How this instrument works
An electron gun trades voltage for speed by plain bookkeeping. Charge q falling through potential difference V gives up qV joules, and with nowhere else for that energy to go it becomes kinetic: qV = ½mv². Rearranged, v = √(2qV/m). Notice what is absent — gap width, plate shape, field strength, transit time. Electrostatic force is conservative, so only endpoints count. A one-millimetre gap and a metre-long column at identical voltage deliver identical speed, which is why gun designers argue about beam optics rather than about final velocity.
Joseph John Thomson could not separate this charge from that mass, and said so plainly. His 1897 Cavendish experiments deflected cathode rays through crossed electric and magnetic fields and returned only their ratio q/m — over a thousand times larger than a hydrogen ion's, which is how he argued for a corpuscle far lighter than any atom. Twelve years on, Robert Millikan's oil drops pinned q by itself, and m followed by division. Today those two constants have quite different standing: since May 2019, q is defined rather than measured, fixed at exactly 1.602176634 × 10⁻¹⁹ C by SI's redefinition, while electron rest mass 9.1093837015 × 10⁻³¹ kg stays an experimental figure carrying roughly three parts in ten billion of uncertainty.
Three assumptions sit under that square root. Speed must stay well below c: weigh qV against an electron's rest energy of 511 keV, and once that ratio passes about 1.3 per cent — near 7 kV — this expression overstates speed by a full one per cent. A scanning electron microscope at 20 kV already runs 3 per cent high, and a colour television tube at 25 kV needed relativity for honest figures. Second, an electron must begin essentially at rest, whereas thermionic cathodes emit with a thermal spread of a few tenths of an electronvolt, blurring results badly below roughly 10 V. Third, flight has to happen in vacuum; above about 10⁻³ mbar, collisions with residual gas scatter a beam before it ever arrives.
- Type your figure into Accelerating voltage and pick V, kV or mV from its unit menu rather than keying an exponent. Guns are usually quoted in kilovolts.
- Read Electron speed in m/s, or switch that field to km/h for figures you can weigh against everyday motion.
- Sanity-check against light speed, 2.998 × 10⁸ m/s. Anything past roughly 3 × 10⁷ m/s means relativity has begun to bite and this reading runs high.
- For protons or heavier ions, multiply Electron speed by √(mₑ/m). Protons at any voltage travel 42.85 times slower.
Worked example — 100 volts across an electron gun
Set Accelerating voltage to 100 V, a bench figure typical of a low-energy diffraction column rather than of anything industrial. Energy first: qV = 1.602176634 × 10⁻¹⁹ × 100 = 1.602176634 × 10⁻¹⁷ J, which particle physicists simply call 100 eV. Divide by half an electron mass, then take a square root: v = √(2 × 1.602176634 × 10⁻¹⁹ × 100 ⁄ 9.1093837015 × 10⁻³¹) = √(3.51764 × 10¹³) = 5,930,969.58 m/s. Switch Electron speed to km/h and it reads 21.35 million.
Just under six million metres per second amounts to 1.98 per cent of light speed, where relativity's correction comes to 0.015 per cent — some 870 m/s, comfortably beneath any measurement such an apparatus could resolve. That margin is precisely why 100 V sits in safe territory for this sheet.
Push Accelerating voltage to 400 V and Electron speed returns 11,861,939.17 m/s — exactly double, not quadruple. Speed tracks a square root of voltage, so every doubling of speed costs four times as much potential, a scaling that quietly sets what every electron microscope ever built has cost its buyer.
Questions
Does gap width between electrodes change this answer?
No. Only total potential difference counts. Electrostatic force is conservative, so work done on an electron depends on where it starts and ends, never on its route or on how quickly it travelled. A 2 mm gap at 100 V and a 200 mm gap at 100 V hand over identical energy and identical Electron speed. Gap width does alter field strength, hence acceleration and transit time, and it decides whether you get sparks instead of beams — but arrival speed stays untouched.
At what voltage does this formula stop being accurate?
Near 7 kV it overstates speed by one per cent, and error climbs from there: 2.9 per cent at 20 kV, 8.6 per cent at 60 kV. Weigh qV against an electron's rest energy, 511 keV — while that ratio stays under one per cent or so, ordinary kinetic-energy bookkeeping holds fine. Above it, switch to v = c·√(1 − 1/(1 + qV/mc²)²), which keeps speed under c however hard you push. Being non-relativistic, this sheet will happily print a figure exceeding light speed past 255.5 kV.
Can I use this for a proton or an ion?
Yes, with one substitution — swap electron mass for your particle's. Speed scales as 1/√m at fixed voltage, so a proton, 1836 times heavier, arrives √1836 = 42.85 times slower: 138 km/s at 100 V against an electron's 5931 km/s. Charge state matters too, since a doubly ionised atom collects 2qV and gains a factor of √2. This sheet has electron mass and a single elementary charge wired in, so anything else wants that general form.
Why do people quote electron energy in volts instead of speed?
Because voltage is what an operator physically sets, and because energy in electronvolts converts to it by nothing more than a square root. One eV is what a single elementary charge gains crossing one volt: 1.602176634 × 10⁻¹⁹ J. A '30 keV beam' therefore means 30,000 V on a gun, and everyone in any laboratory knows what that implies for penetration depth and resolution without ever converting to metres per second. Speed becomes interesting mainly when timing does — in time-of-flight instruments, or when working out how long one pulse takes to cross a drift tube.
How does accelerating voltage relate to electron wavelength?
Inversely, through a shortcut worth memorising: λ ≈ 1.226 nm ⁄ √V. At 100 V an electron measures 0.123 nm, close to atomic spacing, which is exactly why Clinton Davisson and Lester Germer caught diffraction peaks off a nickel crystal at 54 V in 1927. Raising voltage shrinks wavelength and sharpens resolution, but only as a square root — climbing from 100 kV to 400 kV in a transmission microscope buys a factor of two, and relativity has to be folded in besides.
What mistake do people actually make with this?
Entering energy where voltage belongs. A quantity quoted in joules or in keV is not volts: 1.602 × 10⁻¹⁷ J means 100 V, and 30 keV means 30,000 V, not 30. Running a close second is assuming an electron departs from rest when it does not — thermionic cathodes emit with spreads of a few tenths of an eV, which barely register at kilovolts but matter below about 10 V, where that spread is no small slice of total energy.