Acceleration in the Electric Field Calculator
Enter the particle charge, mass, electric field strength, initial velocity, and travel distance to find acceleration, electric force, final velocity, and kinetic energy gained. Choose a preset particle or enter custom values. Results update instantly and a step-by-step panel shows every line of working.
Formula
Worked example
An electron (q = -1.602e-19 C, m = 9.109e-31 kg) enters a 1000 N/C field from rest over 0.01 m: F = qE = -1.602e-16 N; a = F/m = -1.759e+14 m/s^2 (magnitude 1.79e+13 g); v = sqrt(2 * 1.759e14 * 0.01) = 1.876e+6 m/s; Delta KE = 0.5 * 9.109e-31 * (1.876e6)^2 = 1.602e-15 J = 10 eV.
Physics behind acceleration in an electric field
A charged particle placed in a uniform electric field experiences an electrostatic force described by Coulomb's law: F = qE, where q is the particle's charge in coulombs and E is the field strength in newtons per coulomb. Applying Newton's second law (F = ma) gives the acceleration: a = qE / m. Because both force and acceleration are vectors, their direction depends on the sign of the charge. A positive charge accelerates in the direction of the field; a negative charge, such as an electron, accelerates opposite to the field. The magnitude of the charge-to-mass ratio (q/m) determines how strongly a given field can accelerate the particle. An electron has an enormous q/m ratio of about 1.76 x 10^11 C/kg, so even a weak laboratory field produces accelerations billions of times larger than Earth gravity.
Kinematic extension: velocity, energy, and time
Once the acceleration is known, the familiar constant-acceleration kinematics apply across the field region. Starting from rest or from an initial speed v0, the particle reaches a final speed v = sqrt(v0^2 + 2 a d) after traveling a distance d. The kinetic energy gained equals the work done by the field: Delta KE = F * d = q * E * d = 1/2 m (v^2 - v0^2). In electron-volt units (1 eV = 1.602 x 10^-19 J), this matches the voltage equivalent: a particle carrying one elementary charge gains 1 eV for every 1 V of potential difference it crosses. The time to cross the region follows from the positive root of the quadratic d = v0 t + 1/2 a t^2. These relationships underpin particle accelerators, mass spectrometers, cathode-ray tubes, and ion drives.
Particle presets and real-world applications
The calculator includes presets for the electron, proton, and alpha particle because they appear in most classroom and laboratory contexts. Electrons are accelerated in electron microscopes and CRT displays, where field strengths of tens of thousands of volts per metre push them to velocities that are a meaningful fraction of the speed of light. Protons and heavier ions are accelerated in cyclotrons, synchrotrons, and linear accelerators for cancer radiotherapy and fundamental physics research. Alpha particles appear in Rutherford scattering experiments and nuclear decay studies. For any other charged body, including dust grains in plasma physics or macro-ions in electrospray ionization, select Custom and enter the actual charge and mass. The formula is identical regardless of particle type.
When the non-relativistic formula breaks down
The equations used here assume that the particle's speed remains well below the speed of light (c = 3 x 10^8 m/s). As a rough guide, relativistic corrections become noticeable above about 0.1 c (3 x 10^7 m/s) and essential above 0.5 c. An electron accelerated through only 14.5 kV already reaches about 0.23 c, so high-voltage electron optics and synchrotron beams require relativistic mechanics. For those cases, the relativistic momentum equation p = gamma m v replaces F = ma, and "acceleration" loses its simple constant meaning. The non-relativistic version here is accurate for low-voltage ion optics, household-scale electrostatics, and introductory coursework.
Common charged particles: charge, mass, and charge-to-mass ratio
| Particle | Charge (C) | Mass (kg) | q/m ratio (C/kg) | Typical use |
|---|---|---|---|---|
| Electron | -1.602e-19 | 9.109e-31 | -1.759e+11 | CRT displays, electron microscopy |
| Proton | +1.602e-19 | 1.673e-27 | +9.578e+7 | Particle accelerators, MRI |
| Alpha particle | +3.204e-19 | 6.645e-27 | +4.822e+7 | Nuclear physics, Rutherford scattering |
| Deuteron | +1.602e-19 | 3.344e-27 | +4.791e+7 | Nuclear reactions, cyclotrons |
| Muon | -1.602e-19 | 1.884e-28 | -8.535e+8 | Cosmic-ray detection, muon tomography |
Values are CODATA 2018 constants. The charge-to-mass ratio (q/m) determines how strongly a particle accelerates for a given field strength.
Frequently asked questions
What is the formula for acceleration in an electric field?
The acceleration of a charged particle in a uniform electric field is a = qE / m, where q is the particle charge in coulombs, E is the electric field strength in newtons per coulomb (or volts per metre), and m is the particle mass in kilograms. It combines Coulomb's force law (F = qE) with Newton's second law (a = F / m).
Does the sign of the charge matter?
Yes. A positive charge accelerates in the same direction as the electric field vector, and a negative charge accelerates in the opposite direction. The magnitude of the acceleration is the same for equal absolute charges, but the direction is reversed. Electrons, being negatively charged, always move against the field.
How do I convert the acceleration to g-forces?
Divide the acceleration in m/s^2 by 9.80665 (standard gravity). This calculator shows the g-force equivalent automatically. For context, an electron in a 1 N/C field accelerates at about 1.76 x 10^11 m/s^2, or roughly 18 billion g.
How is kinetic energy related to the electric potential difference?
The kinetic energy gained equals the work done by the field: Delta KE = q E d = q V, where V = E d is the potential difference across the distance d. For a particle carrying one elementary charge, this gives exactly 1 eV of energy per volt of potential difference. That is why the electron-volt is such a convenient unit in atomic and nuclear physics.
Can I use this calculator for heavy ions or charged dust?
Yes. Select Custom, enter the charge in coulombs and the mass in kilograms for your ion or particle, and the calculator applies the same formula a = qE / m. For doubly ionised oxygen (O^2+), for instance, q = 2 x 1.602e-19 C and m = 16 x 1.66e-27 kg. The field strength and distance inputs work the same way regardless of particle size.
Why does the electron accelerate so much faster than the proton?
The electron and proton carry the same magnitude of charge, but the proton is about 1836 times heavier. Because acceleration scales as 1/m for a fixed force, the proton's acceleration is 1836 times smaller than the electron's in the same field. This mass difference drives almost every particle-separation technique from mass spectrometry to isotope enrichment.