Lorentz Force Calculator

Physics

The configuration

One set of entries feeds every panel, so the force, its direction, the orbit it implies and the charts all describe the same particle. Switching mode keeps what you have typed.

F = |q| · v · B · sin θ

The classic right-hand-rule problem. The force is perpendicular to both v and B, so it turns the particle without ever changing its speed.

Worked example

Loads a complete scenario in one click.

Solve for

Give the force and the rest, and the missing quantity is recovered.

Decimal places

0–10; extreme magnitudes switch to scientific notation.

The particle and the fields

Particle preset

Fills in the charge and the rest mass together.

Charge q

Charge unit

1836 times heavier than an electron, so the same field bends it 1836 times less sharply at the same speed.

Velocity v

Velocity unit

Flux density B

Field unit

Angle θ between v and B (degrees)

0° and 180° give no magnetic force at all.

Particle mass m

Optional — unlocks the radius, period and frequency.

Mass unit

Relativistic momentumOffUses γmv for the radius and period above 0.1 c.

Show the workingShownFormula, substitution and simplification.

Number of charge carriers

Optional — multiplies the single-particle force for a bunch.

Lorentz force — Magnetic force on a moving charge

1.4019e-7 µN

Force in your unit	1.4019e-13 N
Force in µN	1.4019e-7 µN
Force in N	1.4019e-13 N
Force in dyn	1.4019e-8 dyn
Force in lbf	3.1516e-14 lbf
Force in kgf	1.4295e-14 kgf
Work done by the magnetic force	0.0000 J

Derived quantities

Orbit radius r	0.07456918 m
Cyclotron period T	187.4128 ns
Cyclotron frequency f	5.3358 MHz
Angular frequency ω	3.352592e+7 rad/s
Centripetal acceleration a	8.381479e+13 m/s²
Lorentz factor γ	1.0000348 (not applied)
Charge in your unit	1.0000 e
Particle mass	1.673e-27 kg = 1.6726e-27 kg
Speed	2500000.0000 m/s = 2500000.0000 in your unit
Flux density	350.0000 mT = 0.3500 T

Magnetic forces do no work

A magnetic force is always perpendicular to the velocity, so v · F = 0 and the work it does is exactly zero. A magnetic field can bend a beam but never speed it up; only the electric term can change a particle's kinetic energy.

The working

Formula	F = \|q\| · v · B · sin θ
Substitute	F = \|1.602177e-19\| × 2500000 × 0.35 × sin 90.0000°
Simplify	sin 90.0000° = 1.000000, so F = 1.401905e-13 N
Cyclotron radius	r = m v⊥ / (\|q\| B) = 0.07456918 m
Cyclotron period	T = 2π m / (\|q\| B) = 1.874128e-7 s, f = 1/T = 5335815 Hz

Batch calculation

One row per case as q, v, B, θ in SI units — coulombs, metres per second, tesla and degrees. A malformed row is reported by line number and skipped rather than failing the whole batch.

Rows

Line	q (C)	v (m/s)	B (T)	θ	F
1	1.602e-19	2.500e+6	0.35	90.0000°	1.4017e-7 µN
2	-1.602e-19	1.000e+7	0.02	45.0000°	2.2656e-8 µN
3	2.000e-6	3.000e+4	0.5	90.0000°	30.0000 mN
4	3.204e-19	5.000e+5	1.2	30.0000°	9.6120e-8 µN

Particles

Particle	Charge	Rest mass
Electron	−1 e	9.109384e-31 kg
Proton	+1 e	1.672622e-27 kg
Neutron	0	1.674927e-27 kg
Alpha particle (He²⁺)	+2 e	6.644657e-27 kg
Deuteron (²H⁺)	+1 e	3.343584e-27 kg
Muon (µ⁻)	−1 e	1.883532e-28 kg
Ar-40 ion (singly ionised)	+1 e	6.635910e-26 kg
Ca-40 ion (doubly ionised)	+2 e	6.635944e-26 kg

How strong is that field?

Setting	B	Note
Interstellar space	0.100 nT	Galactic magnetic fields, measured from the polarisation of starlight.
Earth's surface field	50.000 µT	About 50 µT, tilted steeply from horizontal outside the tropics.
Refrigerator magnet	5.000 mT	A few millitesla at the surface, falling off fast.
Loudspeaker gap	1.000 T	Roughly 1 T across the voice-coil gap of a moving-coil driver.
Clinical MRI	3.000 T	1.5 T and 3 T are the common clinical strengths.
Strongest continuous laboratory magnet	45.000 T	A hybrid resistive/superconducting magnet, and a good sanity ceiling.

