How inductance falls out of relativistic electrostatics — a eight-step lecture for the curious reader.
The thesis of this lecture is short: every "magnetic" effect you have ever felt or measured — the click of a relay, the back-kick of an inductor, the attraction between two current-carrying wires — is, at root, the ordinary electric force, just translated through Einstein's special relativity. We will not invoke a magnetic field anywhere in the chain. We will end by deriving the inductance of a real copper coil entirely from accelerating electrons.
Take two long, straight, parallel wires. Run a steady current through both in the same direction. They attract each other. Reverse one, they repel. We call this the magnetic force, write down a Lorentz law F = qv × B, and move on with our lives.
But pause. Each of those wires is electrically neutral. The mobile electrons drifting through the copper are exactly balanced, electron-for-ion, by the positive lattice they sit in. Hold one wire in your hand: no shock. Bring a charged comb near it: nothing. There is no Coulomb force between the two wires, because there is no net charge on either of them. Yet the wires move toward each other when current flows. Where does the attraction come from?
The honest answer used to be: from a new field, the magnetic field, which appears whenever charges move. In 1962, Edward Purcell pointed out something startling. The "new" field is not new at all. It is what the ordinary Coulomb force looks like when you change reference frames. To see this, imagine a tiny test charge drifting along beside the electrons in one of the wires, matching their speed. From its point of view, the electrons in that wire are stationary. But the lattice ions — which were at rest in the lab — are now moving past it. And special relativity tells us that anything moving relative to you appears length-contracted along its direction of motion. The ion lattice, in this moving observer's view, is squeezed: the ions are packed slightly closer together than the electrons. The wire is no longer neutral. It carries a small net positive charge, and that net charge attracts our test electron by ordinary Coulomb's law.
The remarkable thing about this story is not that it works for two parallel wires. It is that it has to work for every configuration of moving charges, anywhere, of any complexity, by the principle of relativity. Whatever a magnetic field does, it does because Coulomb's law in some other reference frame would do the same thing. Magnetism is not optional, but it is not fundamental either. It is bookkeeping.
This raises an irresistible question. The most subtle and useful magnetic phenomenon is not steady attraction between wires — it is induction: the back-EMF of a coil, the kick of a relay opening, the basis of every transformer. Can we derive inductance, too, from electrostatics alone? That is what this lecture builds up to.
Before we tackle a coil, we need a slightly upgraded version of Coulomb's law. The textbook expression — field equals charge over distance squared, pointed radially — is exactly correct for a stationary charge. For a charge that is moving, special relativity slightly distorts the field.
The picture is easiest to absorb visually. A stationary point charge has spherical symmetry: its electric field lines spray out in every direction with equal density. A point charge that is moving fast does not, because length contraction along the direction of motion squeezes the field lines closer together in the perpendicular direction. The faster the charge moves, the more pronounced the squeeze. In the limit where the charge approaches the speed of light, the field collapses into a thin transverse pancake.
If you do the algebra carefully — boost a Coulomb field from the charge's rest frame to the frame of an observer moving past it — you find that the force this distorted field exerts on a second moving charge has exactly two pieces: a part that looks like the ordinary Coulomb force, and a part that, when you squint, looks like the Lorentz magnetic force qv × B. The "magnetic" part is not a separate phenomenon. It is what is left over when you write the relativistic electric force in terms of the lab-frame velocity.
For the rest of this lecture we will use exactly this single idea. The recipe: write down the field of a moving charge with its relativistic correction, compute the force on another charge, and watch what falls out. We will not need B for any step. The constant μ0 that the textbook treatment associates with magnetism will appear at the end automatically — and we will see it is just 1/(ε0c2), the relativistic correction factor wearing a different label.
Before going after a coil, let's exercise the framework on a problem simple enough to do by hand: two electrons whirling around a common axis on the same circular orbit, separated by some fixed angle θ. We want to know the force one exerts on the other in the direction along their motion — the tangential force, the one that, summed up over many electrons, would be the back-reaction we feel as inductance.
