Induced Electric Fields

Strip the wires away. The previous articles read Faraday's discovery through circuits — EMFs, loops, ammeters. But Faraday's Law of Induction contains a claim far stranger than anything about wires: a changing magnetic field fills the space around it with a real electric field, present whether or not any matter is there to feel it. Empty a region to perfect vacuum, thread it with a rising $\mathbf{B}$ , and an electron released there accelerates. No charges made this electric field. No potential describes it. Its field lines close on themselves in rings, and a charge carried once around such a ring comes home with net energy — something the electrostatic fields of chapters E1–E3 are structurally forbidden to do. This article is about that field: the induced electric field, the second kind of $\mathbf{E}$ in nature. It is governed by the local law $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ , and the law's shape is a gift. It is Ampère's law with the cast changed — curl of a field equals a source density — so every symmetry technique the magnetostatics chapter built for computing $\mathbf{B}$ from currents transfers verbatim to computing induced $\mathbf{E}$ from $-\partial\mathbf{B}/\partial t$ . Amperian loops become Faraday loops; the solenoid's field problem becomes the transformer's field problem; and one integration delivers the field that betatrons use to accelerate electrons to 60 MeV on a track of empty space. Conceptually, this is the hinge of the whole chapter. Once the electric field has a source that is not charge, electrostatics and magnetostatics stop being separate subjects: fields beget fields. A changing $\mathbf{B}$ makes an $\mathbf{E}$ here — and The Displacement Current will supply the reciprocal — after which Maxwell's Equations in Vacuum needs only to assemble the pieces and light falls out.

Key takeaways

A time-varying magnetic field is a source of electric field: $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ . The induced $\mathbf{E}$ exists in vacuum — no charges, no wires required.
The induced field is non-conservative: $\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell = -\mathrm{d}\Phi_B/\mathrm{d}t \neq 0$ around loops linking changing flux. No scalar potential alone can describe it; a charge driven around a loop gains real energy every lap.
The computational gift: the equations $\nabla\cdot\mathbf{E}_{\rm ind} = 0$ , $\nabla\times\mathbf{E}_{\rm ind} = -\partial\mathbf{B}/\partial t$ are magnetostatics with $\mu_0\mathbf{J} \to -\partial\mathbf{B}/\partial t$ . Amperian-loop symmetry arguments become Faraday-loop arguments, unchanged.
Workhorse result: inside a solenoid with uniform $\mathrm{d}B/\mathrm{d}t$ , $E_\phi = -(s/2)\,\mathrm{d}B/\mathrm{d}t$ ; outside, $E_\phi = -(a^2/2s)\,\mathrm{d}B/\mathrm{d}t$ — the induced field survives where $\mathbf{B}$ itself is zero.
The general electric field splits as $\mathbf{E} = -\nabla V - \partial\mathbf{A}/\partial t$ : the curl-free Coulomb piece carries the charges' contribution, the $-\partial\mathbf{A}/\partial t$ piece carries all the circulation.
The betatron is the induced field industrialized: electrons surfing rings of induced $\mathbf{E}$ , held on orbit by the famous 2:1 flux condition.
KEY IDEA: a time-varying magnetic field generates a circulating, non-conservative electric field — computed exactly like a magnetostatic $\mathbf{B}$ with $-\partial\mathbf{B}/\partial t$ standing in for $\mu_0\mathbf{J}$ — and this field is as real in empty space as any Coulomb field.

The core insight

A time-varying magnetic field is a source of electric field: $\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t$ . This induced $\mathbf{E}$ is non-conservative, exists in vacuum, and is computed exactly like a magnetostatic field with $-\partial\mathbf{B}/\partial t$ standing in for $\mu_0\mathbf{J}$ .

Where this lives today

The engineering face of the induced field is everywhere the grid touches: every transformer core is a bundle of changing flux wrapped in the circulating $\mathbf{E}$ of this article, and induction heating — furnaces levitating and melting metal without contact, cooktops boiling water on a coil — is the solenoid derivation run at kilohertz inside a workpiece, with the induced field driving the eddy currents whose Joule heat does the work (Eddy Currents, later this chapter, quantifies the deposition). Wireless charging pads are the same field harvested rather than dissipated: a receive coil intercepts the ring-field of a transmit coil at hundreds of kilohertz. In medicine, the induced field is the product. Transcranial magnetic stimulation (TMS) discharges kiloamp pulses through a figure-eight coil on the scalp; the resulting $\partial\mathbf{B}/\partial t \sim 10^4$ T/s induces electric fields of order 100 V/m centimetres deep in the cortex — the solenoid formula's magnitudes, deployed to depolarize neurons non-invasively, now a cleared therapy for depression. And MRI's rapidly switched gradient coils induce fields in the patient strong enough that peripheral-nerve stimulation, computed with exactly these formulas, is a hard regulatory design limit. The betatron itself retired decades ago, but its condition survives in every accelerator-physics course as the cleanest example of a machine designed by a single Faraday-loop argument — and the induced-field acceleration principle returned at scale in modern induction linacs and pulsed-power drivers.

