}{\partial t}$$

If we define the auxiliary fields:

$$\mathbf{D}(\mathbf{r},t)={ϵ}_0\mathbf{E}+\mathbf{P}\mathbf{H}(\mathbf{r},t)=\frac{1}{{\mu }_0}\mathbf{B}(\mathbf{r},t)-\mathbf{M}(\mathbf{r},t)\text{,}$$

the Maxwell equations in matter are:

$$\begin{array}{rlrl}\nabla \cdot \mathbf{D}& ={\rho }_f& \nabla \times \mathbf{E}& =-\frac{\partial \mathbf{B}}{\partial t}\\ \nabla \cdot \mathbf{B}& =0& \nabla \times \mathbf{H}& ={\mathbf{j}}_f+\frac{\partial \mathbf{D}}{\partial t}\end{array}$$

The matching conditions that change:

$${\hat{\mathbf{n}}}_2\cdot ({\mathbf{D}}_1-{\mathbf{D}}_2)={\sigma }_f{\hat{\mathbf{n}}}_2\times ({\mathbf{H}}_1-{\mathbf{H}}_2)={\mathbf{K}}_f$$

The convenience of $\mathbf{D}$ and $\mathbf{H}$.

They absorb the bound sources into the field definitions — the Maxwell equations then involve only the free (extrinsic) charge and current densities, which are typically what we control experimentally.

Constitutive Relations

We have $1+1+3+3=8$ equations but 12 components of $\mathbf{E},\mathbf{B},\mathbf{D},\mathbf{H}$. The gap is closed by constitutive relations $\mathbf{D}=\mathbf{D}\{\mathbf{E},\mathbf{B}\}$ and $\mathbf{H}=\mathbf{H}\{\mathbf{E},\mathbf{B}\}$.

For low field strengths, linear approximations are valid. In the static limit:

$$\mathbf{D}(\mathbf{r})=ϵ\mathbf{E}(\mathbf{r})\mathbf{B}(\mathbf{r})=\mu \mathbf{H}(\mathbf{r})$$

where $ϵ={ϵ}_0(1+{\chi }_e)$ is the dielectric permittivity and $\mu ={\mu }_0(1+{\chi }_m)$ is the magnetic permeability — macroscopic constants that encode how strongly the material polarizes or magnetizes in response to applied fields.

Equivalently, via the electric susceptibility ${\chi }_e$ and magnetic susceptibility ${\chi }_m$:

$$\mathbf{P}={ϵ}_0{\chi }_e\mathbf{E}\mathbf{M}={\chi }_m\mathbf{H}$$

The susceptibilities measure the proportionality between the induced response ($\mathbf{P}$ or $\mathbf{M}$) and the field that drives it. For vacuum, ${\chi }_e={\chi }_m=0$, recovering $ϵ={ϵ}_0$ and $\mu ={\mu }_0$.

These materials are referred to as “simple” media.

Beyond Simple Media

Linear constitutive relations are known to be incorrect on the microscopic scale. More generally, one uses spatially nonlocal dielectric functions $ϵ(\mathbf{r},{\mathbf{r}}^{\prime })$ which permit the field at one point to influence the response at nearby points. These are usually calculated quantum-mechanically.

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Limits of Classical Electrodynamics

The domain of classical EM is vast, but finite.

Semiclassical regime: Phenomena such as the photoelectric effect and absorption/emission of radiation require quantum treatment of matter, but the EM field can often remain classical — this semiclassical approach produces quantitatively accurate results in many cases (Chapter 20).

QED corrections:

• Vacuum polarization: A bare charge polarizes the quantum vacuum (analogous to screening in a dielectric).
• QED calculations predict breakdown of linearity of the vacuum Maxwell equations at field strengths $\sim 10^{18}$ V/m and $\sim 10^9$ T — though these effects have not yet been detected.
• $\mathbf{E}$ and $\mathbf{B}$ are non-commuting vector operators in QED rather than $c$-number fields; from the uncertainty principle they cannot take on sharp values simultaneously. This non-classical regime of quantum optics emerges when classical fluctuations are suppressed to reveal the quantum effects.

Experimental Tests of New Physics

Modified Coulomb’s law: $1/r^{2+ϵ}$ (a signature of nonzero photon mass). Maxwell confirmed $|ϵ|<5\times 10^{-5}$; contemporary experiments give $|ϵ|<6\times 10^{-17}$.

