Annus Mirabilis · Interactive critical edition in preparation
Dynamics of the slowly accelerated electron
Push an electron with a field and follow its speed, energy and path.
Electron dynamics and force conventions
Dynamics of the slowly accelerated electron
Static worked example
Model note
- Longitudinal mass (
longitudinalMass), Transverse mass (comoving) (transverseMassComoving), Transverse mass (laboratory) (transverseMassLaboratory), Kinetic energy (electron) (kineticEnergy), Kinetic energy (Newtonian) (kineticEnergyNewtonian), Accelerating potential (acceleratingPotential), Accelerating potential (Newtonian) (acceleratingPotentialNewtonian), Radius of curvature (magnetic deflection) (radiusCurvatureMagnetic), Radius of curvature (electric deflection) (radiusCurvatureElectric), Lorentz factor (lorentzFactor), Speed ratio (speedRatio) and Positions along the electron's path (trajectoryPositions): Host calculation, by the site's own reference code. Computed inelectron. - Computed from the settings last applied (settings revision 1, result 1).
- Not modeled: radiation reaction from accelerated charges; self-electromagnetic fields and structure of the electron; quantum electrodynamic corrections and spin; space charge interactions in beam ensembles; non-uniform and fringe electromagnetic fields; apparatus-specific geometry of historical deflection experiments.
Predict before the numbers
At 0.6c, is the electron harder to push sideways by a factor of 1.25 or 1.5625?
The result appears when you choose, say you have one in mind, or skip.
Worked example: an electron at 0.6c, γ = 1.25, carries 2.05 × 10⁻¹⁴ J of kinetic energy where Newton's formula gives 1.47 × 10⁻¹⁴ J; reaching this speed from rest takes 1.28 × 10⁵ V, against 9.2 × 10⁴ V by Newton's formula.
Uniform field chamber (E = 1.0 × 105 V/m, B = 0 T)
v₀ = 0.6c · γ = 1.25 · convention: source (1905 masses)
The path over the first 2.0 ns, computed by the model and drawn to scale in the x–y plane.
Switching between Einstein's 1905 convention (comoving force / stationary acceleration, transverse coefficient 1.5625 × m at 0.6c) and Planck's 1906 convention (laboratory force, transverse coefficient 1.25 × m) changes only the coefficient named in the equation of motion. Everything observable, the deflection radii, potentials, kinetic energy and paths, is the same under both.
Values at these settings
| Lorentz factor γ | 1.25 |
|---|---|
| Longitudinal mass (m · γ³) | 1.7792 × 10⁻³⁰ kg |
| Transverse mass: comoving (m · γ²) | 1.4233 × 10⁻³⁰ kg |
| Transverse mass: laboratory (m · γ) | 1.1387 × 10⁻³⁰ kg |
| Relativistic kinetic energy (W) | 2.0468 × 10⁻¹⁴ J |
| Accelerating potential (P = W/e) | 127750 V |
| Magnetic curvature radius (Rm) | Magnetic field magnitude is zero. |
| Electric curvature radius (Re) | 2.2995 m |
An electron pushed toward the speed of light gets harder and harder to speed up, and the energy needed to reach light's speed is infinite. Einstein ended the paper with three measurements that could test these predictions.
§10 assumes that an electron at rest, or moving slowly, obeys Newton's law μ d²x/dt² = εX in its own frame, and transforms that law to the stationary system K by §3 and §6. With force defined as the force measured in the frame moving with the electron, and acceleration measured in K, Einstein finds a longitudinal mass μ/(√(1 − (v/V)²))³ and a transverse mass μ/(1 − (v/V)²), and adds at once that other definitions of force and acceleration would give other numbers, so that theories of the electron must be compared with care. At the lab's default speed, 0.6c, these are 1.779 × 10−30 kg and 1.423 × 10−30 kg for an electron of mass 9.109 × 10−31 kg. With the laboratory convention that Planck introduced in 1906, force as the rate of change of momentum, the transverse mass is 1.139 × 10−30 kg instead, and the trajectories are the same either way. The kinetic energy is W = μV²{1/√(1 − (v/V)²) − 1}, here 2.047 × 10−14 J against 1.474 × 10−14 J from ½mv²; it becomes infinite at v = V, so, as with the earlier results, speeds above light's have no possibility of existence. §10 ends with three relations open to experiment, which the lab computes: the ratio of magnetic to electric deflectability equals v/V, here 0.6; the potential difference needed to reach a speed, here 127.75 kV instead of the Newtonian 91.98 kV; and the radius of curvature in a magnetic field, 0.128 m at 0.01 T. In the default electric field of 105 V/m the path curves with a radius of 2.30 m.