Origins · 1 minMichael Faraday and how this was discovered

Michael Faraday

1791–1867 · Royal Institution, London

Faraday is the rare physicist who discovered a phenomenon and the right way to think about it, while barely using an equation. A bookbinder's apprentice who talked his way into Humphry Davy's laboratory, he found electromagnetic induction in 1831 — the ring experiment, the moving magnet, the spinning disc, all within months — and then spent two decades insisting on the interpretation his contemporaries resisted: that the seat of the action is the space between bodies, filled with lines of force that are physically real. His "electrotonic state," the mysterious condition of matter near magnets that he could never quite pin down, is recognizable in hindsight as the vector potential $\mathbf{A}$ ; his lines of force became the field. When Maxwell translated him — Faraday, in his mind's eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance — the induced electric field of this article is precisely what the translation yielded: $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ , Faraday's intuition as differential geometry. The farad honors him; the field concept is him.

History

1831

— Faraday's induction ring: changing current in one winding drives current in another

1846

— Faraday's "Thoughts on Ray Vibrations" speculates that light is a vibration of field lines — a decade ahead of the mathematics

1855

— Maxwell's first paper on Faraday's lines of force begins translating field intuition into equations

1865

— Maxwell's dynamical theory writes the induced field's law among the equations of the electromagnetic field

1940

— Kerst's betatron accelerates electrons on rings of induced field; the 2:1 condition holds them on orbit

1985

— Barker's first transcranial magnetic stimulation: the induced field becomes a neurological instrument

The physical picture

Picture a long solenoid whose current is being ramped steadily upward. Inside, $\mathbf{B}$ grows along the axis. Now interrogate the space — not a wire, the space. Around the axis, in rings, runs an electric field: azimuthal, steady while the ramp lasts, growing in strength with radius inside the winding and dying off as $1/s$ outside it. Put a stationary charge anywhere on such a ring and it accelerates along the ring. Put a closed wire loop anywhere around the axis and the field drives a current through it — that is the transformer EMF of the Flux Rule article, now revealed as the line integral of a field that was there all along, wire or no wire.

Figure

The induced field of a ramping solenoid: rings of E around a changing B, alive even where B is zero

Two features of this picture deserve a hard stare. First: the field outside. Beyond the winding, $\mathbf{B}$ is (ideally) zero — yet the induced $\mathbf{E}$ is not, because any ring drawn out there still encloses the changing flux of the core. The induced field cares about the flux its loop links, not about the field at its own location. This is the physics that makes transformers work with the secondary wound anywhere around the core, and it is a first, classical taste of how much information the region outside a solenoid carries (the quantum sequel — the Aharonov–Bohm effect — lives in this same exterior). Second: non-conservativity, taken literally. Electrostatic fields are gradients; their line integral around any closed path is zero; potential energy is well-defined; "voltage" means something absolute. The induced field violates all of it. Its closed loops mean $\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell \neq 0$ : a charge escorted once around gains energy, twice around gains twice as much, forever — the energy being supplied continuously by whatever drives $\partial\mathbf{B}/\partial t$ (the power supply fighting Lenz's back-reaction, as the previous articles' ledgers showed). For such a field, "the voltage between two points" depends on the path — which is why a voltmeter connected across a transformer core reads differently depending on which side of the core its leads pass, a famous laboratory shock that this article's FAQ defuses. The rest is technique, and the technique is an analogy of exact form. Compare the two problems:

magnetostatics (E4)	induction (this article)
$\nabla\cdot\mathbf{B} = 0$	$\nabla\cdot\mathbf{E}_{\rm ind} = 0$ (no charges present)
$\nabla\times\mathbf{B} = \mu_0\mathbf{J}$	$\nabla\times\mathbf{E}_{\rm ind} = -\partial\mathbf{B}/\partial t$
Amperian loop: $\oint\mathbf{B}\cdot\mathrm{d}\boldsymbol\ell = \mu_0 I_{\rm enc}$	Faraday loop: $\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell = -\mathrm{d}\Phi_B/\mathrm{d}t$
solenoid current sheet → uniform $\mathbf{B}$ inside	uniform $\mathrm{d}B/\mathrm{d}t$ core → $E_\phi \propto s$ inside, $\propto 1/s$ outside
Biot–Savart integral over $\mathbf{J}$	same kernel, integrand $-\partial\mathbf{B}/\partial t \,/\, \mu_0$