You can do the calculation by writing down the field of moving electron 1 at the location of electron 2, applying the relativistic correction (the angular factor that comes from boosting the rest-frame Coulomb field), and projecting the resulting force onto electron 2's tangent direction. The geometry has a useful symmetry: the angle between electron 1's velocity and the line joining the two electrons is exactly half the angular separation, θ/2. After a few lines of algebra the answer is simply
Two sanity checks make this less alarming than it looks. First, set β → 0 — the slow limit. The big fraction collapses to 1, and what is left is exactly the tangential component of the ordinary Coulomb force between two charges separated by the chord 2R sin(θ/2). Second, in this configuration the "magnetic part" of the force (the piece that the standard textbook would attribute to v×B) turns out to be purely radial — it points toward or away from the orbit center and contributes nothing to the tangential motion. So the tangential force, as we've computed it, is exactly what the full Lorentz treatment would give. The relativistic correction shows up only in the angular factor.
This warm-up matters because it shows the framework actually works. Two moving charges, a quantitative answer, no magnetic field invoked anywhere. Now we turn to the harder problem: charges that are accelerating, which is what conduction electrons do whenever a current changes.
Inductance is the property of a coil that resists a change in current. Push more current through it, and it pushes back with a counter-voltage proportional to how fast you're trying to change things — the back-EMF, written −L dI/dt. The textbook story attributes this to the magnetic field: changing current makes a changing magnetic flux, which (by Faraday's law) induces an electric field, which opposes the change. We're going to derive the same result without ever mentioning a magnetic field.
The new ingredient we need is what an accelerating charge looks like at a distance. A charge moving at constant velocity carries with it the squashed Coulomb field of Section 2 — but the squash pattern is the same from one instant to the next, so an observer far away sees a field that doesn't change in time. Now imagine the charge speeds up. The field pattern it carries also changes from one moment to the next, and the change in the field produces something new: an extra electric field at every distant point, pointing along the direction of the charge's acceleration, with magnitude that falls off as 1/distance. This is the "induction field" of an accelerating charge.
Two things to notice about that formula. The 1/c2 is the relativistic correction — the same factor that built the squashed field from Section 2, now appearing as a velocity-dependent contribution to the potential. And the field points along the direction of the charge's acceleration (with a sign flip if the charge is positive). It does not die off as 1/R2 the way the static Coulomb field does — only as 1/R. That slower fall-off is what allows distant accelerating charges to act on each other meaningfully.
Now place that mechanism into a coil. When the current changes, every conduction electron in the wire — there are about 1023 of them in a typical hand-wound solenoid — accelerates simultaneously, in the same coordinated way along the wire. Each one of those accelerating electrons emits the induced field of Figure 4. At any other electron in the coil, you have to add up the contribution from every single accelerating source. The figure below shows the simplest version of this: just two adjacent turns of the coil, viewed edge-on.
If you carry out the bookkeeping carefully — adding up the induced field from every accelerating electron in the coil at every other electron's position, and integrating around the wire — the answer is
That second line is exactly the standard Neumann formula for self-inductance — the same one taught in every textbook treatment that does invoke a magnetic field. The geometric integral is the same. The constant μ0 in front is the same. We just got it a different way: from the relativistic correction 1/(ε0c2) that we put into the induced field of an accelerating charge, with no B in sight. The "magnetism" of the textbook treatment is hiding entirely inside that μ0 = 1/(ε0c2) — it is the same factor of c−2 wearing different clothes.
For a long, thin solenoid with N turns, length ℓ, and cross-sectional area A, the geometric double integral simplifies to the familiar high-school result:
Here μ0 is just notation. There is no magnetic field anywhere in the derivation; μ0 is the relativistic factor that converts a tiny per-electron acceleration into a measurable EMF when summed over a coil's worth of charges.