Magnetic monopoles: Maxwell equations lack full symmetry — a magnetic charge density ${\rho }_m$ is “missing.” With it, duality transforms would mean it’s a matter of convention whether a particle carries electric charge, magnetic charge, or a mixture, as long as they satisfy:

$$c^2{\rho }_e^2+{\rho }_m^2=c^2{\rho }_e^{\prime 2}+{\rho }_m^{\prime 2}$$

If every particle in the universe has the same ratio ${\rho }_m/(c{\rho }_e)$, we can rotate to make ${\rho }_m=0$ for all of them — recovering the Maxwell–Lorentz equations, consistent with all known experiments. But if a particle with a different ratio were discovered, this “rotate away” option disappears and new fundamental physics is required.

Other interesting possibilities: Charge not exactly conserved; electromagnetism not exactly the same in every inertial frame; violation of rotational and/or inversion symmetry. No experimental evidence for any at present.

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Units

We use SI units, designed so that mechanical and electrical energy are measured in the same units. The base seven units are metre ($\text{m}$), kilogram ($\text{kg}$), second ($\text{s}$), Ampere ($\text{A}$), Kelvin ($\text{K}$), mole ($\text{mol}$) and candela ($\text{cd}$).

Until 1983, $c$ was determined experimentally. Then the General Conference on Weights and Measures defined:

$$c=299792458\text{m/s}\text{(exact)}$$

which demotes the metre to a derived unit (the distance light travels in $1/299792458$ seconds).

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Problems

Foundations

2.4 — Necessity of Displacement Current

The magnetostatic equation $\nabla \times \mathbf{B}={\mu }_0\mathbf{j}$ is not consistent with conservation of charge for a general time-dependent charge density. Show that consistency can be achieved using $\nabla \times \mathbf{B}={\mu }_0\mathbf{j}+{\mathbf{j}}_D$ and a suitable choice for ${\mathbf{j}}_D$.

Solution Sketch

Take the divergence: $0={\mu }_0\nabla \cdot \mathbf{j}+\nabla \cdot {\mathbf{j}}_D$. By the continuity equation and Gauss’s law:

$$\nabla \cdot {\mathbf{j}}_D=-{\mu }_0\nabla \cdot \mathbf{j}={\mu }_0\frac{\partial \rho }{\partial t}={\mu }_0{ϵ}_0\frac{\partial }{\partial t}\nabla \cdot \mathbf{E}=\nabla \cdot \left({\mu }_0{ϵ}_0\frac{\partial \mathbf{E}}{\partial t}\right)$$

Since ${\mu }_0{ϵ}_0=1/c^2$, this is satisfied by:

$${\mathbf{j}}_D={\mu }_0{ϵ}_0\frac{\partial \mathbf{E}}{\partial t}$$

2.11 — Ampère-Maxwell Matching Conditions

A surface current density $\mathbf{K}({\mathbf{r}}_S,t)$ flows in the $z=0$ plane between region 1 ($z>0$) and region 2 ($z<0$). Each region contains arbitrary, time-dependent distributions of charge and current.

• a) Use the theta function method to derive a matching condition from $\nabla \times \mathbf{B}={\mu }_0\mathbf{j}+\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$.
• b) Explain how to adapt this to a non-flat interface.

Solution Sketch

(a): Same theta-function procedure as for Gauss’s law: write $\mathbf{B}={\mathbf{B}}_R\Theta (z)+{\mathbf{B}}_L\Theta (-z)$ and similarly for $\mathbf{E}$. Take the curl — the $\delta (z)$ terms from $\nabla \Theta$ give:

$$\hat{\mathbf{n}}\times ({\mathbf{B}}_1-{\mathbf{B}}_2)={\mu }_0\mathbf{K}$$

The $\frac{1}{c^2}\partial \mathbf{E}/\partial t$ term contributes nothing singular at $z=0$ (it’s a regular function times $\Theta$, no $\delta$), so the matching condition is the same as in magnetostatics.

(b): The matching is done infinitesimally close to the surface — at that scale, any smooth surface looks flat. So the same condition applies locally with $\hat{\mathbf{n}}$ the local surface normal.

Modified Electrostatics

2.12 — A Variation of Gauss’ Law (Podolsky, 1942)

Podolsky proposed replacing Gauss’ law with $(1-a^2{\nabla }^2)\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ while retaining $\nabla \times \mathbf{E}=0$.

• (a) Find the electric field for a point charge by solving the Podolsky–Poisson equation. Use the ansatz $\phi (r)=\frac{q}{4\pi {ϵ}_{0}r}u(r)$.
• (b) Suggest a physical meaning for $a$.

Solution Sketch

Since \nabla \times \mathbf E = 0, we can assign a potential to the electric field.

(a): For a point charge $\rho =q\delta (\mathbf{r})$, write $\mathbf{E}=-\nabla \phi$, so $(1-a^2{\nabla }^2){\nabla }^2\phi =-q\delta (\mathbf{r})/{ϵ}_0$. Integrate over an arbitrary sphere to handle the delta function — extract one $\nabla$ to convert from $V$ to $S$ integral and reduce one Laplacian.