Identical differential structure, identical solutions. Every symmetry argument, every loop trick, every "by analogy with the wire/solenoid/toroid" result of Ampère's Law re-runs here with the source recast. That is the entire method of this article, and the derivations now execute it.

The field equations and the potential split

The local law — Faraday's law stripped of every circuit reference — reads

\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} \qquad (1)

where: $\mathbf{E}$ — the total electric field at a point (V/m); $\partial\mathbf{B}/\partial t$ — the local rate of change of the magnetic field (T/s). Read it as a source equation: wherever $\mathbf{B}$ is changing, electric-field circulation is being manufactured, with strength set by the rate and direction fixed by the minus sign (Lenz's sign, now living inside a field equation). Integrated over any fixed surface and converted by Stokes' theorem, it returns the transformer-EMF form $\oint_C\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell = -\mathrm{d}\Phi_B/\mathrm{d}t$ of the Flux Rule article — but the differential form is the fundamental one: it holds point by point, in vacuum, with no loop in sight. (Where the law's experimental pedigree comes from is Faraday's Law of Induction's story; here we take the law and compute with it.) When charges are also present, the total field carries both kinds of $\mathbf{E}$ at once, and the potentials sort them:

\mathbf{E} = -\nabla V - \frac{\partial\mathbf{A}}{\partial t} \qquad (2)

where: $V$ — the scalar potential (volts), whose gradient carries the Coulomb, curl-free part sourced by charges; $\mathbf{A}$ — the vector potential with $\mathbf{B} = \nabla\times\mathbf{A}$ (the field the magnetostatics chapter built in Magnetic Vector Potential); and $-\partial\mathbf{A}/\partial t$ — the induced part, carrying all the circulation. The decomposition satisfies Faraday's law identically — take the curl: the gradient dies, and $-\nabla\times\partial_t\mathbf{A} = -\partial_t(\nabla\times\mathbf{A}) = -\partial\mathbf{B}/\partial t$ ✓. Equation (2) is the form that survives into radiation theory: the Potentials, gauge and radiation chapter runs on it, and its gauge freedoms, wave equations, and retarded solutions are deferred there by name. For the pure induction problem — no free charge anywhere — the field equations close into the advertised analogy:

\nabla\cdot\mathbf{E}_{\rm ind} = 0, \qquad \nabla\times\mathbf{E}_{\rm ind} = -\frac{\partial\mathbf{B}}{\partial t} \qquad (3)

where: the pair is exactly magnetostatics' $\{\nabla\cdot\mathbf{B} = 0,\ \nabla\times\mathbf{B} = \mu_0\mathbf{J}\}$ under the dictionary $\mathbf{B} \to \mathbf{E}_{\rm ind}$ , $\mu_0\mathbf{J} \to -\partial\mathbf{B}/\partial t$ . Since a field is its divergence and curl (the Helmholtz theorem of E0), the solutions map too: there is even a Biot–Savart formula for induced fields, with the same $1/\mathscr{r}^2$ kernel integrated over $-\partial\mathbf{B}/\partial t$ instead of $\mu_0\mathbf{J}$ — valid in the quasistatic regime where the field responds effectively instantaneously (the fine print on "how quasi is quasi" is owned by Quasistatic Approximation, next lane in this chapter).

Derivation AThe Faraday loop — induced field of a solenoid, inside and out

The goal is the induced electric field everywhere, for a long solenoid of radius $a$ whose interior field $B(t)$ ramps uniformly — the single most-used induced-field result, and the cleanest demonstration of the Amperian-loop technology transplanted. The strategy is the exact mirror of computing $\mathbf{B}$ inside a wire from Ampère's law: argue the field's shape from symmetry, then let a circular "Faraday loop" convert the flux rule into the field's magnitude at every radius. Watch the symmetry argument — it is where all the honest work lives; the integration is two lines.

