Lorentz Force Calculator
Use this Lorentz Force Calculator to compute the magnetic force acting on a charged particle moving through a magnetic field. Enter the particle's charge, velocity, the magnetic field strength, and the angle between the velocity and field vectors to instantly get the resulting Lorentz force in Newtons (metric) or pound-force (American system).
Lorentz Force Law
The Lorentz force is the combined effect of electric and magnetic fields on a moving charged particle. This calculator focuses on the magnetic component of the Lorentz force.
The Lorentz force law states that a magnetic field exerts a force on any charged particle that is moving through it. The magnitude of this force depends on three factors:
- The electric charge of the particle (q)
- The velocity of the particle (v)
- The strength of the magnetic field (B)
What makes the Lorentz force unique is its direction: it is always perpendicular to both the particle's velocity and the magnetic field direction. This means the force does not accelerate the particle along its path — it only curves the trajectory.
Importantly, if a charged particle moves parallel to the magnetic field lines (angle = 0° or 180°), the Lorentz force is zero. The force is maximum when the particle moves perpendicular to the field (angle = 90°).
Lorentz Force Equation
The Lorentz force equation is:
F = q × v × B × sin(α)
Where:
- F — Resulting force (Newtons in SI; pound-force in American units)
- q — Electric charge of the particle (Coulombs, C)
- v — Speed of the particle (m/s in SI; ft/s in American units)
- B — Magnetic field strength (Tesla in SI; Gauss in American units)
- α — Angle between the velocity vector and the magnetic field vector (degrees)
The sin(α) factor accounts for the angle between velocity and field:
- α = 90° → sin(90°) = 1 → maximum force
- α = 0° → sin(0°) = 0 → zero force (particle moves along field lines)
- α = 45° → sin(45°) ≈ 0.707 → about 70.7% of maximum force
Unit Systems
Metric (SI) System
- Charge (q): Coulombs (C)
- Velocity (v): meters per second (m/s)
- Magnetic field (B): Tesla (T)
- Force (F): Newtons (N)
American (Imperial) System
- Charge (q): Coulombs (C) — same as SI, no equivalent imperial unit
- Velocity (v): feet per second (ft/s); 1 ft/s = 0.3048 m/s
- Magnetic field (B): Gauss (G); 1 G = 10⁻⁴ T
- Force (F): pound-force (lbf); 1 N = 0.224809 lbf
Lorentz Force Applications
The Lorentz force has numerous practical and scientific applications:
Scientific Instruments
- Cyclotrons and Particle Accelerators — The Lorentz force bends the path of charged particles into circular orbits, allowing them to be accelerated in a spiral path. This is fundamental to particle physics research.
- Mass Spectrometers — By measuring how much a charged particle's trajectory curves in a magnetic field, scientists can determine the mass of atoms and molecules. The Lorentz force makes this identification possible.
- Cathode Ray Tubes (CRTs) — Older television sets and oscilloscopes used magnetic fields to deflect electron beams via the Lorentz force to draw images on screen.
Everyday Technology
- Electric Motors — In an electric motor, current-carrying conductors experience a Lorentz force in the magnetic field of the motor, producing rotational motion. This is the core operating principle of virtually all electric motors.
- Loudspeakers — The voice coil (carrying alternating current) sits in a permanent magnetic field. The Lorentz force pushes the coil back and forth, vibrating the speaker cone to produce sound.
- Railguns — An electromagnetic weapon that uses the Lorentz force on a current-carrying projectile between two conducting rails to accelerate it to extremely high velocities.
- Magnetohydrodynamic (MHD) Drives — Propulsion systems for ships that use the Lorentz force on seawater (a conductor) to push the vessel without moving parts.
Natural Phenomena
- Aurora Borealis / Australis — Charged particles from the solar wind are deflected by Earth's magnetic field through the Lorentz force, spiraling into the polar regions and creating the northern and southern lights.
- Van Allen Radiation Belts — Charged particles are trapped in Earth's magnetic field by the Lorentz force, forming toroidal belts of radiation around Earth.
Frequently Asked Questions
What is the Lorentz force?
The Lorentz force is the force experienced by a charged particle moving through electric and magnetic fields. The magnetic part of this force is F = qvB sin(α), where q is the charge, v is the velocity, B is the magnetic field strength, and α is the angle between velocity and field.
What happens when α = 0° (particle moves parallel to the field)?
When the angle is 0°, sin(0°) = 0, so the Lorentz force is zero. A particle moving exactly along the magnetic field lines experiences no magnetic force. This is why particle accelerators inject particles perpendicular to the magnetic field.
Does the Lorentz force change the speed of the particle?
No. Because the Lorentz force is always perpendicular to the velocity, it does no work on the particle and cannot change its speed or kinetic energy. It only changes the direction of motion, causing circular or helical trajectories.
What is a Tesla (T)?
The Tesla is the SI unit of magnetic flux density. 1 T = 1 kg/(A·s²). For reference: Earth's magnetic field is about 25–65 microtesla (μT); a typical MRI machine uses 1.5–3 T; the strongest continuous laboratory magnets reach ~45 T.
What is the relationship between Gauss and Tesla?
1 Tesla = 10,000 Gauss (10⁴ G). Gauss is the CGS unit commonly used in the American system for smaller magnetic fields. For example, Earth's field is about 0.5 Gauss = 0.00005 Tesla.
How is the Lorentz force related to the cross product?
In vector form, the Lorentz force is F = q(v × B), where × is the vector cross product. The magnitude of this cross product is |v||B|sin(α), which is why sin(α) appears in the scalar formula. The direction is given by the right-hand rule.