Calculate the acceleration of charged particles (electrons, protons, alpha) in electric fields using F=qE and a=qE/m with energy and velocity outputs.
When a charged particle enters an electric field, it experiences a force F = qE, where q is the particle's charge and E is the electric field strength. This force produces an acceleration a = qE/m according to Newton's second law. This principle is fundamental to particle accelerators, cathode ray tubes, mass spectrometers, and ion propulsion systems.
This calculator determines the acceleration, force, final velocity, and kinetic energy of various charged particles in a uniform electric field. Choose from common particles — electrons, protons, alpha particles, deuterons, and muons — or specify the electric field directly or via voltage and plate separation.
The tool also warns you when relativistic speeds are reached, provides a comparison table showing how different particles behave in the same field, and displays the kinetic energy in both joules and electron-volts (eV), the standard energy unit in particle physics. Check the example with realistic values before reporting.
Calculating particle dynamics in electric fields by hand involves juggling very small numbers and unit conversions between joules and electron-volts. This calculator handles all that instantly, correctly showing results in scientific notation for the extreme values typical in particle physics.
The multi-particle comparison table lets you quickly see how different particles respond to the same field, which is essential for understanding mass spectrometry, ion beam technology, and particle detector design.
Force: F = qE. Acceleration: a = F/m = qE/m. Final velocity (from rest): v = √(2ad). Kinetic energy: KE = ½mv². Energy in eV: KE / 1.602×10⁻¹⁹.
Result: 1.759 × 10¹⁴ m/s²
An electron (m = 9.109 × 10⁻³¹ kg, q = 1.602 × 10⁻¹⁹ C) in a 1000 V/m field experiences a force of 1.602 × 10⁻¹⁶ N, producing an enormous acceleration of 1.759 × 10¹⁴ m/s².
When a charged particle is placed in an electric field, it experiences a Coulomb force proportional to its charge and the field strength. For a uniform field between parallel plates, E = V/d where V is the voltage and d is the plate separation. This simple setup is the basis for many important technologies.
**Cathode ray tubes** (CRTs) accelerated electrons using electric fields to create images on phosphor screens. **Mass spectrometers** use electric and magnetic fields to separate ions by mass-to-charge ratio. **Particle accelerators** like the Large Hadron Collider use oscillating electric fields to push protons to 99.9999991% the speed of light. **Ion thrusters** on spacecraft accelerate xenon ions to produce thrust in the vacuum of space.
The classical formula a = qE/m works perfectly for everyday speeds. However, as particles approach the speed of light, their effective mass increases according to special relativity. The relativistic mass is γm where γ = 1/√(1 − v²/c²). At the LHC, protons have a Lorentz factor of about 7,500, making them behave as if they were thousands of times heavier than at rest.
Electrons have the same charge magnitude as protons but are about 1,836 times lighter. Since a = F/m, the same force produces a much larger acceleration for the less massive electron.
An eV is the kinetic energy gained by an electron accelerated through a potential difference of 1 volt. It equals 1.602 × 10⁻¹⁹ joules. Particle physicists use eV because joule values are inconveniently tiny.
When a particle reaches a significant fraction of the speed of light (typically >1%), classical mechanics becomes inaccurate. At 10% of c, the error is about 0.5%. The calculator flags when relativistic corrections are needed.
This calculator assumes a uniform electric field. For non-uniform fields (like those near point charges), the acceleration varies with position and requires integration.
Particle accelerators use sequences of electric fields to repeatedly accelerate charged particles. The same F = qE principle applies, but fields are switched on and off to provide cumulative acceleration.
Neutral particles (zero net charge) experience no net force from a uniform electric field. Only charged particles are accelerated.