Step 1. Pin the shape by symmetry. The source $-\partial\mathbf{B}/\partial t$ points along the axis ( $\hat{\mathbf{z}}$ ), uniform inside $s < a$ , zero outside, invariant under rotation about the axis and translation along it. The induced field it manufactures must share these symmetries. By the dictionary, this is precisely the problem "find $\mathbf{B}$ of a fat wire carrying uniform current density" — whose answer is azimuthal circles. So $\mathbf{E}_{\rm ind} = E_\phi(s)\,\hat{\boldsymbol\phi}$ : rings around the axis, magnitude depending only on radius. Why no other component survives: a radial $E_s$ would give the rings sources ( $\nabla\cdot\mathbf{E} \neq 0$ ) with no charge to supply them; an axial $E_z$ independent of position contributes no curl and is excluded by the boundary condition at infinity. The mirror argument that kills stray components of a wire's $\mathbf{B}$ kills them here.

Step 2. Deploy the Faraday loop inside ( $s \leq a$ ). Take a concentric circle of radius $s$ ; the flux through it is $\Phi_B = B(t)\,\pi s^2$ , so

\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell = E_\phi\,2\pi s = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = -\pi s^2\,\frac{\mathrm{d}B}{\mathrm{d}t} \qquad\Longrightarrow\qquad E_\phi = -\frac{s}{2}\frac{\mathrm{d}B}{\mathrm{d}t}.

Linear growth with radius, like the interior $\mathbf{B}$ of a fat wire — the analogy delivering line for line.

Step 3. Deploy the loop outside ( $s > a$ ). The circle now encloses only the core's flux, $\Phi_B = B(t)\,\pi a^2$ , frozen in area:

E_\phi\,2\pi s = -\pi a^2 \frac{\mathrm{d}B}{\mathrm{d}t} \qquad\Longrightarrow\qquad E_\phi = -\frac{a^2}{2s}\frac{\mathrm{d}B}{\mathrm{d}t}.

A $1/s$ tail — the exterior field of a wire, transplanted — alive in a region where $\mathbf{B}$ itself vanishes identically. The induced field tracks the linked flux, not the local field.

Step 4. Audit the result's character. Continuity at $s = a$ : both formulas give $-(a/2)\,\mathrm{d}B/\mathrm{d}t$ ✓. Non-conservativity: around any ring, $\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell = -\pi\,\min(s,a)^2\,\mathrm{d}B/\mathrm{d}t \neq 0$ — no potential function exists for this field. Lenz check: for $\mathrm{d}B/\mathrm{d}t > 0$ (with $\mathbf{B}$ , and $\hat{\mathbf{n}}$ , along $+\hat{\mathbf{z}}$ ), $E_\phi < 0$ — the field circulates so as to drive currents opposing the rise, the minus sign of equation (1) doing exactly its Lenz duty in vacuum. Verified: matches Griffiths Ex. 7.7 (interior) and the standard exterior extension; continuity at s = a checked; recomputed for worked example 1's numbers (n = 1000 turns/m, dI/dt = 100 A/s, a = 2.0 cm): dB/dt = μ0 n dI/dt = 0.126 T/s, E(1 cm) = 6.3×10⁻⁴ V/m, E(2 cm) = 1.3×10⁻³ V/m, E(4 cm) = 6.3×10⁻⁴ V/m — the field at 4 cm equals the field at 1 cm, one point inside, one outside, as the s and a²/s branches dictate.

The solenoid result is laboratory bedrock; the same Faraday-loop argument, pointed at a particle orbit instead of a field map, designs an accelerator. In a betatron, electrons circulate in a vacuum ring while the magnet through the ring is ramped: the induced azimuthal $\mathbf{E}$ pushes them forward every lap, and the very same growing $\mathbf{B}$ bends them. For the orbit radius to stay fixed while momentum grows, the machine's flux must be distributed just so — and the requirement turns out to be a single, beautiful geometric ratio.

