Electron Speed Calculator

The classical formula comes from equating the work done by the electric field to the kinetic energy of the electron: eV = ½m₀v². Solving for v gives v = √(2eV/m₀), where e is the electron charge (1.602×10⁻¹⁹ C), V is the accelerating voltage, and m₀ is the rest mass of the electron (9.109×10⁻³¹ kg). This formula is accurate only when the resulting speed is much less than the speed of light.

The relativistic speed is derived from the total energy gained by the electron: eV = (γ − 1)m₀c², where γ is the Lorentz factor. Rearranging gives v = c × √(1 − 1/(1 + eV/m₀c²)²). This formula correctly accounts for the fact that an electron's effective mass increases as it approaches the speed of light, preventing it from ever reaching c.

Relativistic effects become non-negligible when the electron's speed exceeds roughly 10% of the speed of light (about 3×10⁷ m/s), which corresponds to an accelerating voltage of around 2,500 V. At 10% of c, the classical formula underestimates the true speed by about 0.5%, but the error grows rapidly at higher voltages.

No. According to special relativity, as a particle with mass approaches the speed of light, the energy required to accelerate it further becomes infinite. No matter how large the accelerating voltage, the relativistic formula always yields a speed strictly less than c (≈ 299,792,458 m/s).

The key constants are: electron rest mass m₀ = 9.10938×10⁻³¹ kg, elementary charge e = 1.60218×10⁻¹⁹ C, and the speed of light c = 2.99792×10⁸ m/s. These are CODATA-recommended values used in standard physics calculations.

CRT televisions typically accelerated electrons through 10,000–30,000 V, producing speeds of roughly 10–30% of the speed of light — well into the relativistic regime. Electron microscopes can use voltages of 100 kV to 3 MV, where relativistic corrections are essential for accurate beam control and de Broglie wavelength calculations.

The kinetic energy gained by an electron accelerated through voltage V is simply K = eV, where e is the electron charge. In electron-volt units, an electron accelerated through 1 V gains exactly 1 eV of kinetic energy. For example, a 10,000 V accelerating voltage gives the electron 10,000 eV (10 keV) of kinetic energy.

No — a magnetic field exerts a force perpendicular to an electron's velocity, changing its direction but not its speed, so it cannot increase kinetic energy. Only an electric field (via an accelerating voltage) does work on the electron and increases its speed. Magnetic fields are used in devices like cyclotrons to curve the electron's path, while electric fields provide the actual acceleration.