Reading the diagram

A field pointing out of the screen is drawn as ⊙ — the tip of an arrow coming toward you — and one pointing into the screen as ⊗, its tail feathers going away. In this tool the +z axis is the out-of-screen direction, and it is drawn toward the lower left so that all three axes stay visible at once.

Directly out of the screen (+z) is what a force along +z reads as; Directly toward −y (down the page) is the usual answer for a positive charge moving along +x through a field along +z.

Lorentz Force Calculator – F = q(E + v × B) in Every Form You Need

The Lorentz force is the single relation that ties electricity and magnetism to motion. Every electric motor, cathode-ray tube, mass spectrometer, cyclotron and Hall sensor is an application of one equation: F = q(E + v × B). This Lorentz force calculator evaluates it in all the forms a student or engineer actually meets — the scalar magnetic force on a moving charge, the electric force on a charge at rest, both fields together, the macroscopic force on a current-carrying wire, and the full three-component cross product — and then reports the direction in plain language, because a sign error is far more common than an arithmetic one.

The magnetic force on a moving charge

When only a magnetic field acts, the magnitude reduces to F = |q| · v · B · sin θ, where θ is the angle between the velocity and the field. A proton (q = 1.602176634 × 10⁻¹⁹ C) travelling at 2.5 × 10⁶ m/s straight across a 0.35 T field feels 1.4019 × 10⁻¹³ N. That looks vanishingly small until you divide by the proton mass: the acceleration is 8.38 × 10¹³ m/s², which is why the particle turns on a centimetre scale rather than continuing in a straight line.

The sin θfactor is the whole story of the geometry. At 90° the force is at its maximum; at 0° or 180° — motion straight along the field line — it is exactly zero, which is how charged particles from the solar wind funnel down the Earth's field lines into the auroral ovals instead of being turned aside.

The electric force and the combined law

The electric term, F = qE, does not care how fast the charge is moving. It points along E for a positive charge and against it for a negative one, so an electron in a 1.2 × 10⁵ V/m deflection field is pushed the opposite way from a proton in the same field. Put both terms together and you get a velocity selector, or Wien filter: arrange E and v × B to oppose each other and only particles at the speed v = E/B pass through undeflected. With E = 6.0 × 10⁴ V/m and B = 0.20 T that speed is exactly 3.0 × 10⁵ m/s, and the calculator flags the balance when your entered speed matches it.

The cross product, term by term

For geometry that is not a clean 0° or 90°, the vector form is unavoidable:

(v × B)ₓ = v_y·B_z − v_z·B_y, (v × B)_y = v_z·Bₓ − vₓ·B_z, (v × B)_z = vₓ·B_y − v_y·Bₓ

A 2.0 µC charge moving at (3.0 × 10⁴, 1.0 × 10⁴, 0) m/s through B = (0, 0, 0.50) T has v × B = (5.0 × 10³, −1.5 × 10⁴, 0), so F = (1.0 × 10⁻², −3.0 × 10⁻², 0) N with magnitude 3.16 × 10⁻² N. The tool prints each determinant separately so you can check your own working line by line.

Force on a current-carrying wire

Summing the microscopic force over every carrier in a conductor gives the macroscopic form F = B · I · L · sin θ. A motor conductor carrying 4.5 A through 25 cm of a 0.60 T gap feels 0.675 N — multiply by the number of turns and the armature radius and you have the torque. This is also the relation behind loudspeaker voice coils and railguns.

Circular motion, cyclotrons and the relativistic correction

When the magnetic force is the only force and v is perpendicular to B, it acts as a pure centripetal force, qvB = mv²/r. That gives the three quantities that matter more than the force itself:

r = mv / (|q|B), T = 2πm / (|q|B), f = |q|B / (2πm)

For the proton above, r = 7.46 cm, T = 187 ns and f = 5.34 MHz. Notice that the period does not contain the speed at all — a faster particle simply traces a wider circle in the same time, which is exactly why a cyclotron can drive its dees at one fixed frequency. Above about a tenth of the speed of light that stops being true, because the momentum becomes γmv with γ = 1/√(1 − v²/c²); the relativistic toggle applies that factor to the radius and the period.

A magnetic field can never speed a particle up

The magnetic force is perpendicular to the velocity at every instant, so the work it does is exactly zero and the kinetic energy never changes. Only the electric term can accelerate a charge in the everyday sense — the magnet just steers.

