Special relativity · Electrodynamics §10

Force conventions and dynamics of the slowly accelerated electron.

What force, work, energy, and deflection relations follow for a slowly accelerated electron, and why do two different “transverse masses” appear?

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 in electron.
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.

JavaScript is off. This is a complete worked example calculated when the site was built. Kinetic energy, accelerating potential, mass coefficients, and deflection curves remain available; changing the parameters requires JavaScript.

Predict before the numbers

At 0.6c, is the electron harder to push sideways by a factor of 1.25 or 1.5625?

Three relations the model could have It depends on which frame's force you compare to acceleration (1.25 in lab, 1.5625 in comoving), but both predict the identical physical deflection. 1.25 (gamma) 1.5625 (gamma^2)

The result appears when you choose, say you have one in mind, or skip.

Set field strengths, initial speed, and conventions

Section 10 examines the motion of a slowly accelerated electron in electromagnetic fields. By combining rest-frame dynamics with stationary coordinate measurements, Einstein derives longitudinal mass μγ³ and transverse mass μγ². Planck's laboratory force convention yields transverse mass μγ. Both conventions make identical predictions for physical deflections, potentials, and trajectories.

Initial speed β = v/c

Force convention

Experiment settings electric and magnetic field strengths, mass language, a historical dataset overlay

Electric field Ey (V/m)

Magnetic field Bz (Tesla)

Mass language

Historical dataset overlay

Changes here apply with Apply settings.

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.

Longitudinal mass (m · γ³)

1.953125 × m

Einstein's §10: μ / (√(1 − v²/V²))³, that is, γ³m

Transverse mass, comoving force (Einstein 1905)

1.5625 × m

Comoving force / stationary acceleration: F′_y / a_y = γ²m

Transverse mass, laboratory force (Planck 1906)

1.25 × m

Laboratory force / stationary acceleration: F_y / a_y = γm

Relativistic kinetic energy W

127.75 keV

mc²(γ − 1)

Newtonian kinetic energy

91.98 keV

½mv²

Accelerating potential P

127.75 kV

W / e (exact)

Newtonian potential

91.98 kV

½mv² / e

Magnetic curvature radius (R_m)

Straight path (B = 0)

γmv / (|q|B)

Electric curvature radius (R_e)

2.2995 m

γmv² / (|q|E)

The convention changes a name, not what is observed

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

Relativistic electron dynamics quantities computed under current 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 (R_m)	Magnetic field magnitude is zero.
Electric curvature radius (R_e)	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.

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),
  });
}

Worked case (readable without JavaScript)

Consider an electron of mass m = 9.109 × 10⁻³¹ kg and charge −e = −1.602 × 10⁻¹⁹ C moving at initial speed v = 0.6c (β = 0.6, γ = 1.25) through a transverse electric field E_y = 10⁵ V/m.

In §10, Einstein transforms Newton's second law from the electron's instantaneous rest frame back to the stationary coordinate system. When defining force as the field times charge in the comoving frame (the source convention), the equations of motion in stationary coordinates become:

\begin{gathered}\frac{d^2x}{dt^2} = \frac{\varepsilon}{\mu}\frac{1}{\beta^3}X, \\ \frac{d^2y}{dt^2} = \frac{\varepsilon}{\mu}\frac{1}{\beta}\left(Y - \frac{v}{V}N\right)\end{gathered}

Explain this equation

Comparing comoving force components to stationary accelerations gives the longitudinal mass mγ³ = 1.953125m and the transverse mass mγ² = 1.5625m. In contrast, Planck's 1906 laboratory convention (F = dp/dt) yields transverse mass mγ = 1.25m.

Crucially, both definitions predict the exact same physical trajectory and radius of curvature in a transverse electric field:

\begin{gathered}R_e = \frac{\gamma m v^2}{|q| E} \approx 2.2995\text{ m} \\ (\text{Newtonian } R_{e,\text{newt}} = \frac{m v^2}{|q| E} \\ \approx 1.8396\text{ m})\end{gathered}

The relativistic kinetic energy required to accelerate the electron from rest to 0.6c is:

\begin{gathered}\begin{aligned}W &= m c^2 (\gamma - 1) = 0.25 m c^2 \\ &\approx 127.75\text{ keV}\end{aligned} \\ \implies P = \frac{W}{e} \approx 127.75\text{ kV}\end{gathered}

Because γ − 1 grows without bound as v → c, an infinite accelerating potential would be required to reach the speed of light: superluminal velocities have no physical possibility of existence for material particles.

Return to paper section 10

Before you draw a conclusion

A common wrong turn this instrument speaks to

The relativity paper derives E = mc².
The June paper computes the kinetic energy of an electron and shows it becomes infinite at the speed of light. The relation between mass and energy content comes three months later, in a separate paper.
Read the full entry, with its sources

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