Let's plug in real numbers and watch the chain of magnitudes work out. Take a hand-wound copper solenoid with the following specifications:
| Quantity | Value |
|---|---|
| Number of turns, N | 100 |
| Length, ℓ | 10 cm = 0.10 m |
| Loop radius, r | 1 cm = 0.01 m |
| Cross-sectional area, A = πr2 | 3.14 × 10−4 m2 |
| Wire cross-section, Awire | 1 mm2 = 10−6 m2 |
| Copper electron density, n | 8.5 × 1028 m−3 |
Suppose we ramp the current at a moderate rate — say dI/dt = 100 A/s. We are going to follow one accelerating electron in the wire all the way to the back-EMF the coil produces, with nothing magnetic in the chain.
The current is I = n e Awire vdrift, so the rate at which any one electron's drift speed changes is
That is 7 thousandths of a meter per second per second. A snail's acceleration. Nothing dramatic happens to any one electron.
From Eind = e a / (4πε0c2R) with R = 0.01 m:
If we tried to detect the contribution from one electron, we'd never have a chance.
The wire has length Lwire = N · 2πr = 100 · 2π · 0.01 ≈ 6.3 m. The total number of conduction electrons in the coil is
Every single one of these electrons accelerates simultaneously when I changes. Every single one contributes its tiny induced field at every other electron's location. The geometric "Neumann" sum that pulls all of these together gives, for a 100-turn solenoid of these dimensions,
and the back-EMF when we ramp I at 100 A/s is
You can verify this is the same 4 mV the textbook would compute. The textbook starts with L = μ0N2A/ℓ, multiplies by dI/dt, and reads off the answer. We started with Eind = −qa/(4πε0c2R) for an accelerating electron, summed over the coil, and got the same number because μ0 = 1/(ε0c2). The two derivations are mathematically the same calculation written in two languages.
There are three ideas worth carrying away from this lecture.
Magnetism is relativistic electrostatics. The magnetic field is not a separate physical entity. It is the bookkeeping device we invented to handle the relativistic correction to Coulomb's law for moving sources. Purcell's parallel-wire argument shows this for the steady case. The accelerating-charge derivation shows it for inductance. The same logic extends to every "magnetic" phenomenon you can name.
The 1/c2 in μ0 is the entire magnetic content. Whenever you see μ0 in a formula — in inductance, in the Biot–Savart law, in the speed of light, in the magnetic constant of the universe — what you are looking at is just 1/(ε0c2). The relativistic correction factor was given a separate name and a unit and a place in the SI base for historical reasons. It is not a separate constant of nature.
You should not actually do calculations this way. The textbook treatment with B and Faraday's law is enormously easier than the all-electric one. For any practical coil design, plug into L = μ0N2A/ℓ and move on. The point of doing it the hard way once is conceptual reassurance: nothing new is happening in a magnetic phenomenon. Every measurement made in classical electromagnetism could, in principle, be predicted from Coulomb's law and special relativity alone.
The classic exposition of magnetism-as-relativistic-electrostatics is Edward Purcell's Electricity and Magnetism (third edition with David Morin), particularly chapter 5. The "induced field of an accelerating charge" picture used in Section 4 is the slow-motion limit of the Liénard–Wiechert potentials, treated in any graduate electromagnetism textbook (Jackson, Zangwill, Griffiths chapter 10). For the unified Lagrangian formulation in which all of these results fall out of a single principle of least action, see the Darwin Lagrangian discussion in Jackson's chapter 12.
Lecture compiled May 2026. Figures designed for color rendering; black-and-white printing preserves all information through line weight and labeling.
We have now used relativistic electric interaction to explain the back-EMF of one coil. Let us use the same language for another familiar experiment: place two solenoids face to face, send current through both, and they either pull toward each other or push away from each other depending on how the currents circulate.
In the old vocabulary one would say that each solenoid produces something around itself, and the two somethings interact. We will not say that. We will say only this: each solenoid is a large organized collection of moving electrons and stationary positive lattice ions. The ordinary static charge forces almost cancel because each wire is neutral. What remains is the tiny velocity-dependent correction required by relativity. Add that correction over all the moving charges in both coils, and a macroscopic force appears.
