Case Study 1 — Maxwell's Equations and the Discovery of Radio Waves

Domain: Physics and electrical engineering. The advance: In the 1860s, vector calculus predicted the existence of radio waves and revealed that light is electromagnetism — two decades before anyone built an antenna. This is the cleanest example in all of science of pure calculus discovering a new piece of reality.

A problem stitched together from fragments

By 1860, electricity and magnetism were a pile of separate empirical laws. Coulomb had measured the force between charges. Gauss had related the electric field's outflow to the charge inside a surface. Faraday had shown that a changing magnetic field pushes a current around a loop. Ampère had related magnetic fields to the currents that make them. Each law worked. None of them obviously belonged together, and nobody could say what, if anything, the collection meant.

James Clerk Maxwell, a Scottish physicist with an unusual gift for geometric imagination, set out to write the whole pile in one mathematical language — the vector calculus you learned in Part VII of this book. In modern differential form, the four laws become four equations:

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}, \qquad \nabla \cdot \mathbf{B} = 0,$$ $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \qquad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}.$$

Read them with the eyes Chapter 37 gave you. The divergence $\nabla\cdot$ measures how much a field spreads out from a point; the first equation says electric field lines spread out from charge, and the second says magnetic field lines never do (there are no magnetic monopoles). The curl $\nabla\times$ measures how much a field circulates around a point; the third equation (Faraday) says a changing magnetic field makes the electric field circulate, and the fourth (Ampère, with Maxwell's crucial addition) says currents and changing electric fields make the magnetic field circulate. Every operator here is from Chapter 37; the time derivatives are the partials of Chapter 29.

Maxwell's one inspired addition

The pre-Maxwell version of the fourth equation read simply $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$ — magnetic circulation comes only from real current. Maxwell noticed this was mathematically broken. Take the divergence of both sides. The divergence of a curl is always zero (a identity you proved in Chapter 37), so the left side gives $0 = \mu_0\,\nabla\cdot\mathbf{J}$, forcing $\nabla\cdot\mathbf{J}=0$. But charge conservation requires $\nabla\cdot\mathbf{J} = -\partial\rho/\partial t$, which is not zero whenever charge is piling up — as it does between the plates of a charging capacitor.

Maxwell repaired the equation by adding the term $\mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$, the displacement current. It was not demanded by any experiment; it was demanded by consistency of the vector calculus. A changing electric field, Maxwell asserted, makes magnetism just as a real current does. That single term — added to make the algebra honest — is the term that produces radio.

The Key Insight. Maxwell did not add the displacement current because he saw it in a lab. He added it because the divergence-of-a-curl identity from Chapter 37 said the equations were inconsistent without it. Calculus told him the laws were incomplete, and told him exactly how to complete them. This is mathematics doing physics' job.

Taking the curl of the curl

Now comes the miracle, and it is a pure vector-calculus exercise you could carry out yourself. Work in a vacuum: no charges ($\rho=0$, so $\nabla\cdot\mathbf{E}=0$) and no currents ($\mathbf{J}=0$). Take the curl of Faraday's law:

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

The left side simplifies by the vector identity $\nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E}$ (Chapter 37). Since $\nabla\cdot\mathbf{E}=0$ in vacuum, the first term vanishes and the left side is just $-\nabla^2\mathbf{E}$. On the right, substitute Ampère's vacuum law $\nabla\times\mathbf{B} = \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$:

$$-\nabla^2\mathbf{E} = -\frac{\partial}{\partial t}\left(\mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}\right) = -\mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}.$$

Cancel the minus signs and you are left with

$$\nabla^2\mathbf{E} = \mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}.$$

This is the wave equation — the same partial differential equation that governs a vibrating string or a sound wave, here governing the electric field. The identical derivation gives the same equation for $\mathbf{B}$. Out of four static-sounding laws about charges and currents, the calculus has produced waves that travel through empty space.

