Put a charged particle in an electric field and it feels a force equal to its charge times the field strength. That force produces an acceleration, the acceleration produces a speed, and the whole process is the basis of the cathode ray tube, the mass spectrometer, the electron microscope and every particle accelerator ever built. The charge acceleration calculator above works the complete chain from the field or voltage you specify.
Arb Digital publishes free physics calculators that carry a problem through to a usable answer rather than stopping at the first intermediate quantity. The electric field calculator on this site gives field strength, potential and force at a location; it deliberately stops before particle motion. This page picks the problem up from there and takes it to a velocity and a transit time.
What This Charge Acceleration Calculator Does
You describe the drive in whichever terms you have. Enter a potential difference and a gap and the tool derives the field. Enter a field strength and a gap and it derives the potential difference. Either way it computes the force on your particle, the resulting acceleration, the kinetic energy gained, the final speed and the time spent in the gap.
Particle presets cover the electron, the proton and the alpha particle with their published charge and mass, and a custom option lets you enter any charge in units of the elementary charge and any mass in kilograms or unified atomic mass units. That covers ions in a mass spectrometer as readily as electrons in a display tube.
The tool also watches the speed. Once the final velocity passes about a tenth of the speed of light, the classical expression starts to overstate it, and the page switches to the relativistic energy-momentum result and tells you it has done so. That threshold is reached by an electron at only a few kilovolts, which is why a purely classical treatment of this problem is misleading in exactly the situations people most often calculate.
How to Use It
- Choose the drive mode that matches your data. Accelerator specifications are usually quoted as a voltage; laboratory plate geometries are usually quoted as a field and a separation.
- Pick the particle before editing charge or mass. Selecting a preset overwrites both boxes with published values, so set it first and then adjust if you need a custom species.
- Enter charge as a multiple of e. A singly charged ion is 1, a doubly charged ion is 2, an alpha particle is 2. There is no need to type the elementary charge itself.
- Use atomic mass units for ions. Isotope masses are tabulated in u, and converting them by hand into kilograms is an unnecessary opportunity for error.
- Read the relativity note. If the tool says the result is relativistic, the classical formula would have given a materially different and wrong answer.
The Formula: How the Acceleration Is Calculated
The force on a charge q in a field E is F = qE, and Newton's second law gives a = F ÷ m = qE ÷ m. Across a uniform gap the field and the potential difference are related by E = V ÷ d, so the acceleration can equally be written a = qV ÷ (md).
Energy is simpler still, because it does not care about the geometry. The work done on the particle is W = qV, and by the work-energy theorem that is the kinetic energy gained from rest. OpenStax University Physics Volume 2, section 7.2 on electric potential and potential difference, sets out this relation and defines the electronvolt as the energy a single elementary charge gains through one volt.
Classically the final speed is v = √(2qV ÷ m). Relativistically it is v = c√(1 − 1 ÷ γ²) with γ = 1 + W ÷ (mc²), and the tool uses this form whenever the classical answer would exceed a tenth of the speed of light. The particle masses and the elementary charge come from the NIST CODATA value for the electron mass and its companion entries in the same reference on constants, units and uncertainty.
Work the defaults. An electron crossing 1,000 V over a 0.01 m gap sees a field of 100,000 V/m. The force is 1.602 × 10−19 × 105 = 1.602 × 10−14 N, and dividing by the electron mass of 9.109 × 10−31 kg gives an acceleration of 1.759 × 1016 m/s². The energy gained is exactly 1 keV, and the classical speed √(2 × 1.602 × 10−16 ÷ 9.109 × 10−31) = 1.876 × 107 m/s, which is 6.3 per cent of the speed of light. The relativistic value is 1.873 × 107 m/s, about 0.15 per cent lower.
Why the Gap Length Changes Nothing About the Final Energy
This is the result that surprises people and it is worth stating carefully. Halving the gap while holding the voltage constant doubles the field, doubles the force and doubles the acceleration. But the particle now travels only half as far, and work is force times distance, so the work done is unchanged. It arrives with exactly the same kinetic energy, just sooner.
That is why accelerator energies are quoted in electronvolts and megaelectronvolts rather than in newtons or in metres per second. The electronvolt is a unit that encodes the only thing the geometry cannot change. A 10 keV electron is a 10 keV electron whether it crossed a millimetre or a metre.
It is also why the field shape does not matter for energy. The electrostatic field is conservative, so the work done moving a charge between two points depends only on the potential difference between them, not on the path or on how the field varies along it. The uniform-gap assumption in this tool affects the acceleration and the transit time, which are path-dependent, and leaves the energy exact.
