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?
SR-13 · Electron dynamics and force conventions
Dynamics of the slowly accelerated electron
Section 10 derives the equations of motion for a slowly accelerated electron by transforming from its instantaneous rest frame back to the stationary system, yielding longitudinal and transverse mass coefficients and unbounded kinetic energy as speed approaches the speed of light.
Einstein's source convention compares comoving force to stationary acceleration, giving a transverse coefficient m·gamma^2; Planck's laboratory convention (F = dp/dt) gives m·gamma. Both conventions yield identical observable trajectories, potentials, and deflection radii.
The relativistic kinetic energy W = mc^2(gamma - 1) approaches infinity as v -> c, demonstrating that superluminal velocities have no possibility of existence for ponderable material points.
Einstein lists three relations accessible to experiment: the velocity-dependent accelerating potential, the magnetic deflection radius, and the ratio of magnetic to electric deflectability A_m / A_e = v/V. Historical measurements by Kaufmann and Bucherer overlay directly onto the predicted curves.
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.
Switching between Einstein's 1905 convention (comoving force / stationary acceleration, transverse coefficient 1.5625m at 0.6c) and Planck's 1906 convention (laboratory force, transverse coefficient 1.25m) alters only the named coefficient in the equation of motion. All physical observables (deflection radii, potentials, kinetic energy, and spatial trajectories) remain strictly identical under both conventions.
Accepted snapshot
| Lorentz factor γ | 1.25 |
|---|---|
| Longitudinal mass (m · γ³) | 1.7792e-30 kg |
| Transverse mass: comoving (m · γ²) | 1.4233e-30 kg |
| Transverse mass: laboratory (m · γ) | 1.1387e-30 kg |
| Relativistic kinetic energy (W) | 2.0468e-14 J |
| Accelerating potential (P = W/e) | 127750 V |
| Magnetic curvature radius (R_m) | Magnetic field magnitude is zero. m |
| Electric curvature radius (R_e) | 2.2995 m |
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.
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 Ey = 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:
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:
The relativistic kinetic energy required to accelerate the electron from rest to 0.6c is:
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.