Calculus names the speed of light

A wave equation $\nabla^2 u = \frac{1}{v^2}\,\partial^2 u/\partial t^2$ always describes a disturbance traveling at speed $v$. Matching our equation to that template, the electromagnetic wave must travel at

$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}.$$

Here $\mu_0$ and $\varepsilon_0$ are constants you measure with a tabletop apparatus of coils and capacitors — they have nothing, on their face, to do with light. Plug in the measured values $\mu_0 = 4\pi\times10^{-7}$ and $\varepsilon_0 \approx 8.854\times10^{-12}$ (in SI units):

$$\mu_0\varepsilon_0 \approx (1.2566\times10^{-6})(8.854\times10^{-12}) \approx 1.113\times10^{-17},$$ $$c \approx \frac{1}{\sqrt{1.113\times10^{-17}}} \approx 3.00\times10^8 \ \text{m/s}.$$

That number was already known — it was the measured speed of light. Maxwell saw at once what it meant. "We can scarcely avoid the inference," he wrote, "that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena." Light was not a separate thing from electricity and magnetism. It was an electromagnetic wave, and the visible spectrum was just the narrow band our eyes happen to detect.

Real-World Application — the entire wireless world (electrical engineering). Maxwell's prediction implied that electromagnetic waves should exist at every frequency, not only the visible band. In 1887 Heinrich Hertz built a spark-gap transmitter and a loop receiver across his lab and detected exactly the invisible waves Maxwell's calculus had predicted — what we now call radio. Within twenty years Marconi was sending signals across the Atlantic. Today every antenna in your phone, every Wi-Fi router, every radar dish, and every fiber-optic cable is engineered by solving Maxwell's equations under the boundary conditions the hardware imposes. The vector calculus that felt most abstract in Part VII is the part that runs the wireless world.

Why this story belongs in a calculus book

Trace the calculus in what just happened. The four equations are statements about divergence and curl (Chapter 37). Maxwell found them inconsistent using the identity that the divergence of a curl is zero (Chapter 37). The repair turned the system into a wave equation — a partial differential equation, a forward glimpse of the PDE course of §40.7. And the speed $c$ emerged from nothing but the structure of the equations, a vivid instance of theme 5: calculus appears in every quantitative field, and the field does not get to choose what the mathematics says.

It also embodies theme 1, calculus is the mathematics of change, in its starkest form. Faraday's and Ampère's laws are statements about rates of change — a changing $\mathbf{B}$ makes a circulating $\mathbf{E}$, a changing $\mathbf{E}$ makes a circulating $\mathbf{B}$. Couple two rate-of-change laws together and they feed each other forward through space, each field's change generating the other, and a self-sustaining wave rolls outward at the speed of light. The whole of radio, radar, and optics is two derivatives chasing each other through a vacuum.

Discussion Questions

1. Where exactly did calculus, rather than experiment, drive the discovery? Trace the two moments: the displacement-current term (forced by the divergence-of-a-curl identity) and the wave equation (produced by taking the curl of the curl). Neither came from a measurement.

2. Why does $\nabla\cdot\mathbf{B}=0$ encode "no magnetic monopoles"? Recall what divergence measures (Chapter 37). What would a nonzero divergence of $\mathbf{B}$ at a point physically require?

3. The constants $\mu_0$ and $\varepsilon_0$ are measured with capacitors and coils, with no reference to light. Why is it remarkable that $1/\sqrt{\mu_0\varepsilon_0}$ equals the speed of light? What does this say about theme 5?

4. Hertz, who first detected radio waves, reportedly saw "no use whatsoever" for them. What does the gap between his discovery (1887) and Marconi's transatlantic transmission (1901) suggest about the relationship between fundamental and applied science?

5. The wave equation here is a PDE. Section 40.7 names PDEs as a next course. What new questions would a PDE course let you ask about Maxwell's wave equation that this book did not?

A Short Annotated Reading

• Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.). Cambridge. The standard undergraduate path from vector calculus to Maxwell's equations and the wave derivation above. If you want to do the curl-of-the-curl with full rigor, this is where to go.
• Maxwell, J. C. (1865). "A Dynamical Theory of the Electromagnetic Field." Phil. Trans. R. Soc. The original paper in which the speed of light falls out of the equations. Dense, but the inference about light is thrilling to read in Maxwell's own words.
• Forbes, N., and Mahon, B. (2014). Faraday, Maxwell, and the Electromagnetic Field. Prometheus. An accessible narrative history of how the experimental laws became four equations.
• Hertz, H. (1893). Electric Waves. The record of the experiments that confirmed Maxwell's calculus by detecting the predicted radio waves.