How this instrument works
J. J. Thomson worked out in 1881 that a moving charge ought to feel a sideways push from a magnetic field, and got his coefficient too small by half. Oliver Heaviside repaired that arithmetic in 1889. Hendrik Antoon Lorentz then set this magnetic term beside its electric partner while building electron theory, in papers of 1892 and 1895, and F = q(E + v × B) has carried his name since. Only that magnetic half is computed here — what a bare charged particle feels purely by crossing field lines.
Direction is what makes this force peculiar. It stands perpendicular to velocity and perpendicular to field at once, so it can never do work: magnets bend trajectories, they never speed anything up. Whatever slice of velocity runs parallel to B gets ignored outright — that is all sin θ is doing — while a surviving perpendicular slice wheels its particle around a circle of radius mv/(|q|B). Combine both and you get a helix, which is why solar wind protons corkscrew down geomagnetic field lines into polar skies instead of arriving head-on.
Some scale to calibrate against: Earth manages roughly 25 to 65 microtesla at ground level, a fridge magnet a few millitesla, a clinical MRI bore 1.5 or 3 tesla, LHC steering dipoles 8.3 tesla, and strongest steady laboratory fields about 45 tesla. Neutron stars reach 10⁸ tesla; magnetars go a thousand times beyond. Assumptions are tighter than they appear. A point charge is presumed, sitting in field uniform across whatever space it occupies, and radiation from its own acceleration goes unaccounted — harmless until accelerations turn violent enough for radiation reaction to bite. Relativistic speeds are fine, provided you abandon F = ma for dp/dt with p = γmv. Neutral particles carrying magnetic moments lie outside altogether; a neutron feels forces this expression cannot see.
- Enter your value under Charge, choosing C, mC, µC or nC from its unit menu. One electron carries 1.602 × 10⁻¹⁹ C, so single-particle work wants scientific notation instead.
- Enter Particle speed in m/s or km/h. Sign has no bearing on the answer; magnitude and angle carry everything.
- Type Magnetic flux density (T) straight in tesla — that field has no unit menu. Convert gauss yourself: 1 T = 10 000 G, so 50 µT of geomagnetic field is 0.5 G.
- Set Angle between velocity and field in deg or rad, measuring from v round to B. Not from v to the force, which sits 90° off both no matter what.
- Magnetic force reports in N, mN or kN. Orientation stays yours to work out: right hand for positive charge, mirrored for anything negatively charged.
Worked example — one coulomb, one tesla, one newton
Set Charge to 1 C, Particle speed to 1 m/s, Magnetic flux density (T) to 1, and Angle between velocity and field to 90°. Every factor equals unity and sin 90° = 1, giving F = 1 × 1 × 1 × 1 = 1 newton, exact to every digit your display can hold. That line simply restates a tesla's operational meaning backwards: one tesla is whatever flux density shoves a one-coulomb charge, crossing it squarely at one metre per second, with one newton.
Which makes this quartet a useful bench test whenever you suspect a units bug. Wind Angle between velocity and field down to 45° and output falls to 0.707 N; at 30° you get 0.5 N; at 0° it disappears completely, because charge travelling along field lines goes undisturbed. For something nearer bench reality, try this sheet's own defaults — 1 mC of charge at 1000 m/s across a 0.5 T ferrite magnet, squarely on, which also lands at 0.5 N.
Questions
Why can a magnetic field never change a particle's speed?
Because force comes out perpendicular to velocity, always. Power delivered is F·v, and perpendicular vectors dot to zero, so kinetic energy holds steady while direction swings. Cyclotrons exploit exactly this split: magnets supply steering only, and every bit of acceleration comes from an electric field across gaps between dees. Anything that genuinely speeds charge up is electric somewhere in its lineage, however magnetic the hardware looks.
Which angle belongs in θ?
Measured from velocity round to magnetic field, anywhere between 0 and 180°. Since sin θ is symmetric about 90°, entering 120° gives what 60° gives. Two slips dominate here. First, feeding in an angle read off to the force direction, which is meaningless — force lies 90° from both vectors by construction. Second, typing radians while that field still reads deg: 1.57 degrees returns roughly 0.027 of what you expected, which is a suspicious enough number to catch.
How does this relate to F = B·I·L·sin θ for a current-carrying wire?
One is a sum of the other. A wire holds n charge carriers per cubic metre drifting at v, so current I = nqvA, and adding up |q|vB sin θ over every carrier inside a length L collapses neatly to BIL sin θ. Use the wire version for motors, rails and busbars where current is what your meter reads; use this page when you have an individual particle with a known charge and speed, as in beam optics or mass spectrometry.
When does the magnetic term outweigh the electric one?
Compare vB against E directly, since both terms sit in the same bracket. An electron at 10⁶ m/s through Earth's 50 µT field sees vB = 50 V/m — trivial beside the megavolts per metre inside an accelerating structure, but decisive over the vast field-free stretches a beam actually spends its life in. Slow charges barely notice magnetic fields at all, which is why electrostatics governs chemistry while magnetism governs particle beams and plasma.
Does this still work at relativistic speeds?
The force expression itself is exact at any speed — it was built for a relativistic theory before relativity had a name. Newton's second law is what breaks. Swap F = ma for F = dp/dt with p = γmv and everything stays honest. One consequence has real hardware behind it: orbital frequency falls as γ climbs, so a fixed-frequency cyclotron slips out of step with its own beam past about 20 MeV for protons. Synchrocyclotrons and synchrotrons exist to chase that drift.
Why does an electron and a proton give the same answer here?
Because this sheet takes |q|, a magnitude, and returns a magnitude. Charges of −1.602 × 10⁻¹⁹ C and +1.602 × 10⁻¹⁹ C at matching speed through matching field feel pushes of identical strength in exactly opposite directions. Point a right hand along v, curl toward B, and your thumb gives the direction for positive charge; flip it for negative. Velocity selectors and mass spectrometers depend on that opposition, since beams of either polarity bend apart into separate detectors.