Take a small length element dℓ1 from coil 1 and a small length element dℓ2 from coil 2. Let the distance between those two pieces be R. Each piece of wire is neutral overall, so the large static attraction and repulsion terms cancel almost perfectly: electron-electron, ion-ion, electron-ion, and ion-electron contributions balance.
But the cancellation is not exact when charges are moving. The moving electrons have slightly altered electric fields, and observers moving with one set of electrons do not agree with lab observers about the charge density of the other wire. The leftover interaction is smaller than the static electric force by a factor of order v^2/c^2, but there are enormously many charges, so the total is measurable.
For two current elements, the surviving relativistic electric coupling has the form
The dot product is the important part. If the two little pieces of wire carry current in the same direction, dℓ1 · dℓ2 is positive. If they carry current in opposite directions, it is negative. If the pieces are perpendicular, their direct contribution is zero.
A solenoid is just a wire wound into many turns. So to get the coupling of two solenoids, we add the little contribution above for every piece of coil 1 against every piece of coil 2:
This expression contains only geometry and the relativistic factor 1/(ε0c2). There is no new force hidden inside it. It says: nearby parallel current elements contribute strongly; distant elements contribute weakly; opposite-directed elements contribute with the opposite sign.
For two solenoids with many turns, the same double integral is simply repeated for every pair of turns. If coil 1 has N1 turns and coil 2 has N2 turns, the coupling roughly scales like N1N2. This is why two coils with many turns can pull or push much more strongly than two single loops carrying the same current.
Now let the distance between the solenoids be x. For two face-to-face coaxial solenoids, the coupling M12(x) is larger when the coils are closer and smaller when they are farther apart. In other words, as x increases, M12 decreases:
If the currents are held fixed by their power supplies, the mechanical force along the line between the coils is
This one line gives both attraction and repulsion.
If the currents circulate the same way, then I1I2 > 0. Since dM12/dx < 0, the force Fx is negative: it points toward smaller separation. The coils attract.
If one current is reversed, then I1I2 < 0. The same derivative is still negative, so now Fx is positive: it points toward larger separation. The coils repel.
It is worth translating the formula back into particles. In two neighboring turns with currents flowing the same way, the moving electrons in one wire see the charge balance of the other wire slightly distorted by relativity. The positive lattice and the moving electrons no longer cancel in quite the same way they do when the charges are stationary. The residual electric force pulls the neighboring current elements sideways toward each other.
In a pair of solenoids, every turn of coil 1 has such a residual interaction with every turn of coil 2. Some pieces are angled, some are farther away, and some partially cancel. But for two aligned solenoids the dominant contributions cooperate. The front face of one coil interacts most strongly with the front face of the other. When the circulations match, most of the nearby current elements contribute with the same sign, and the coils pull together. When one circulation is reversed, the signs reverse, and the coils push apart.
Nothing measurable changed. The force is the same. The direction is the same. The dependence on current is the same. The dependence on number of turns is the same. The only thing that changed is the bookkeeping. Instead of naming a separate force, we kept the moving charges, applied relativistic electric interaction, and summed the result over the geometry of the two coils.
So two solenoids attract or repel for the same reason two straight current-carrying wires attract or repel. A neutral wire is not perfectly neutral in every moving frame. The tiny mismatch demanded by relativity produces a real electric force. A solenoid is simply a way of arranging a very long current-carrying wire so that millions of tiny relativistic electric interactions add instead of canceling.
Now we can explain a transformer in the same language. Put two solenoids near each other. Call them the primary coil and the secondary coil. Drive a changing current through the primary. Even though no metal connects the two coils, a voltage appears around the secondary. If the secondary circuit is closed, that voltage drives a current.
