Lorentz Force Calculator
The Lorentz force is the combined electric and magnetic force felt by a charged particle moving through electromagnetic fields. Enter charge, velocity, magnetic field strength, and the angle between the velocity and field to get the magnetic force. Switch to "solve for" mode to work backwards from any known quantity. Particle presets for electrons, protons, and alpha particles fill in the charge automatically.
Formula
Worked example
An electron (q = -1.602e-19 C) moving at 1e7 m/s through a 0.5 T field at 90 degrees: F = 1.602e-19 x 1e7 x 0.5 x sin(90) = 8.01e-13 N. The negative charge means the actual force is opposite to the right-hand rule result.
What is the Lorentz force?
The Lorentz force is the total electromagnetic force on a charged particle moving through electric and magnetic fields. It was formulated by Hendrik Lorentz in the 1890s and is fundamental to electrodynamics. The full expression is F = q(E + v x B), where E is the electric field, v is the particle velocity, and B is the magnetic field. This calculator focuses on the magnetic component F = qvB sin(θ), which depends on how the particle direction relates to the field direction. The force is always perpendicular to both the velocity and the magnetic field, meaning it curves the path without doing work on the particle.
How to use this calculator
Select "Solve for" to choose which quantity you need. In the default "Force (F)" mode, enter the charge, velocity, field strength, and the angle between them; the magnetic force appears instantly. To work backwards, switch to "Charge", "Velocity", "Magnetic field", or "Angle" mode and enter the force plus the three known quantities. Use particle presets (electron, proton, alpha particle, muon) to fill in the charge automatically. All five quantities support multiple units: charge can be in coulombs, microcoulombs, nanocoulombs, or elementary charges; velocity in m/s, km/h, ft/s, or mph; field in tesla, millitesla, microtesla, or gauss; force in newtons, millinewtons, micronewtons, or dynes. The steps panel shows the full worked calculation with your exact numbers.
The right-hand rule and force direction
The Lorentz force formula gives the magnitude, but the direction requires the right-hand rule. Point your right-hand fingers along the velocity vector v, then curl them toward the magnetic field B. Your thumb points in the direction of the force on a positive charge. For a negative charge such as an electron, the force is opposite: reverse the thumb direction or use the left hand. When the velocity is parallel to the field (angle = 0 or 180 degrees), sin(θ) = 0 and the magnetic force is zero. Maximum force occurs at 90 degrees perpendicular - the particle is deflected into circular motion at that point.
Applications: cyclotrons, mass spectrometry, and auroras
The Lorentz force underpins many technologies. In a cyclotron, charged particles spiral outward under a magnetic field, gaining energy each half-orbit from an electric field. The radius of the circular path (r = mv/qB) grows with speed, letting engineers accelerate particles to high energies in a compact device. Mass spectrometers use the same principle: different masses curve differently in a magnetic field, separating ions by mass-to-charge ratio. Hall effect sensors detect current or position by measuring the sideways voltage the Lorentz force creates across a conductor. On a cosmic scale, Earth's magnetic field deflects the solar wind, funneling charged particles toward the poles where they create the aurora borealis and aurora australis.
Common particles and magnetic field strengths
| Particle / Field | Value | Notes |
|---|---|---|
| Electron charge | -1.602e-19 C | Elementary charge, negative |
| Proton charge | +1.602e-19 C | Elementary charge, positive |
| Alpha particle charge | +3.204e-19 C | 2 x elementary charge |
| Earth surface field | 25-65 µT | Varies by latitude |
| Refrigerator magnet | ~5 mT | Typical permanent magnet |
| MRI scanner | 1.5-3 T | Clinical diagnostic range |
| Lab electromagnet | 1-20 T | Research-grade magnets |
| Strongest continuous lab field | 45.5 T | National High Magnetic Field Lab, 2019 |
| Neutron star surface | ~1e8 T | Extreme astrophysical field |
Reference values for Lorentz force calculations.
Frequently asked questions
Why is the Lorentz force zero when the particle moves along the field?
The magnetic force depends on sin(θ), the sine of the angle between the velocity and the field. When θ = 0 or 180 degrees, sin(θ) = 0, so the force vanishes. Only the component of velocity perpendicular to the field produces a force. A particle moving exactly along a field line continues in a straight line without deflection.
Does the magnetic force do work on a charged particle?
No. The Lorentz magnetic force is always perpendicular to the velocity, so it has no component along the direction of motion. Work is force times displacement in the direction of force (W = F · d), and a perpendicular force contributes nothing. The magnetic force changes the direction of motion but not the speed or kinetic energy. This is why a particle in a uniform magnetic field follows a circular or helical path at constant speed.
What is the difference between the electric and magnetic components of the Lorentz force?
The full Lorentz force is F = q(E + v x B). The electric component qE acts along the electric field direction regardless of the particle speed, and it does work, accelerating or decelerating the particle. The magnetic component q(v x B) depends on the particle velocity and the magnetic field, acts perpendicular to motion, does no work, and is zero for a stationary particle. In most textbook problems "Lorentz force" refers to the magnetic component only.
How do I find the direction of the force on an electron?
Apply the right-hand rule to get the direction for a positive charge, then reverse it for the electron. Point fingers in the velocity direction, curl toward B; your thumb points in the force direction for positive charges. Since the electron has negative charge, the actual force is exactly opposite. Alternatively, use the left-hand rule directly: left hand fingers along v, curl toward B, and the left thumb points toward the force on the electron.
What units should I use for a calculation involving an electron in a 2 T MRI field?
Use the electron preset (charge = -1.602e-19 C automatically), set B to 2 T, and enter the electron velocity in m/s. A typical diagnostic MRI scanner produces a 1.5-3 T field. At 90 degrees and 1e6 m/s, the force would be about 3.2e-13 N, far too small to feel but significant for quantum-scale particles. Using the elementary charge unit (e) for the input and nanonewtons or piconewtons for the output often gives more readable numbers at particle scales.
Can this calculator handle the full Lorentz force including the electric field?
This calculator computes the magnetic part F = |q|vB sin(θ). For the full force F = qE + qvB sin(θ), add the electric force qE separately. At typical macroscopic scales, electric fields are specified in V/m, and the electric force is simply charge times field strength (q x E). The two components add vectorially, but they are often orthogonal in practical setups so magnitudes can be added directly.