When Relativity Takes Over
The classical kinetic energy expression is the first term of a series, and it stops being a good approximation surprisingly early. For an electron, whose rest energy is 511 keV, an accelerating voltage of just 5 kV puts the speed near 14 per cent of the speed of light, and by 100 kV the classical formula overstates the speed by more than a fifth. Push the classical formula hard enough and it returns speeds above the speed of light, which is the clearest possible signal that it has failed.
Protons are far more forgiving because their rest energy is 938 MeV, nearly two thousand times larger. A 1 MV proton reaches only about 4.6 per cent of the speed of light and the classical answer is accurate to a fraction of a per cent. This is why electron optics engineers reach for relativistic corrections routinely while ion beam work at similar voltages often does not need them.
The tool reports the correct value in either regime and says which one it used. For the full energy-momentum treatment, including the momentum and the Lorentz factor, the relativistic kinetic energy calculator is the more detailed page, and the de Broglie wavelength calculator turns the resulting momentum into the wavelength that matters for electron microscopy.
Where This Sits Among the Other Electromagnetism Tools
This page is about a charge moving under an electric force. For the field itself, the electric field calculator and the Coulomb's law calculator handle sources and static forces, and the electric potential calculator deals with potentials without motion. For a charge in a magnetic field, the Lorentz force calculator covers the velocity-dependent force that curves a beam rather than speeding it up. Downstream, the kinetic energy calculator and the velocity calculator handle the mechanics once the particle is free, the acceleration calculator covers uniform acceleration generally, and the energy converter moves between electronvolts and joules.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the classical speed formula for electrons — it overstates the answer measurably above about 5 kV and becomes nonsense above 511 kV, where it can return speeds greater than light.
- Entering charge in coulombs when the box asks for multiples of e — that error scales the answer by nineteen orders of magnitude and is immediately visible in the force figure.
- Assuming a shorter gap gives more energy — it gives more force over less distance and exactly the same energy. Only the voltage sets the energy.
- Mixing atomic mass units and kilograms — an isotope mass of 4 in the kilogram box describes a particle heavier than a grain of sand.
- Forgetting that the field must be uniform — the acceleration and transit time here assume a constant field across the gap. Fringing fields and non-parallel electrodes change both.
Related Free Tools From Arb Digital
Work the source field with the electric field calculator, the Coulomb's law calculator and the electric potential calculator. Curve the beam instead of speeding it up with the Lorentz force calculator. For the high-energy regime, use the relativistic kinetic energy calculator and the de Broglie wavelength calculator. Ordinary mechanics is handled by the kinetic energy calculator, the velocity calculator and the acceleration calculator, and units by the energy converter. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
About 1.87 times ten to the seventh metres per second, which is roughly 6.3 per cent of the speed of light. The classical formula gives 1.876 times ten to the seventh and the relativistic treatment gives 1.873, so the correction is already visible at this modest voltage.
Acceleration equals charge times field strength divided by mass. If you know the potential difference and the gap instead of the field, substitute field equals voltage divided by gap, which gives acceleration equal to charge times voltage divided by mass times gap.
No, provided the potential difference is unchanged. A narrower gap raises the field and the force but shortens the distance over which it acts, and the two effects cancel exactly. The gap changes the acceleration and the transit time, not the energy or the final speed.
Whenever the classical speed exceeds roughly a tenth of the speed of light. For electrons that happens by a few kilovolts, because the electron rest energy is only 511 kiloelectronvolts. For protons, whose rest energy is 938 megaelectronvolts, classical mechanics remains adequate to several megavolts.
It is the kinetic energy one elementary charge gains when accelerated through a potential difference of one volt, equal to about 1.602 times ten to the minus nineteen joules. It is used because the energy of an accelerated particle depends only on the voltage it crossed.
Only in direction. The magnitude of the force, the acceleration, the energy and the final speed are identical for equal charge magnitudes. A negative charge simply accelerates toward higher potential while a positive one accelerates toward lower potential.
Because it is about 1,836 times heavier. Both gain identical kinetic energy from the same voltage, but speed goes as the square root of energy divided by mass, so the proton ends up roughly 43 times slower for the same accelerating voltage.
This tool is provided for educational and study use. It assumes a uniform field, a particle starting from rest, and no collisions, space charge or magnetic effects, so treat its output as an idealised physics result rather than a prediction for a real beamline.
