How Maxwell Unified Electricity and Magnetism

Your phone right now is swimming in electromagnetic fields. Wi-Fi, 5G, Bluetooth, the GPS quietly triangulating your position — all of it is riding on a piece of mathematical infrastructure that James Clerk Maxwell assembled in the 1860s. Before him, electricity and magnetism were two separate departments of physics that happened to share a building. After him, they were one thing. And the domino that started it all was a compass needle moving when it had no business doing so.

In 1820, Hans Christian Ørsted noticed that running a current through a wire deflected a nearby compass needle. Reverse the current, the needle flips the other way. Kill the current, north again. The Physics Explained channel's new deep-dive into Maxwell's equations opens here, and rightly so — this is where the whole story turns. As the video puts it: "This was one of the first clear signs that electricity and magnetism were not separate phenomena after all."

That sounds tidy. But let's not let the tidiness flatten what was actually a pretty chaotic intellectual situation.

The lopsided middle

Ørsted's finding kicked off a generation of investigation. André-Marie Ampère showed that current-carrying wires exert forces on each other — same direction, they attract; opposite directions, they repel. Biot and Savart worked out the math for how magnetic field strength drops off with distance from a wire. Ampère then went deeper and formalized the relationship between the circulation of a magnetic field around a closed loop and the current threading through that loop — what we now call Ampère's law.

Then came Faraday. In 1831, he discovered electromagnetic induction: a changing magnetic field can drive a current in a nearby coil, even with no battery, no wire connecting them, nothing. The rate matters — move a magnet toward a coil slowly, you get a small current; move it fast, you get a bigger one. The induced current always fights back against whatever is causing it (that's Lenz's law, baked into the minus sign in Faraday's equation).

Here's where the history gets more interesting than the usual telling admits. Faraday's conceptual framework — field lines, flux, the idea that fields themselves are the fundamental actors — was viewed with suspicion by much of the mathematical physics establishment of his day. Faraday had almost no formal mathematical training. His way of visualizing electromagnetic phenomena through physical field lines struck many of his contemporaries as hand-wavy at best. It wasn't until Maxwell came along and translated Faraday's geometric intuitions into rigorous mathematics that the field picture earned institutional respect. The progression from Faraday to Maxwell wasn't a clean relay race; it was closer to one generation's intuition being quietly vindicated by another generation's algebra.

What Faraday's law established, at the field level, was this: a changing magnetic field creates a circulating electric field. Not attached to any wire, not caused by any battery — just there, in space, because the magnetic field is changing. "If we remove the wire, the current disappears, but the circulating electric field remains," the video explains. The wire doesn't make the field; it just gives the charges somewhere to go.

So by the time Maxwell enters the picture, the situation is structurally awkward. Faraday's law says: changing magnetic field → circulating electric field. But Ampère's law, in its original form, only says: electric current → circulating magnetic field. The relationship is lopsided. One direction has a field-level mechanism; the other only has a charge-level one. Maxwell noticed the asymmetry, and it bothered him.

The capacitor problem, and the fix that changed everything

Maxwell's entry point was a specific, almost administrative-looking contradiction. Imagine a capacitor — two plates separated by a gap — charging up. Draw an imaginary loop around the wire leading to one plate. Ampère's law asks: how much current threads through a surface bounded by that loop?

If you stretch a flat surface through the wire: current passes through, law works. But you can stretch any surface across the same boundary loop — including one that balloons out and passes through the gap between the plates. No charge jumps across that gap. So the original Ampère's law would give you zero, even though the magnetic field around the loop is identical in both cases. One law, same loop, two different answers depending on which imaginary surface you use. That's not physics — that's a broken accounting system.

Maxwell's fix was to say: okay, no charge crosses that gap, but the electric field in there is changing as the capacitor charges, which means the electric flux through that surface is changing. He proposed that a changing electric flux contributes to the circulation of the magnetic field just like a real current does — and he called this contribution the "displacement current."

One nuance worth flagging: Maxwell didn't arrive at displacement current through pure field intuition the way we'd now present it. His original derivation was embedded in a mechanical ether model — he was thinking about literal physical displacements in an imagined medium. The modern reading, where displacement current is understood as a mathematical term encoding the effect of changing electric flux (not a physical current of any kind), came later. The video presents the modern interpretation, which is the right one for understanding the physics — but it's worth knowing that Maxwell got to the right answer through conceptual scaffolding he eventually had to tear down.

The correction lands with satisfying symmetry. After Maxwell: a changing electric field creates a circulating magnetic field. Faraday had already shown the reverse. "Now at the level of changing fields, the dynamical symmetry between electric and magnetic fields is complete," the video notes.

What self-sustaining fields actually mean for you ⚡

Once the symmetry is complete, something wild becomes possible. Strip away all the charges, all the wires, all the currents. In empty space, the two revised equations say: a changing electric field begets a circulating magnetic field, which is itself changing, which begets a circulating electric field, which is changing... and the whole structure can propagate. The fields sustain each other across empty space. That's an electromagnetic wave.

When Maxwell worked out the predicted speed of that wave using the constants in his equations — the permittivity and permeability of free space — the number he got was very close to the measured speed of light at the time, within the measurement precision of mid-19th-century optics. Close enough to make the identification compelling; not a mathematically exact identity, since the precision of the speed of light continued to be refined for over a century afterward. But close enough that Maxwell could write, with justified excitement, that light is an electromagnetic disturbance. Electricity, magnetism, and light: one thing.

The next video in the Physics Explained series promises to pull the wave equation directly out of Maxwell's four equations — worth watching if the math interests you.

The part we don't talk about enough

Here's what I keep sitting with: the electromagnetic spectrum that Maxwell's equations describe is not neutral infrastructure. Radio waves let Marconi transmit across the Atlantic by 1901. Radar let the Allies track aircraft in WWII. And then the same physics — same equations, same propagating fields — became the substrate for everything that now competes for your attention at 2am.

The Wi-Fi carrying TikTok to your phone, the cell towers pinging your location to advertisers, the Bluetooth beacons in retail stores tracking foot traffic — it's all Maxwell. The equations don't care whether the electromagnetic wave is carrying a symphony or a targeted ad optimized by a behavioral model. They propagate either way.

That's not an argument against the physics. It's just a fact that the history of Maxwell's equations is also a history of what humans decided to do with a self-propagating field. Ørsted's compass needle moved in 1820, and we've been figuring out what that means ever since — not just mathematically, but politically, economically, and ethically.

There's something both thrilling and quietly unsettling about the fact that it all started with a bookkeeping error in Ampère's law. Maxwell noticed that using two different surfaces gave two different answers for the same physical quantity. He fixed the math. And buried in the fix was the entire electromagnetic infrastructure of modernity.

I don't think that's a reason to feel awe and nothing else. I think it's a reason to pay attention to what the next "bookkeeping fix" unlocks — and who gets to decide what we do with it.

— Mei Zhang, Buzzrag Biotech & Genetics