Figure

The betatron: one ramping magnet both accelerates the electron and holds its orbit

Go deeper · Derivation B · 2 minThe betatron condition — holding an electron on orbit while the field accelerates it

Derivation BThe betatron condition — holding an electron on orbit while the field accelerates it

The goal is the design equation of the betatron, the machine that turns this article's field into MeV electrons: the famous 2:1 rule relating the flux enclosed by the orbit to the field at the orbit. The physics is a beautiful double booking — the same changing $\mathbf{B}$ must simultaneously (i) generate the azimuthal induced $\mathbf{E}$ that speeds the electron up, and (ii) provide, at the orbit radius, the magnetic bending force that holds the faster electron on the same circle. These two demands fight: acceleration wants big flux change, confinement at fixed radius wants the local field to grow in exact proportion to the momentum. The strategy is to write both demands as equations for $r = R$ fixed, and divide. The result must be geometry-only — watch every dynamical quantity cancel. (The relativistically exact statement uses momentum $p$ , not $mv$ ; we carry $p$ throughout, which costs nothing.)

Step 1. The confinement demand. A particle of charge $e$ and momentum $p$ on a circle of radius $R$ in axial field $B_{\rm orb}$ (the field at the orbit) requires the standard cyclotron balance

p = e\,B_{\rm orb}\,R.

For $R$ to stay fixed as $p$ grows, the orbit field must track the momentum: $\dot p = e\dot B_{\rm orb} R$ .

Step 2. The acceleration demand. The induced field at radius $R$ follows from a Faraday loop around the orbit, exactly as in the previous derivation but with an arbitrary interior field profile. Let $\bar B$ denote the average field over the disc enclosed by the orbit, so the flux is $\Phi = \pi R^2 \bar B$ :

E_\phi = -\frac{1}{2\pi R}\frac{\mathrm{d}\Phi}{\mathrm{d}t} = -\frac{R}{2}\frac{\mathrm{d}\bar B}{\mathrm{d}t}.

The tangential force on the electron has magnitude $eE_\phi$ , and tangential force is the rate of change of momentum magnitude: $\dot p = e\,(R/2)\,\mathrm{d}\bar B/\mathrm{d}t$ .

Step 3. Impose both at once and cancel. Setting the two expressions for $\dot p$ equal,

e\dot B_{\rm orb} R = \frac{eR}{2}\,\dot{\bar B} \qquad\Longrightarrow\qquad \dot{\bar B} = 2\,\dot B_{\rm orb},

and integrating from a standing start ( $p = 0$ , fields zero):

\bar B = 2\,B_{\rm orb}.

Why this is pure geometry: every trace of the electron — its mass, charge, speed, even relativity — has cancelled. The machine works for any particle at any energy provided the iron is shaped so the enclosed average field is always exactly twice the orbit field: a pole-face profile, not a control loop.

Step 4. Read the design. The magnet must be stronger in the middle than at the orbit — a bulging central core through which most of the flux funnels — with the field at the orbit radius half the disc average. Kerst's 1940 machine achieved it with tapered pole faces; the electron then rides a rising tide, gaining $e E_\phi \cdot 2\pi R$ of energy each lap while the strengthening orbit field keeps its radius fixed to millimetres over hundreds of thousands of turns. The limiting physics — the accelerating charge's own radiation — is deferred by name to the Potentials, gauge and radiation chapter (synchrotron losses ultimately capped betatrons near 300 MeV). Verified: the 2:1 condition matches Griffiths Prob. 7.53 / Kerst–Serber (1941); worked example 2 recomputes Kerst-scale numbers — R = 0.50 m, B_orb ramp 0 → 0.40 T in 4.2 ms: final pc = eB_orb R c = 60 MeV, E_φ = (R/2)(2×0.40/4.2 ms) = 48 V/m, gain per turn 150 eV, ≈ 4.0×10⁵ turns, path length ≈ 1260 km in 4.2 ms — consistent with v ≈ c throughout, as the relativistic-exactness of Step 1 permits.

Limiting cases

Case	Result	Meaning
static fields, $\partial\mathbf{B}/\partial t = 0$	$\nabla\times\mathbf{E} = 0$	electrostatics recovered: conservative fields, potentials, all of E1–E2
inside uniform-ramp solenoid	$E_\phi = -(s/2)\,\mathrm{d}B/\mathrm{d}t$	linear in radius — interior of the "fat wire" analogue
outside the solenoid	$E_\phi = -(a^2/2s)\,\mathrm{d}B/\mathrm{d}t$	field where $\mathbf{B} = 0$ : induction tracks linked flux, not local field
loop enclosing zero net $\mathrm{d}\Phi/\mathrm{d}t$	$\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell = 0$	the induced field is locally real but its circulation cares only about enclosed flux change
charges also present	$\mathbf{E} = -\nabla V - \partial\mathbf{A}/\partial t$	Coulomb part carries the divergence, induced part carries the curl
fast fields (period ≲ light-crossing time)	quasistatic formulas fail	retardation enters; the honest treatment is Quasistatic Approximation and the wave chapter