Start with γ = 1/√(1 − 0.6²) = 1.25. Einstein's longitudinal mass is μγ³ = 9.109 × 10−31 × 1.953 = 1.779 × 10−30 kg, and his transverse mass is μγ² = 9.109 × 10−31 × 1.5625 = 1.423 × 10−30 kg. Where do the powers come from? Across the motion, lengths are unchanged, but each second of the electron's own time is γ seconds in K, so an acceleration seen in K is smaller by γ² than in the electron's frame; dividing the force measured in the electron's frame by it gives μγ². Measure the force in K instead, as Planck did in 1906: across the motion that force is smaller by γ, and the transverse mass becomes μγ = 1.139 × 10−30 kg. Along the motion the force is the same in both frames, and the acceleration seen in K is smaller by γ³, the extra factor coming from the change of simultaneity along the motion, so the longitudinal mass is μγ³ in both conventions. The motion itself is the same in both descriptions; only the word mass is defined differently. Next the energy. The work done by the field is ∫εX dx, and with the first equation of motion this is μV²(γ − 1). The electron's rest energy is μc² = 9.109 × 10−31 × (2.998 × 108)² = 8.187 × 10−14 J, and γ − 1 = 0.25, so W = 2.047 × 10−14 J. Newton's ½μv² gives 0.5 × 9.109 × 10−31 × (1.799 × 108)² = 1.474 × 10−14 J. As v approaches c, γ grows without limit, and so does W. The three testable relations follow. Dividing W by the electron's charge gives the potential needed: 2.047 × 10−14/1.602 × 10−19 = 127 750 V, against 91 980 V for Newton. An electron moving at v across a magnetic field B curves with radius R = γμv/(εB); at 0.01 T that is 1.25 × 9.109 × 10−31 × 1.799 × 108/(1.602 × 10−19 × 0.01) = 0.128 m. Across an electric field E the radius is γμv²/(εE), 2.30 m at 105 V/m. A magnetic force and an electric force deflect the electron equally when E = vB, which is why the ratio of the two deflectabilities is v/c, 0.6 here.
Modern qualifications
§10 writes μ and ε for the electron's mass and charge, V for the speed of light and β for the modern γ, and defines force by a spring balance at rest in the electron's moving frame. Einstein notes that the masses hold for any ponderable point, since a small charge can be added to one. Kaufmann's deflection measurements of fast β-ray electrons, made from 1901 to 1906, seemed at first to favour Abraham's rival electron theory over the Lorentz–Einstein formulas, while Bucherer's of 1908 favoured the latter, and the question stayed open for some years. None of these data are digitized here yet, so the lab draws no measured points. The paper ends by thanking M. Besso and is dated Bern, June 1905; the Annalen received it on 30 June.
Not modeled: radiation reaction from accelerated charges; self-electromagnetic fields and structure of the electron; quantum electrodynamic corrections and spin; space charge interactions in beam ensembles; non-uniform and fringe electromagnetic fields; apparatus-specific geometry of historical deflection experiments.
Show the code
The site’s own reference code for this law, in TypeScript. It computes the numbers above when the page runs it, and it is what the page uses wherever a FrankenSim result is not available.
evaluateSr13 · src/physics/reference/electron.ts · this site’s source, as built · sha256:bb62046b9943f0fd874f6cf4023f1baf428e741c5f1cbd01923fbbab5921bad3
This function computes the listed outputs when it runs.
In words
It returns the longitudinal mass, the transverse mass in the comoving convention and the transverse mass in the laboratory convention, along with the kinetic energy, the accelerating potential, the radius of curvature in a magnetic and in an electric field, and the trajectory. At six tenths of the light speed an electron's three masses are 1.7792, 1.4233 and 1.1387 times 10^-30 kg: the longitudinal is the rest mass times gamma cubed, the comoving transverse times gamma squared, and the laboratory transverse times gamma.
