How this instrument works
This is Newton's second law with the pushing force supplied entirely by the electric term of the Lorentz force: F = qE, then a = F ⁄ m collapses the two steps into one division. A charge sitting in a field feels a force proportional to how much charge it carries and how strong the field is; divide that force by the particle's own mass and inertia turns it into a rate of speeding up. Nothing here needs the particle already moving — an electric field pushes on a charge at rest exactly as hard as on one in flight, which is what separates it from the field's magnetic sibling.
The formula is deliberately blunt about scale. Multiply charge and field together and the result is newtons of force no matter how tiny either factor is, because coulombs and volts-per-metre are both large units by atomic standards; a nanocoulomb sitting in a modest bench field still generates a measurable push. What tips the balance toward a big or small acceleration is mass sitting in the denominator — a fleck of soot or a droplet of ink weighs so little that a field most people would call unremarkable can out-accelerate gravity several times over, which is exactly why electrostatic precipitators and inkjet deflection plates work on modest voltages rather than needing a spark-gap field.
The formula assumes a field that is uniform along the line of travel and a particle small enough that its own charge does not noticeably distort that field — true inside the parallel-plate gap of a deflection unit or precipitator collector, less true near a sharp corona wire where the field varies from point to point. It also stays non-relativistic: push a light enough charge hard enough for long enough and this straight division stops matching reality once speed climbs toward a meaningful fraction of light speed, the same ceiling that limits every constant-force calculation in mechanics.
- Enter the particle's charge under Particle charge, choosing nC, µC or C from the unit menu; the default of 50 nC suits a small charged flake or droplet.
- Enter Electric field strength, V ⁄ m directly in volts per metre — this field carries no unit menu, so convert first if your source figure is in V/mm or kV/m.
- Enter Particle mass in mg or g; use the mass of the object actually being pushed, not its weight.
- Read Acceleration in m/s²; a built-in check blocks zero or negative mass, since the division has no answer without it.
Worked example — a 50 nC fleck in a 2,000 V/m field
Take a charged dust fleck carrying 50 nC (5 × 10⁻⁸ C) with a mass of 5 mg (5 × 10⁻⁶ kg), sitting in a 2,000 V/m field — roughly what two plates a centimetre apart produce at a modest 20 volts across them. a = qE ⁄ m = (5 × 10⁻⁸ × 2,000) ⁄ (5 × 10⁻⁶) = 20.0 m/s². That is delivered by a field weak enough to touch safely and a charge too small to feel as a static shock.
Twenty metres per second squared sounds modest beside a lightning bolt, but set against gravity's 9.8 m/s² it wins outright — this fleck accelerates twice as hard as it falls. That is precisely the trick electrostatic precipitators use to pull soot out of flue gas onto collector plates, and the same trick continuous inkjet printers use to steer charged ink droplets toward or away from the page, with no moving parts beyond the field itself.
Questions
Why is there no magnetic term in this formula?
Because the full Lorentz force is F = q(E + v × B), and this sheet supplies only the electric half. The magnetic term needs the particle already moving across a field it did not create; with B taken as zero, or the particle momentarily at rest, qE is the entire force. Add a magnetic field and crossing velocity and the path curves instead of running straight, which this formula alone cannot describe.
How is this different from the electric field of a point charge?
That covers the other half of the same physics. A point-charge field calculator finds E itself — the push per coulomb that some source charge creates in the space around it, E = kQ ⁄ r². This sheet starts from an E you already know or have measured and asks what it does to a different charged particle sitting inside it: multiply by that particle's own charge for force, then divide by its mass for acceleration.
Does the sign of the charge matter?
Yes, for direction, not size. A positive charge accelerates along the field lines; a negative charge of equal magnitude accelerates just as fast but the opposite way, because the qE product simply flips sign. Enter charge as negative and the instrument returns a negative acceleration — read that as same size, opposite direction, not as an error or a weaker push.
Why does a light particle accelerate so much faster than a heavy one?
Because mass sits in the denominator, and inertia is literally the property that resists a push. Two particles carrying identical charge in an identical field feel identical force, F = qE, but a particle with a thousandth of the mass converts that same force into a thousand times the acceleration. It is the same reason a struck ping-pong ball leaps away while a struck bowling ball barely rolls, replayed here with an electric push instead of a mallet.
What happens if the charge or field is entered as zero?
The acceleration reads zero — with no force acting, there is nothing to overcome the particle's inertia, so it stays put or continues at whatever constant velocity it already had. That differs from setting mass to zero, which the instrument blocks outright: zero mass would demand dividing by nothing, which has no defined answer, while zero charge or zero field simply describes a particle the electric part of the world is not touching.
Where does a formula like this actually get used?
In anything that steers or sorts charged particles with a field instead of a mechanical part: electrostatic precipitators pulling charged ash and soot out of flue gas onto collector plates, continuous inkjet printers deflecting charged ink droplets toward or away from the page, mass spectrometers accelerating ions down a flight tube, and the Millikan oil-drop experiment, which balanced this exact force against gravity to measure the electron's charge in 1909.