Tools / Physics
100% client-side · CODATA constants · non-relativistic modelLorentz Force Calculator
The force on a charged particle in electric and magnetic fields. Solve the electric part (F = qE), the magnetic part (F = qvB·sinθ), or the combined Lorentz force — plus cyclotron radius and orbital frequency as bonus readouts.
Mode 1 — Electric force F = q·E
Mode 2 — Magnetic force F = q·v·B·sinθ
| Fmax = qvB (θ = 90°) | — |
|---|---|
| Direction note | — |
Mode 3 — Combined fields: F = q(E + v×B)
Documented assumption: E is taken perpendicular to v×B, so the electric and magnetic forces add in quadrature: Ftot = √(FE² + FB²). For the general 3-D case, add the vector components yourself — the physics is in the FAQ.
| FE = qE (electric part) | — |
|---|---|
| FB = qvB·sinθ (magnetic part) | — |
| Resultant direction φ from E axis | — |
| FB/FE ratio | — |
Bonus — cyclotron motion readouts
| Orbit radius r = mv⊥/(qB) | — |
|---|---|
| Orbital frequency f = qB/(2πm) | — |
For the full resonance treatment (magnet design, m/q, relativistic check), see our Cyclotron Frequency Calculator.
Worked examples
Frequently asked questions
What is the Lorentz force?
The Lorentz force is the total force on a charged particle in electric and magnetic fields: F = q(E + v × B), where q is the charge, E the electric field, v the velocity and B the magnetic field. It has two parts: the electric part F = qE, which pushes along (or against) the field, and the magnetic part F = qvB·sinθ, which is always perpendicular to both v and B.
What is the formula for the magnetic force on a moving charge?
F = qvB·sinθ, with q in coulombs, v in m/s, B in tesla and θ the angle between v and B. Example: a proton (q = e) at v = 10⁶ m/s crossing B = 1 T at θ = 90° feels F = 1.602176634×10⁻¹³ N = 160.218 fN. Mode 2 solves this five ways: F, q, v, B or θ.
How do I solve for the angle θ between v and B?
Invert the formula: θ = arcsin(F / qvB). It only works if F ≤ qvB — a larger F is physically impossible and the calculator rejects it. Note there are two solutions: θ and 180° − θ give the same sinθ. Example: a force at 75% of the qvB maximum means θ = 48.590378° (or 131.409622°).
Is the proton cyclotron frequency 42.58 MHz per tesla?
No — that is a common mix-up. 42.57747892 MHz/T is the proton spin-flip (NMR) frequency, which comes from the magnetic moment, not the orbital motion. The orbital cyclotron frequency f = qB/(2πm) of a proton at 1 T is 15.24518644 MHz (a deuteron gives 7.62637793 MHz at the same field). Different physics, different number — this calculator uses the orbital one.
What is the cyclotron radius, and why is it a bonus readout?
A charge with velocity perpendicular to B circles with radius r = mv/(qB). Example: a proton at v = 5×10⁶ m/s in B = 2 T orbits with r = 2.609921 cm, circling at the orbital frequency above. Modes 2 and 3 show r and f whenever the particle mass is known (electron, proton, deuteron, alpha or a custom ion).
What does the “E ⊥ v×B assumed” note in combined mode mean?
Mode 3 adds the electric and magnetic parts in quadrature: Fₜₒₜ = √(Fᴱ² + Fᴮ²), with direction φ = atan2(Fᴮ, Fᴱ) from the E axis. That is exact only when E is perpendicular to v × B. In the general 3-D case the fields are not orthogonal and you must add the vector components: F = qE + q(v × B) component by component. The assumption is stated on screen so the number is never silently wrong.
Does the sign of the charge matter?
For the magnitude, no — the formulas use |q|, so an electron feels the same 1.602176634×10⁻¹³ N as a proton at the same v and B. For the direction, yes: the electric force on a negative charge points opposite to E, and the magnetic force follows the right-hand rule flipped (electron preset q = −e fills the sign automatically, and the result panel notes the direction).
Why is the magnetic force zero when v is parallel to B?
Because F = qvB·sinθ and sin 0° = 0: a charge moving along the field lines feels nothing. More deeply, the magnetic force is always perpendicular to v, so it does zero work — it can bend a trajectory into a circle but can never change the particle’s speed or kinetic energy. Only the electric part (qE) can accelerate or decelerate a charge.
What units can I use?
Charge: C, mC, µC, nC, pC or elementary charges e. Electric field: V/m, kV/m, V/cm or N/C. Velocity: m/s or km/s. Magnetic field: T, mT, µT or G (gauss; 1 T = 10,000 G). Angle θ in degrees. Forces auto-scale from kN down to aN, and every result also shows the exact SI value.
Lorentz force vs Coulomb force — what is the difference?
Coulomb’s law (F = kq₁q₂/r²) is the force between two charges — it is the microscopic origin of the electric field E. The Lorentz force F = q(E + v × B) is what a single charge feels once the fields exist, adding the magnetic term that Coulomb’s law lacks. Use our Coulomb’s Law calculator for charge–charge forces and this one for charge–in–field forces.
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