With the ansatz $\phi =\frac{q}{4\pi {ϵ}_{0}r}u(r)$, the equation reduces to:

$$a^2\frac{{\mathrm{d}}^{2}u}{\mathrm{d}{r}^2}=u-1$$

with $u(0)=0$ (regularity) and $u(\infty )=1$ (Coulomb limit). Solution:

$$\phi (r)=\frac{q}{4\pi {ϵ}_{0}r}\left(1-e^{-r/a}\right)$$

This is finite at the origin — the Coulomb divergence is eliminated.

(b): The parameter $a$ has dimensions of length. By analogy with meson theory, it can be regarded as the de Broglie wavelength of a particle mediating the modified Coulomb interaction.

Source: B. Podolsky, Physical Review 62, 68 (1942).

2.13 — If the Photon Had Mass

If the photon had mass $m$, the modified Poisson equation becomes ${\nabla }^2\phi =-\rho /{ϵ}_0+\phi /L^2$ with $L=\hslash /(mc)$. Two concentric conducting shells (radii $r_1<r_2$) at common potential $\Phi$. Find the charge on the inner shell and show it vanishes as $L\to \infty$.

Solution Sketch

(a): Between the shells ($\rho =0$), the substitution $\phi (r)=u(r)/r$ gives:

$$\frac{{\mathrm{d}}^{2}u}{\mathrm{d}{r}^2}=\frac{u}{L^2}$$

Solution: $u=A{e}^{r/L}+B{e}^{-r/L}$, so $\phi (r)=\frac{1}{r}\left(A{e}^{r/L}+B{e}^{-r/L}\right)$. Apply BCs $\phi (r_1)=\phi (r_2)=\Phi$:

$$\phi (r)=\Phi \left\{\frac{r_2}{r}\frac{\sinh [(r-r_1)/L]}{\sinh [(r_2-r_1)/L]}+\frac{r_1}{r}\frac{\sinh [(r_2-r)/L]}{\sinh [(r_2-r_1)/L]}\right\}$$

Defining $\Delta =(r_2-r_1)/L$, the electric field $\mathbf{E}=-\nabla \phi =-{\phi }^{\prime }(r)\hat{\mathbf{r}}$:

$$\mathbf{E}(r)=\frac{\Phi \hat{\mathbf{r}}}{\sinh \Delta }\left\{r_2\left[\frac{\sinh [(r-r_1)/L]}{r^2}-\frac{\cosh [(r-r_1)/L]}{rL}\right]+r_1\left[\frac{\sinh [(r_2-r)/L]}{r^2}+\frac{\cosh [(r_2-r)/L]}{rL}\right]\right\}$$

(b): Integrate the modified Gauss’s law $(1-L^2{\nabla }^2)(\nabla \cdot \mathbf{E})=\rho /{ϵ}_0$ over a sphere of radius $r_1$ — the first term gives the enclosed charge via the surface integral at $r_1$ where the field is known.

(c): To leading order when $L\to \infty$:

$$Q\approx \frac{2\pi {ϵ}_0\Phi }{3}\frac{r_1}{L^2}{\left(\frac{r_2}{L}\right)}^2\left(1+\frac{r_1}{r_2}\right)$$

This vanishes as $L\to \infty$ ($m\to 0$): with massless photons, all charge is on the outer shell.

2.14 — A Variation of Coulomb’s Law

Suppose $\phi (r)=\frac{q}{4\pi {ϵ}_0}\frac{1}{r^{1+\eta }}$ with $0<\eta <1$. Compute the potential inside and outside a spherical shell of radius $R$ with uniform surface charge density $\sigma =Q/(4\pi R^2)$.

Solution Sketch

By symmetry, evaluate on the $z$-axis. The distance from a surface element to the observation point is $\sqrt{r^2+R^2-2rR\cos \theta }$. Integrating over the shell:

$$\phi (r)=\frac{Q}{4\pi {ϵ}_0}\frac{1}{1-\eta }\frac{1}{2Rr}\left\{|r+R{|}^{1-\eta }-|r-R{|}^{1-\eta }\right\}$$

Check ($\eta \to 0$): $|r+R{|}^1-|r-R{|}^1=2R$ for $r>R$ and $2r$ for $r<R$, giving $Q/(4\pi {ϵ}_{0}r)$ outside and $Q/(4\pi {ϵ}_{0}R)$ inside — the standard Coulomb results.

TODO: A blog post or something similar with this method, e.g., how we test validity of our theories.