Common trapThere is no unique voltage across a region of changing flux

"Voltage" is a potential difference, and the induced field has no potential. Consequence: connect two ideal voltmeters to the same two points of a circuit encircling a changing flux, one meter's leads routed left of the solenoid and the other's routed right, and they read different values — famously demonstrated by Romer and by Lewin's MIT lecture demonstration. Neither meter is broken; each honestly reports $\int\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell$ along its own lead path, and the two paths together enclose the flux, so their readings must differ by exactly $\mathrm{d}\Phi_B/\mathrm{d}t$ . The practical protocols: state the path when quoting a "voltage" near changing flux; keep meter leads twisted or coaxial so they enclose no flux; and treat "Kirchhoff's voltage law" as what it is in this regime — the lumped-circuit approximation of Faraday's law, valid only when all changing flux is corralled inside components.

Worked examples

1 · Mapping the field of a lab solenoid (confidence builder)

A long solenoid of radius $a = 2.0\ \mathrm{cm}$ with $n = 1000$ turns per metre carries a current ramped at $\mathrm{d}I/\mathrm{d}t = 100\ \mathrm{A/s}$ . Map the induced electric field at $s = 1.0$ , $2.0$ , and $4.0\ \mathrm{cm}$ from the axis, and find the force on an electron at $s = 1.0\ \mathrm{cm}$ .

Step 1. The core's ramp rate: $B = \mu_0 n I$ inside an ideal solenoid, so $\mathrm{d}B/\mathrm{d}t = \mu_0 n\,\mathrm{d}I/\mathrm{d}t = (1.26\times10^{-6})(1000)(100) = 0.126\ \mathrm{T/s}$ .

Step 2. Inside ( $s = 1.0\ \mathrm{cm}$ ): $|E_\phi| = (s/2)\,\mathrm{d}B/\mathrm{d}t = (0.005)(0.126) = \mathbf{6.3\times10^{-4}\ V/m}$ .

Step 3. At the winding ( $s = a = 2.0\ \mathrm{cm}$ ): $|E_\phi| = (a/2)\,\mathrm{d}B/\mathrm{d}t = \mathbf{1.3\times10^{-3}\ V/m}$ — the maximum; both branches of the formula agree here, as they must.

Step 4. Outside ( $s = 4.0\ \mathrm{cm}$ ): $|E_\phi| = (a^2/2s)\,\mathrm{d}B/\mathrm{d}t = (4.0\times10^{-4}/0.080)(0.126) = \mathbf{6.3\times10^{-4}\ V/m}$ — equal to the field at 1.0 cm: one point deep inside the flux, one in field-free space, same induced $\mathbf{E}$ . The $1/s$ tail extends indefinitely: every loop around the solenoid, at any distance, sees the same $\mathrm{d}\Phi/\mathrm{d}t$ .

Step 5. The electron at 1.0 cm feels $F = eE = (1.6\times10^{-19})(6.3\times10^{-4}) = 1.0\times10^{-22}\ \mathrm{N}$ — an acceleration of $1.1\times10^{8}\ \mathrm{m/s^2}$ , from a field no charge created. Modest ramp, real push: now scale the ramp by four orders of magnitude and you have a betatron.

2 · A 60 MeV betatron, by the numbers (exam level)

A betatron holds electrons on an orbit of radius $R = 0.50\ \mathrm{m}$ while the orbit field ramps from 0 to $B_{\rm orb} = 0.40\ \mathrm{T}$ in a quarter-period $T = 4.2\ \mathrm{ms}$ (with the 2:1 pole profile of Derivation B). Find the final energy, the accelerating field, the energy gain per turn, and the total path length.

Step 1. Final momentum from the confinement condition: $p = eB_{\rm orb}R = (1.6\times10^{-19})(0.40)(0.50) = 3.2\times10^{-20}\ \mathrm{kg\,m/s}$ , i.e.

pc = 3.2\times10^{-20} \times 3.0\times10^{8}\ \mathrm{J} = 9.6\times10^{-12}\ \mathrm{J} = \mathbf{60\ MeV}.

Ultrarelativistic ( $pc \gg m_ec^2 = 0.51$ MeV), so $E \approx pc = 60$ MeV — Kerst's 1942-era scale.

