Electron Speed Calculator - Voltage to Velocity, KE, and v/c

The electron speed calculator takes a single accelerating voltage V (in volts) and returns the resulting electron velocity in metres per second, the fraction of the speed of light (v/c), the kinetic energy in electron-volts, and the Lorentz factor. It runs the relativistic relation v = c sqrt(1 - 1/gamma^2) with gamma = 1 + e V / (m_e c^2), which agrees with the classical v = sqrt(2 e V / m_e) to better than 1% for V below about 10 kV and stays correct up to multi-MeV electron-beam and X-ray voltages.

• • Cathode-ray and oscilloscope physics: Confirm the 5.93e6 m/s speed of an electron accelerated by 100 V in a CRT or old-school oscilloscope problem.
• • Electron-beam and X-ray tube energy: Read the velocity at 100 kV to 1 MeV accelerating voltages used in electron microscopes and medical linacs.
• • Photoelectron and thermionic-emission problems: Convert a stopping potential in electron-volts directly to a velocity in m/s for a modern-physics lab.
• • Drift-velocity sanity check: Compare the single-electron speed from a known V against the millimetres-per-second drift velocity in a copper wire.

An electron accelerated from rest through a potential difference V picks up a kinetic energy equal to e times V. For small V the kinetic energy is tiny compared with the electron rest energy m_e c^2 (about 511 keV), so v = sqrt(2 e V / m_e) is a good approximation. As V grows past a few kilovolts the relativistic relation v = c sqrt(1 - 1/(1 + e V / (m_e c^2))^2) replaces the classical one, and the page returns a continuous answer from sub-volt photoelectron voltages up to multi-MeV linac energies.

For the bound-electron counterpart inside a hydrogen-like atom, the Bohr Model Calculator returns the Bohr radius, energy level, and transition wavelength from the same principal quantum number.

The electron speed calculator converts the accelerating voltage V to kinetic energy K = e V in joules, divides by the electron rest energy m_e c^2 to get a dimensionless ratio, then plugs that ratio into the relativistic gamma and v/c relations.

• voltage V: Accelerating potential difference in volts, must be non-negative. Internally converted to joules via K = e V.
• elementary charge e: Elementary charge, exactly 1.602176634e-19 C since the 2019 SI redefinition. Converts volts to joules.
• electron rest mass m_e: Electron rest mass, 9.1093837015e-31 kg from NIST CODATA 2018. Sets the rest energy m_e c^2 = 511 keV.
• speed of light c: Speed of light in vacuum, exactly 299792458 m/s. Sets the ceiling for the electron velocity at any V.

The output is intentionally unit-rich. The primary speed row is in metres per second, the secondary row is in kilometres per second for a more readable magnitude, and the third row expresses the velocity as a fraction of c so the user can see at a glance whether relativity matters. The kinetic energy row shows the same energy in electron-volts, and the Lorentz factor row gives gamma so the relativistic correction is explicit.

According to NIST CODATA 2018, the electron rest mass is 9.1093837015e-31 kg, the value used here for the rest energy m_e c^2 that sets the relativistic scale of the speed calculation.

When the same problem starts from a charge and a distance instead of a voltage, the Electric Potential Calculator turns point-charge or parallel-plate geometries into the potential difference V that this page takes as input.

Four ideas carry the calculator from an accelerating voltage to a fully relativistic speed: the elementary charge, the electron rest energy, the Lorentz factor, and the classical-versus-relativistic crossover.

These four ideas also explain why the kinetic energy in electron-volts is numerically equal to the accelerating voltage in volts. The definition of the electron-volt sets 1 eV equal to the kinetic energy an electron picks up across a 1 V gap.

After the kinetic energy and speed are in hand, the Electron Configuration Calculator returns the orbital notation, noble gas shorthand, and block for the same atom you accelerated.

Five steps turn an accelerating voltage into the electron speed, the kinetic energy, and the Lorentz factor on the same page.

1. 1 Open the input panel: Scroll to the top of the page to find the accelerating voltage input. The default is 100 V, which reproduces a textbook cathode-ray-tube electron.
2. 2 Enter the accelerating voltage: Type the voltage in volts that the electron falls through from rest. Values from 0 V up to 10 MV are accepted. Negative values are rejected with an inline error.
3. 3 Read the speed in m/s and km/s: The first two result rows show the electron velocity in metres per second and in kilometres per second from the relativistic relation.
4. 4 Check the fraction of c and gamma: The third row shows v/c and the fifth row shows gamma. v/c below 0.1 means the classical formula is good; above 0.2 the relativistic branch is the right model.
5. 5 Use the kinetic energy row for eV problems: The fourth row reports the electron kinetic energy in electron-volts, numerically equal to the input voltage. Read this row for photoelectron or thermionic problems.

If a modern-physics problem gives the stopping potential of a photoelectron as 2.0 V and asks for the maximum kinetic energy and speed, the electron speed calculator returns KE = 2 eV, v = 8.39e5 m/s, v/c = 0.0028, and gamma = 1.0000039, the textbook photoelectron answer without any unit-conversion steps.

When the same electron scatters off a photon next, the Compton Wavelength Calculator returns the wavelength shift from the scattering angle so the photon-electron energy exchange matches the kinetic energy just computed.

Five practical benefits show up when the same accelerating voltage feeds downstream quantities.

• • Single relativistic formula: The page always runs v = c sqrt(1 - 1/gamma^2), so the result is correct from sub-volt photoelectron voltages up to multi-MeV linac energies without switching formulas.
• • Four complementary outputs: Speed in m/s, speed in km/s, fraction of c, kinetic energy in eV, and gamma are all returned together from one input.
• • Electron-volt friendly: The kinetic energy row is in electron-volts, so a problem stated in eV can be entered as volts and read back in the same unit.
• • Lorentz factor readout: The gamma row makes the relativistic correction explicit: below 1.02 the classical formula is fine; above 1.10 the relativistic branch is doing real work.
• • Sanity check against drift velocity: The m/s row makes it obvious that a 100 V electron is faster than 1% of c, while drift velocity in copper is on the order of millimetres per second.

For a one-off homework problem a single v = sqrt(2 e V / m_e) on the back of an envelope is enough. The benefits compound when a problem set or lab report needs the same accelerating voltage to feed downstream quantities such as the de Broglie wavelength or the bending radius in a magnetic field.

For the much slower drift motion of conduction electrons in a metal or semiconductor, the Electrical Mobility Calculator returns the drift velocity from the electric field and the carrier mobility on the same voltage-style input.

Four factors affect the speed the page returns, and two limitations tell you where it stops matching a more elaborate model.

• • The page models a free electron in a uniform electric field. In a real CRT or electron microscope the field is non-uniform and bremsstrahlung radiation is ignored.
• • Spin and magnetic-moment effects are not included. The result is the speed of a structureless point particle, so the page is the right tool for a kinematics problem but not for spin-orbit coupling or fine-structure calculations.

These limitations are deliberate. The electron speed calculator is built for the introductory and intermediate physics problems where the electron behaves like a classical or relativistic point particle in an external field, and it stays in that regime on purpose so the same input always returns the textbook answer.

According to Hyperphysics, the total relativistic energy of an accelerated electron is gamma times its rest energy m_e c^2, and the difference gamma - 1 times m_e c^2 is the kinetic energy that drives the speed v = c sqrt(1 - 1/gamma^2) used by this calculator.

When the accelerating voltage is set by an RC charging curve rather than a fixed supply, the Capacitor Charge Time Calculator returns the time constant and the capacitor voltage as a function of time that feeds the calculator as V(t).