Hook: In today's digest, a short line from a GetAClass article on Habr flashed by — "the transformer paradox: how a wire outside 'knows' what's happening inside." Promising title, but the substance was just a sketch: "magnetic permeability μ doesn't appear in the final formulas, what matters is only that it be sufficient." I dug into what exactly lay behind that sketch and stumbled on the most beautiful illusion in classical electrodynamics — textbooks have been lying about transformers for two hundred years. Or rather, not lying, but telling only half the story. The other half is a wave that quietly leaks from the core into the surrounding space, obeys Maxwell's equations from 1865, and physically connects AirPods charging on the subway, Tesla's dream of a worldwide wireless network, and the forty-watt induction cooktop in your kitchen. And here's what really hooked me: the same "paradox" explains why wireless charging will always be less efficient than a wire — and why Tesla went bankrupt in 1905 on a project that physics couldn't refute but couldn't confirm either.
A classical engineering course tells you about transformers roughly like this: "alternating current in the primary winding creates alternating magnetic flux in the core, this flux penetrates the secondary winding and induces an EMF by Faraday's law ε = −dΦ/dt." Period. Everything works, nothing wobbles. You can go design power substations.
Now — a thought experiment that the GetAClass team carefully laid out in their article. Take an ideal core and gradually increase its magnetic permeability μ, letting it approach infinity. What happens?
With DC — catastrophe. The H field inside the core is set by currents and doesn't depend on μ, but magnetic induction B = μH shoots to infinity. The density of magnetic energy w = BH/2 goes to infinity with it. No physics can handle that, a real ferrite will saturate long before such a regime, and textbooks usually close the topic there.
With AC — much more interesting. In steady state, current amplitude is determined by inductive reactance: I ≈ U/(ωL). And inductance L grows with μ, so as μ→∞ the inductance also goes to infinity, current drops to zero, and nothing flows through the coil except infinitesimal current through infinite inductance. Energy doesn't accumulate in the core: magnetic flux Φ remains constant (it must create a back-EMF balancing the applied voltage), while field strength H = B/μ = Φ/(μS) drops to zero. The product BH → 0, energy density → 0. An ideal core with infinite μ under AC excitation doesn't store energy at all. That's the first non-obvious fact.
And here the actual paradox begins. Take an external conducting loop surrounding the core. Faraday's law requires that an EMF be induced in it — the flux through the loop is changing. But in the limit μ→∞ there's no magnetic field outside the core. H = 0, B = 0, everything "collapsed" inward. What in that case "pushes" on the electrons in the external wire? Where does the EMF come from if there's no field?
The answer sounds absurd, but it's the only one if you honestly solve the entire system of Maxwell's equations, not its quasistatic approximation: the core radiates an electromagnetic wave. That same "near field" that modern electrodynamics describes as a non-propagating reactive component turns out to be a propagating field, just with negligible amplitude.
Here's the key estimate. Take a characteristic transformer size a = 6 cm, mains frequency 50 Hz, wavelength in vacuum λ = c/f = 6·10⁸ cm. The ratio (a/λ)² ≈ 10⁻¹⁶. The magnetic field B* outside the core differs from field B inside by a factor of 10¹⁶. That's not zero. It's a very small number, but it exists, and it's precisely what closes the system of equations. The electric field outside is determined by the flux through the core and the frequency of its change, with μ not appearing in the expression at all — what matters is only that the permeability be "sufficient," and beyond that you can increase it arbitrarily without changing anything.
Roughly speaking, a transformer core at 50 Hz is a monstrously poor antenna. Its radiation is 10¹⁶ times weaker than an ideal dipole antenna of the same size. But in physics there's no concept of "zero," there's a concept of "vanishingly small." The paradox resolves as soon as we acknowledge: any alternating current in a closed loop with finite inductance radiates, the question is only how efficiently. A transformer radiates 10¹⁶ times worse than an ideal antenna — but that's 10¹⁶ times better than nothing.
This is that very "antenna" that the GetAClass team carefully highlighted in their article. They honestly write: "The language of waves gives some understanding" — and that's a cautious, nocturnal formulation, because the authors understand it's hard to believe. A core with μ → ∞ and supposedly "no field" outside actually shines an electromagnetic wave with amplitude in femtotesla. And this wave, catching the secondary winding, creates the EMF.