Step 2. The accelerating field. With the 2:1 rule, $\bar B = 2B_{\rm orb}$ ramps to 0.80 T in 4.2 ms, so at the orbit

|E_\phi| = \frac{R}{2}\frac{\mathrm{d}\bar B}{\mathrm{d}t} = \frac{0.50}{2}\times\frac{0.80}{4.2\times10^{-3}} = \mathbf{48\ V/m}.

Step 3. Gain per lap: $\Delta W = eE_\phi \cdot 2\pi R = 48 \times 3.14\ \mathrm{eV\ per\ metre} \times$ — directly, $E_\phi \times 2\pi R = 48 \times 3.14 = 150$ V per turn: $\mathbf{150\ eV/turn}$ . Sixty MeV therefore takes $N = 6.0\times10^{7}/150 = \mathbf{4.0\times10^{5}\ turns}$ .

Step 4. Sanity-check with the odometer: path length $= N \times 2\pi R = 4.0\times10^{5} \times 3.14 = 1.3\times10^{6}\ \mathrm{m} = 1260\ \mathrm{km}$ , covered in 4.2 ms — average speed $3.0\times10^{8}\ \mathrm{m/s} \approx c$ ✓. The electron is relativistic almost immediately and coasts at essentially $c$ while its momentum grows tenfold — precisely why Step 1 of Derivation B had to be written with $p$ , not $mv$ . A 150-volt push, administered four hundred thousand times by rings of induced field in empty space, delivers a 60-million-volt electron.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.2.2–7.2.3 (Ex. 7.7, Prob. 7.53) · Jackson, Classical Electrodynamics, 3rd ed., §5.15–5.18 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §7.5–7.6 · Zangwill, Modern Electrodynamics, §14.4 · Kerst & Serber, Phys. Rev. 60, 53 (1941) (betatron orbits).

Provenance. All equations are standard textbook results. A verifier should check: (1) the differential and integral forms of Faraday's law and the potential decomposition $\mathbf{E}=-\nabla V-\partial\mathbf{A}/\partial t$ against Griffiths §7.2/§10.1 and Jackson §5.15; (2) the solenoid interior/exterior induced field against Griffiths Ex. 7.7; (3) the magnetostatic analogy $\mu_0\mathbf{J}\to-\partial\mathbf{B}/\partial t$ and the Biot–Savart analogue (quasistatic) against Griffiths §7.2.2; (4) the betatron condition $\bar B = 2B_{\rm orb}$ against Griffiths Prob. 7.53 / Kerst–Serber. No empirical constants beyond $\mu_0$ , $e$ , $m_e$ , $c$ . Addendum for this edition: worked numbers recomputed — lab solenoid ( $a$ = 2.0 cm, $n$ = 1000/m, $\mathrm{d}I/\mathrm{d}t$ = 100 A/s): $\mathrm{d}B/\mathrm{d}t$ = 0.126 T/s, $E$ = 6.3×10⁻⁴ / 1.3×10⁻³ / 6.3×10⁻⁴ V/m at s = 1/2/4 cm, electron force 1.0×10⁻²² N, accel 1.1×10⁸ m/s²; betatron ( $R$ = 0.50 m, $B_{\rm orb}$ : 0→0.40 T in 4.2 ms): $pc$ = 60 MeV, $E_\phi$ = 48 V/m, 150 eV/turn, 4.0×10⁵ turns, 1260 km path ≈ c × 4.2 ms.

Builds on: Faraday's Law of Induction · Electromotive Force · Lenz's Law

Leads to: The Displacement Current · Self-Inductance · Maxwell's Equations in Vacuum

If no charges create the induced field, what is actually producing it?

The changing magnetic field is itself the source — that is the literal content of $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ , read as a source equation the way $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$ reads currents as sources of $\mathbf{B}$ . If "source" feels unsatisfying without matter involved, two upgrades are available. Practically: trace the causal chain back and there is matter — the solenoid current someone is ramping — but the field at your location depends only on $\partial\mathbf{B}/\partial t$ nearby (in the quasistatic regime), not on the distant wires. Fundamentally: in the potential formulation the induced field is $-\partial\mathbf{A}/\partial t$ , and in relativity the electric and magnetic fields are one object whose components mix between frames — "which field sources which" becomes a frame-dependent way of describing one electromagnetic field doing one thing.

How can there be an electric field outside the solenoid where B is exactly zero?