Getting the direction right

Point the fingers of your right hand along v, curl them toward B, and your thumb gives v × B. For a positive charge that is the force direction; for an electron it is the exact opposite. On a diagram, a field coming out of the page is drawn ⊙ and one going into the page ⊗. The calculator states the result both as a unit vector and as a sentence — "out of the screen", "toward −y" — so the answer can be checked without redrawing the geometry.

Where the idealisation stops

These formulas assume uniform fields, a point particle, no radiation and no space charge. Real magnets have fringe fields, real beams repel themselves, and an accelerating charge radiates energy away. Use the result as the first-order answer a full field solve should agree with.

Working with units

Everything is normalised to SI before any arithmetic happens. Charge can be entered in coulombs down to picocoulombs or in elementary charges; velocity in m/s, km/h, mph or fractions of c; the magnetic field in tesla, millitesla, microtesla or gauss; the electric field in V/m, N/C, kV/m, MV/m, V/cm or V/mm; and mass in kilograms, grams, unified atomic mass units or MeV/c². The force is reported simultaneously in newtons, dynes, pounds-force and kilograms-force, so 1.4019 × 10⁻¹³ N also reads as 1.4019 × 10⁻⁸ dyn for anyone working in CGS.

Is the Lorentz Force Calculator free?

Yes, Lorentz Force Calculator is totally free :)

Can I use the Lorentz Force Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Lorentz Force Calculator?

Yes, any data related to Lorentz Force Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this Lorentz force calculator work?

Pick the form of the law you need — magnetic force on a moving charge, electric force, both fields together, a current-carrying wire, or the full vector cross product — and every entry is normalised to coulombs, metres per second, tesla and volts per metre before F = q(E + v × B) is evaluated. A proton (q = 1.602176634 × 10⁻¹⁹ C) at 2.5 × 10⁶ m/s crossing 0.35 T at right angles gives 1.4019 × 10⁻¹³ N. That same unrounded number then drives the orbit radius, the period, the cyclotron frequency and the multi-unit restatement, so nothing is ever rebuilt from a rounded intermediate.

Why does a charge moving along the field line feel no force?

Because the magnetic term is a cross product, and a cross product of parallel vectors is zero. F = |q|vB·sin θ vanishes at θ = 0° and θ = 180° no matter how strong the field is, which is why a charged particle spiralling down a field line at the Earth's poles keeps going instead of being deflected. The tool says so explicitly at those angles rather than quietly printing a zero, and in vector mode it reports the angle it recovered from your components so you can see when v and B are accidentally parallel.

Why do magnetic forces never do work?

The force is perpendicular to the velocity at every instant, so v · F = 0 and the power delivered is exactly zero. A magnetic field can turn a particle, curve it into a circle or wind it into a helix, but it cannot change its speed or its kinetic energy — only the electric term qE can do that. This is why a cyclotron needs an alternating electric field across the dees to accelerate the beam; the magnet alone would keep the particle circling at the same speed forever.

How do I get the direction right for a negative charge?

Point your right hand along v, curl toward B, and your thumb gives the direction of v × B. That is the force direction for a positive charge; for an electron or any negative charge the force is the exact opposite, because multiplying by a negative q flips the vector. The tool applies the sign for you and states the answer in plain language — "out of the screen (+z)" or "mostly toward −y, tilted toward +x" — alongside the unit vector, because sign errors are by far the most common mistake with this law.

When do I need the relativistic option?

Above roughly a tenth of the speed of light. The force itself is unchanged, but the radius and period follow from the momentum, and above 0.1 c the classical mv understates the true γmv. At 2.5 × 10⁶ m/s γ is only 1.0000348, so the correction is a thirty-thousandth and safely ignorable; at 0.9 c γ is 2.294 and the real orbit is more than twice as wide as the classical estimate. The tool prompts you when your speed crosses that threshold and rejects any speed at or beyond c outright.

How accurate is this for a real beamline or motor?

The arithmetic is exact for the numbers you type, but the model is the textbook one: uniform fields, a point particle, no radiation and no space charge. Real magnets have fringe fields at their ends, real beams repel themselves, an accelerating charge radiates away energy, and a real motor conductor sits in a field that varies along its length — so F = BIL·sin θ gives the force on an idealised straight segment in a uniform gap rather than the true torque of a wound armature. Treat the results as the first-order answer that a full field solve should be checked against.