It's worth stopping here and taking a breath, because behind this "μ doesn't appear" lies a fundamental engineering principle that's not obvious from the textbook. Magnetic permeability is an internal parameter of the medium, and to an external observer it manifests only through how well the core contains magnetic flux. In the limit μ → ∞ the core contains flux perfectly, and further increasing μ adds nothing to capacity — the flux simply has nowhere to grow, it hits the Maxwell boundary conditions at the core edges. That's precisely why there's no μ in formulas for external field: the internal parameter of the medium dropped out of the description as soon as the medium stopped being the "bottleneck" of the system.
This is a beautiful analogy with conductivity in electrical circuits. In an ideal conductor σ → ∞, and resistance also doesn't appear in formulas for current — it zeroed out. In an ideal core μ → ∞, and permeability drops out of formulas for external field. The system's bottleneck determines its behavior, and when the bottleneck is eliminated, the system becomes insensitive to the parameter that once was the bottleneck. This, by the way, is a general principle of asymptotic physics: limiting cases often simplify the system beyond recognition.
But real ferrites are far from this limit. The best industrial cores have μ_r ≈ 10⁴–10⁵, permalloy up to 10⁵–10⁶. This means in a real 50 Hz transformer μ still constrains the system, and part of the energy actually leaks through "insufficient" permeability — hysteresis losses, eddy currents, residual leakage fields. A power transformer design engineer fights to raise μ as high as possible not for aesthetic reasons, but because every percent of magnetic field "leakage" is a loss in efficiency, winding heating, and noise.
Now flip the problem. If the core radiates a field, that radiation can be captured. That's exactly how wireless charging works — Qi, AirFuel, induction cooktops, charging pads for electric vehicles. The whole difference is that the engineer deliberately throws out the core or makes it open, so most of the magnetic flux leaks outward.
Take the classic Qi standard: frequency 100–205 kHz, two coils several centimeters in diameter, distance between them 4–10 mm. Characteristic size a ≈ 5 cm, wavelength λ = c/f ≈ 1.5–3 km. The ratio (a/λ)² ≈ 10⁻¹⁰. That's a million times better than a 50 Hz power transformer — but still monstrously poor by radio engineering standards. And yet, in this near-field regime the inductive coupling between two coils reaches 70–80% efficiency at short distances. Precisely because (a/λ)² is no longer "negligibly small" but "noticeable."
Here's the historical irony. All classical electrical engineering hides from transformer radiation, builds closed magnetic circuits, closes flux inside the core, carefully minimizes field outside. But modern wireless charging does everything to make the field leak outward as strongly as possible. This is the same physical effect — coupling through near field — but with different objective functions. Siemens and Westinghouse in the 1890s dreamed of field staying inside, and Kuok in 2008 (inventor of Qi) dreamed of it leaking out. Physics didn't change. The problem changed.
And here we can't avoid mentioning Nikola Tesla, because he was the first to realize the duality of the transformer-as-antenna. In 1899 in Colorado Springs he built his famous "magnifying transmitter" — a giant resonant transformer without a closed core. Air core, open magnetic circuit, huge leakage field. In modern terminology, he built a powerful near-field radiator, counting on powering the whole Earth through standing waves in the earth's crust.
In 1901 construction began on Long Island of Wardenclyffe Tower — a 57-meter wooden tower with a 55-ton copper dome, a 37 m deep shaft, and sixteen iron pipes driven another 90 m into the ground. A 200 kW Westinghouse generator. Goal — to transmit not only telegraph signals but electrical power wirelessly over industrial distances.
Money ran out in 1905, J.P. Morgan refused additional financing, the tower was demolished for scrap metal in 1917, Tesla died in debt in 1943. Classic story of "mad inventor bankrupt on castles in the air." But the physics here isn't as simple as it seems.
Tesla's problem wasn't "madness" or ignorance of electrodynamics. The problem was that to transmit power you need energy transfer, and for energy transfer you need a radiating antenna with high efficiency. Tesla tried to build a giant low-frequency antenna with a/λ ≈ 10⁻³ — that's a trillion times better than a power transformer, but still negligible by the standards of what's needed for efficient long-range transmission. Antenna theory (which was just being born then — work by Lee de Forest, Ernst Alexanderson, then Reuter and Smith in the 1930s) would have shown him that his giant tower operates in "monstrously poor antenna" mode, where energy transmission efficiency falls as (a/λ)⁶ in the far zone. That is, to transmit 1 kW over 100 km, you need an antenna with diameter of tens of kilometers, otherwise no transmitter power will save you.