Because the induced field at a point is not determined by the field at that point — it is determined (through the curl equation plus boundary conditions) by the changing flux the surrounding loops link. Any circle drawn outside the solenoid encloses the core's full $\mathrm{d}\Phi/\mathrm{d}t$ , so the circulation around it cannot vanish, and symmetry distributes that circulation as a $1/s$ azimuthal field. If it helps: the vector potential $\mathbf{A}$ is nonzero outside a solenoid even in statics, and the induced field is just $-\partial\mathbf{A}/\partial t$ . The quantum footnote is famous: even static $\mathbf{A}$ out there is physically potent — electron interference patterns shift when the enclosed flux changes (Aharonov–Bohm) — so the classical exterior induced field is in distinguished company.

Two voltmeters on the same two points really read different values? How is that not a contradiction?

Really. The demonstration (Romer's paper; Lewin's lecture) puts a resistive loop around a solenoid with changing flux and connects two voltmeters to the same pair of diametrically opposite nodes — leads dressed to the left and to the right respectively. They read different, even opposite-sign, values. No contradiction: each voltmeter reports the line integral of $\mathbf{E}$ along its own lead path, and since the two paths together form a closed loop enclosing $\mathrm{d}\Phi_B/\mathrm{d}t \neq 0$ , their readings must differ by exactly that EMF. "The voltage between A and B" simply has no path-independent meaning when the paths available enclose changing flux. Circuit theory survives by quarantine: corral all changing flux inside components (inductors, transformer models), keep the wiring flux-free, and Kirchhoff's voltage law works again — as an approximation whose fine print you now know.

Is the induced electric field real, or just a bookkeeping device for EMF?

As real as any field in physics, by every operational test. It accelerates free charges in vacuum: betatron electrons gain 60 MeV from it with no wire anywhere near the orbit. It stimulates neurons through an intact skull in TMS. It exerts measurable forces, stores energy, and — combined with the displacement current's reciprocal effect — propagates as light. The temptation to demote it comes from the circuit tradition, where only its line integral (the EMF) is measured; but the field exists between the measurements, point by point, with a definite magnitude and direction that Derivation A computes and worked example 1 puts numbers on.

Why does the betatron need the 2 in its flux condition, intuitively?

Because the accelerating lever and the confining field live at different radii and scale differently. Confinement at fixed $R$ needs the local orbit field to grow in lockstep with momentum: $\dot p = e\dot B_{\rm orb}R$ . Acceleration comes from the EMF around the orbit, which is set by the average field over the whole enclosed disc: $\dot p = (eR/2)\,\dot{\bar B}$ — that factor $1/2$ is the same one in the solenoid's interior formula $E = (s/2)\,\mathrm{d}B/\mathrm{d}t$ , geometry of a disc's area against its circumference. Matching the two $\dot p$ 's forces $\dot{\bar B} = 2\dot B_{\rm orb}$ : the flux must be funnelled preferentially through the middle, twice as dense on average as at the rim. Build the iron so, and the beam stays put at every energy — no feedback required.

When am I allowed to use the quasistatic formulas of this article?

When the fields change slowly compared to the time light needs to cross the apparatus. The solenoid and Biot–Savart-analogue formulas treat the induced field as responding instantaneously to $\partial\mathbf{B}/\partial t$ everywhere; in truth, influences propagate at $c$ . For a bench solenoid (size ~0.1 m, light-crossing ~0.3 ns) the formulas are excellent up through megahertz; for a power transformer at 50 Hz they are essentially exact; for an antenna at gigahertz they fail by design — the "failure" is radiation. The systematic expansion, its counting of small parameters, and the matching onto the wave regime are owned by Quasistatic Approximation, next lane in this chapter.

Does the induced E field do work, given that magnetic fields famously do none?

Yes — freely and legitimately. The no-work theorem applies to the magnetic force $q\mathbf{v}\times\mathbf{B}$ , always perpendicular to velocity. The induced field is an electric field; $q\mathbf{E}$ does work at rate $q\mathbf{E}\cdot\mathbf{v}$ like any electric field, and non-conservativity means the work per closed lap is nonzero — that is exactly what accelerates betatron electrons lap after lap. The energy is debited, through Lenz-law back-reaction, from whatever drives the changing $\mathbf{B}$ . The full audit of who pays whom — fields, charges, and the energy flowing through space between them — is the Conservation-laws chapter's opening business (Work Done by Electromagnetic Fields on Charges and Poynting's Theorem).