Acceleration in an Electric Field Calculator (a = qE/m for electrons, protons, ions)

Understanding the electric field acceleration calculator — concept and principle

This calculator finds how fast a charged particle (an electron, proton, alpha particle or a custom ion) accelerates in a uniform electric field, and how fast and how far it goes from rest in a given time. It is used in physics courses and for rough estimates in electron guns, ion sources and particle detectors.

How it is calculated

F = qE; a = qE ÷ m

v = a t; d = ½ a t²; kinetic energy = ½ m v² (shown in eV)

• E is the field strength in V/m, t the time in nanoseconds (converted to seconds).
• q and m are the particle's charge and mass: e = 1.602176634 × 10⁻¹⁹ C and mₑ = 9.1093837139 × 10⁻³¹ kg for an electron.
• Positive charges accelerate along the field, negative charges against it.

The principle

A field of E volts per metre pushes on each coulomb of charge with E newtons. Newton's second law turns that force into an acceleration, and in a uniform field the acceleration is constant, so the usual constant-acceleration formulas apply. The work done by the field, qEd, becomes kinetic energy. An electron is about 1,836 times lighter than a proton, so the same field accelerates it about 1,836 times faster.

Worked example

Default inputs: electron, field 1,000 V/m, 10 ns from rest.

1. Force = 1.602 × 10⁻¹⁹ × 1,000 = 1.602 × 10⁻¹⁶ N.
2. a = 1.602 × 10⁻¹⁶ ÷ 9.109 × 10⁻³¹ = 1.759 × 10¹⁴ m/s².
3. v = 1.759 × 10¹⁴ × 10⁻⁸ s = 1,759,000 m/s, which is 0.5867% of the speed of light.
4. d = ½ × 1.759 × 10¹⁴ × (10⁻⁸)² = 0.008794 m.
5. Kinetic energy = 1,000 V/m × 0.008794 m = 8.794 eV. The electron moves against the field.

How to read the result

The kinetic energy in electronvolts always equals the field times the distance travelled, so it tells you the voltage the particle has crossed. Check the speed against the speed of light: above about 10% the non-relativistic formulas start to overestimate the speed, and they can even exceed c, which is impossible.

Common mistakes

• Entering the time in seconds instead of nanoseconds.
• Applying the formulas to fields that are not uniform.
• Ignoring relativity at high speeds.
• Forgetting that an electron moves opposite to the field direction.

Frequently asked questions

Why does a proton accelerate so much less than an electron?

The charges are equal in size, but the proton is about 1,836 times heavier.

How fast is an electron after crossing 1 V?

It gains 1 eV, which corresponds to about 593 km/s from rest.

Does gravity matter?

For electrons and ions in typical lab fields, gravity is many orders of magnitude smaller than the electric force.

Sources: CODATA 2022 recommended values of the fundamental physical constants (NIST) · OpenStax, University Physics Volume 2 (2016), chapters on electric fields