Implementation
export function evaluateSr13(input: Sr13Input): Readonly<Record<string, ScientificResult>> {
const charge =
input.particle === "custom" && input.customCharge !== undefined
? input.customCharge
: -ELEMENTARY_CHARGE;
const mass =
input.particle === "custom" && input.customMass !== undefined
? input.customMass
: ELECTRON_MASS;
const beta = Math.abs(input.initialSpeed);
if (beta >= 1) {
return Object.freeze({
longitudinalMass: asOutside(
"longitudinalMass",
"kg",
"mass",
"superluminal-speed",
"Speed must be less than c.",
),
transverseMassComoving: asOutside(
"transverseMassComoving",
"kg",
"mass",
"superluminal-speed",
"Speed must be less than c.",
),
transverseMassLaboratory: asOutside(
"transverseMassLaboratory",
"kg",
"mass",
"superluminal-speed",
"Speed must be less than c.",
),
kineticEnergy: asOutside(
"kineticEnergy",
"J",
"energy",
"superluminal-speed",
"Speed must be less than c.",
),
kineticEnergyNewtonian: asOutside(
"kineticEnergyNewtonian",
"J",
"energy",
"superluminal-speed",
"Speed must be less than c.",
),
acceleratingPotential: asOutside(
"acceleratingPotential",
"V",
"electric-potential",
"superluminal-speed",
"Speed must be less than c.",
),
acceleratingPotentialNewtonian: asOutside(
"acceleratingPotentialNewtonian",
"V",
"electric-potential",
"superluminal-speed",
"Speed must be less than c.",
),
radiusCurvatureMagnetic: asOutside(
"radiusCurvatureMagnetic",
"m",
"length",
"superluminal-speed",
"Speed must be less than c.",
),
radiusCurvatureElectric: asOutside(
"radiusCurvatureElectric",
"m",
"length",
"superluminal-speed",
"Speed must be less than c.",
),
lorentzFactor: asOutside(
"lorentzFactor",
"1",
"dimensionless",
"superluminal-speed",
"Speed must be less than c.",
),
speedRatio: asOutside(
"speedRatio",
"1",
"ratio",
"superluminal-speed",
"Speed must be less than c.",
),
trajectoryPositions: asOutside(
"trajectoryPositions",
"m",
"trajectory-xy",
"superluminal-speed",
"Speed must be less than c.",
),
});
}
const gRes = gamma(beta);
const g = gRes.status === "value" ? gRes.value : 1;
const longM = longitudinalMass(mass, beta);
const transMComov = transverseMassComoving(mass, beta);
const transMLab = transverseMassLaboratory(mass, beta);
const ke = kineticEnergy(mass, beta);
const pot = acceleratingPotential(beta, Math.abs(charge), mass);
const bMag = Math.hypot(input.magneticFieldX, input.magneticFieldY, input.magneticFieldZ);
const rm =
bMag > 0
? magneticRadius(beta, bMag, Math.abs(charge), mass).exact
: asOutside(
"radiusCurvatureMagnetic",
"m",
"length",
"zero-field",
"Magnetic field magnitude is zero.",
);
const eMag = Math.hypot(input.electricFieldX, input.electricFieldY, input.electricFieldZ);
const re =
eMag > 0
? electricRadius(beta, eMag, Math.abs(charge), mass).exact
: asOutside(
"radiusCurvatureElectric",
"m",
"length",
"zero-field",
"Electric field magnitude is zero.",
);
return Object.freeze({
longitudinalMass: longM,
transverseMassComoving: transMComov,
transverseMassLaboratory: transMLab,
kineticEnergy: ke.exact,
kineticEnergyNewtonian: ke.newtonian,
acceleratingPotential: pot.exact,
acceleratingPotentialNewtonian: pot.newtonian,
radiusCurvatureMagnetic: rm,
radiusCurvatureElectric: re,
lorentzFactor: asValue("lorentzFactor", "1", "dimensionless", g),
speedRatio: asValue("speedRatio", "1", "ratio", beta),
trajectoryPositions: sr13Trajectory(input, beta, charge, mass),
});
}The explanation
Full explanation
The electron's motion follows from its equation of motion in its momentary rest frame. The lab integrates the path and sets the paper's longitudinal and transverse masses beside those of the modern force convention, which gives a different transverse mass.
Show every step of the investigation
Choose the fields and the starting speed, then read the work done and the kinetic energy; the speed never reaches that of light. Compare Einstein's transverse mass with the modern one: the difference comes from how force is defined, not from the physics.
An explanatory model, not an observation of nature. This embed starts from the laboratory’s worked defaults, not a saved run. Presentation options change the surrounding guide, never the numerical inputs.