We will describe this without introducing any new kind of force. The primary coil is a long ordered line of charges. When its current changes, its conduction electrons accelerate. Accelerating charges produce a changing electric influence at distant points. The electrons in the secondary coil feel that influence along the direction of the secondary wire. Add the effect around the entire secondary loop, and the result is an EMF.
Take a tiny length element dℓ1 in the primary and another tiny length element dℓ2 in the secondary. Let their separation be R. When the current in the primary changes, the electrons in dℓ1 accelerate. Their changing relativistic electric field reaches dℓ2 and produces a small electric push along the secondary wire.
The size of that little contribution depends on four things:
For one pair of wire elements, the geometric part of the coupling is
The dot product tells us whether the two current paths cooperate. Parallel pieces contribute positively. Opposite-facing pieces contribute negatively. Perpendicular pieces contribute almost nothing to the loop voltage.
A transformer has many wire elements, not just two. So we add every primary element against every secondary element:
This number depends only on geometry: the number of turns, the coil areas, their spacing, their alignment, and what material guides the electric interaction between them. Bring the coils closer and M12 grows. Add more turns and it grows. Put the secondary far away or sideways and it shrinks.
The important point is that the secondary does not respond to the primary current itself. It responds to the change of the primary current. A steady current means the primary electrons drift at a steady speed. Their pattern is steady. The secondary may feel static stresses, but there is no continuous loop-driving voltage.
When the primary current changes, the primary electrons accelerate. That changing electric influence produces a loop voltage in the secondary:
That is the transformer’s central law in our language. No contact is needed between the coils. The primary’s accelerating charges act across space on the secondary’s charges, and the geometry of the two coils determines how much of that action survives after the whole loop is summed.
The minus sign matters. Suppose the primary current is increasing. The induced electric push in the secondary drives charge in whichever direction makes the secondary’s own relativistic electric reaction oppose that increase. If the primary current is decreasing, the induced push reverses and tries to keep the old situation going.
This is not a mysterious rule. It is energy bookkeeping. If the secondary current helped the primary change for free, the two coils would create energy from nowhere. Instead, when the secondary is connected to a load, energy must be supplied by the primary power source. The induced current direction is the one that makes the source work harder.
A voltage around an open secondary coil separates charge but cannot produce a sustained current. Close the circuit, and the secondary electrons can circulate. If the total resistance of the secondary circuit is Rload, then approximately
In a real transformer, the secondary current also pushes back on the primary through the same coupling. That back-reaction is exactly why drawing current from the secondary makes the primary draw more power from its source.
Each turn of the primary contributes to each turn of the secondary. If the coils are arranged so the contributions cooperate, the coupling roughly scales like
So doubling the primary turns gives the secondary twice as many accelerating source loops to hear from. Doubling the secondary turns gives twice as much wire around which the induced electric push can accumulate. This is why transformers use many turns rather than single loops.
For two tightly coupled coils, nearly the same changing electric influence threads each turn. Then the total loop voltage is almost just the voltage per turn multiplied by the number of turns. That gives the familiar transformer ratio:
If the secondary has more turns than the primary, the same per-turn induced push is added around more loops, so the secondary voltage is larger. If the secondary has fewer turns, the voltage is smaller.
But energy is still conserved. A higher secondary voltage comes with lower available secondary current. A lower secondary voltage comes with higher available secondary current. In the ideal case, ignoring losses,
Finally, notice what happens if the primary current becomes perfectly steady. Then dI1/dt = 0, so
The primary electrons are still moving, but they are no longer accelerating. The secondary no longer receives a changing loop-driving electric influence. A transformer therefore needs changing current. Alternating current works naturally because the primary electrons are constantly speeding up, slowing down, stopping, and reversing.
So a transformer is not magic action through empty space. It is a many-particle electric machine. The primary’s accelerating electrons produce a changing relativistic electric influence. The secondary’s electrons feel that influence around their wire. The loop sum becomes voltage. If the circuit is closed, voltage becomes current. The entire effect is the same electric interaction we have used throughout this lecture, now summed twice: once over all the turns of the primary, and once around all the turns of the secondary.