In this sense Wardenclyffe Tower was a physically correct experiment but economically impossible device. Tesla didn't go bankrupt because the idea was wild — he went bankrupt because nature has a minimum size for an efficient antenna at a given frequency, and at his needed frequency that size was larger than a continent. Modern science knows this precisely: for wireless power transmission over any significant distances you need either microwaves (a/λ ~ 1, which allows beam focusing), or lasers, or superconducting resonators (which narrows the working space to the near zone and means "long-range" transmission is millimeters). Tesla in 1901 had neither microwave electronics, nor lasers, nor superconductivity. He only had a dream and Maxwell's equations, which he knew better than most of his contemporaries, by the way.
The most astounding thing in this story — Maxwell's equations from 1865 already contain everything needed to understand the transformer, wireless charging, and Tesla's dream. The question was only which terms of the equations we solve. A 1900 textbook solved only the static part (rot E = −∂B/∂t, div B = 0) and got "field stays in the core." A 1950 textbook added rot H = J + ∂D/∂t and started talking about "skin effect" and "parasitic capacitance." A 2025 textbook fully solves the system and says: "radiation exists, it's finite, its amplitude scales as (a/λ)², and for a power transformer it's negligibly small, while for wireless charging it's dominantly large."
And here's another non-trivial point. Maxwell's equations are linear. This means if you have a "real" transformer with μ = 10⁴, and you place a tiny resonator next to it tuned to the same frequency, that resonator will catch radiation, even if everything is "hidden" in the core. Actually, that's exactly how all modern contactless current sensors (current transformers) and RFID readers work — they catch a tiny fraction of what "by the textbook" should stay inside the winding. And in this sense Tesla wasn't a crank: he just understood very early that "radiation is always there," and tried to build a system where "always" turns into "enough."
This story is about three things that are boring separately but together form a complete picture.
First — about physics. Maxwell's equations work. They're not an "approximation," not a "model," they're a description of reality where field either exists or doesn't. The transformer paradox isn't a paradox of physics, it's a paradox of our ignorance of physics. Solve the entire system honestly once and the paradox evaporates. The core radiates. Field leaks. External wire catches it. Magnetic permeability μ is nothing more than a parameter determining how much field leaks, not whether it leaks at all.
Second — about engineering. The limiting case μ → ∞ isn't a theoretical abstraction but a working designer's tool. When the system's bottleneck is eliminated, the parameter that created it drops out of final formulas. This, by the way, is a universal engineering principle: if you optimize a system to the limit, key parameters often disappear from final expressions, and design reduces to "sufficiency" not "optimality." In programming this sounds like "if your service processes a request in 1 ms, you don't need to think what language it's written in — you need to think about the bus, network, and database." In electrical engineering it sounds like "if μ is sufficient, design frequency, not the core."
Third — about history of ideas. Tesla, Wardenclyffe, wireless power transmission — this isn't "mad dream," it's a physically correct but economically premature idea. What he tried to do in 1901 is in principle possible — a resonant transformer works in the near field, every Qi charger in the world confirms this. But on industrial scale — impossible, because radiation efficiency scales as (a/λ)⁶ in the far zone, and you need either microwaves, or lasers, or superconductors. Tesla died 80 years before the invention of the magnetron, 100 years before injection lasers, and 70 years before the discovery of high-temperature superconductivity. He was ahead of engineering infrastructure by a century. That's not madness. That's tragic luck — being born too early.
And here's what personally hooked me in this story. The transformer paradox is a trap that catches everyone who learns physics from a textbook. The textbook says "field stays in the core," and the student believes. The textbook says "magnetic permeability determines coupling," and the student believes. And then it turns out "stays in the core" is an approximation, and μ determines only how good that approximation is. And the student who didn't dig in to solve Maxwell's equations himself will never learn that a transformer is an antenna. He'll design power substations and charge AirPods without understanding he's holding the same physical effect in his hands.
Maxwell's equations from 1865 contain everything. And that's perhaps the deepest engineering lesson: never trust a textbook that doesn't show which terms of the equations it threw out. The textbook doesn't lie, but the textbook leaves things unsaid. And in the unsaid — all the beauty.
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