Physics · Electromagnetism · Magnetism
Magnetic Force Calculator
Calculates the magnetic force on a moving charged particle using the Lorentz force law.
Calculator
Formula
F is the magnetic force in Newtons (N); q is the electric charge of the particle in Coulombs (C); v is the speed of the particle in meters per second (m/s); B is the magnetic field strength in Tesla (T); θ is the angle between the velocity vector and the magnetic field vector in degrees.
Source: Griffiths, D.J. — Introduction to Electrodynamics, 4th Edition, Equation 5.1. Also codified in NIST's electromagnetic force definitions.
How it works
When a charged particle moves through a magnetic field, it experiences a force perpendicular to both its velocity and the direction of the field. This interaction is described by the Lorentz force law, one of the cornerstones of classical electromagnetism. Unlike an electric force, the magnetic force does no work on the particle — it only changes the direction of motion, not the speed. This is why charged particles in uniform magnetic fields follow circular or helical paths rather than accelerating in a straight line.
The governing formula is F = qvB sin(θ), where q is the particle's charge in Coulombs, v is its speed in meters per second, B is the magnetic field strength in Tesla, and θ is the angle between the velocity vector and the magnetic field vector. The force reaches its maximum value when the particle moves perpendicular to the field (θ = 90°), and drops to zero when the particle moves parallel to the field (θ = 0° or 180°). The sign of the force depends on the sign of the charge — a positive and negative charge moving in the same direction will be deflected in opposite directions.
Practical applications of this calculation span a wide range of technologies and scientific fields. Cyclotrons and synchrotrons rely on the magnetic force to bend particle beams into circular paths. Mass spectrometers use it to separate ions by mass-to-charge ratio. The Van Allen radiation belts around Earth trap energetic particles through this same mechanism. In everyday engineering, Hall effect sensors exploit the magnetic force to measure current and field strength, and electromagnetic braking in electric vehicles uses it to convert kinetic energy back to electrical energy.
Worked example
Consider a proton moving through a uniform magnetic field inside a cyclotron. The proton has a charge of q = 1.602 × 10⁻¹⁹ C and is traveling at a speed of v = 3.0 × 10⁶ m/s. The magnetic field strength is B = 0.5 T, and the proton's velocity is directed perpendicular to the field, so θ = 90°.
Applying the Lorentz force formula:
F = qvB sin(θ) = (1.602 × 10⁻¹⁹ C) × (3.0 × 10⁶ m/s) × (0.5 T) × sin(90°)
F = (1.602 × 10⁻¹⁹) × (3.0 × 10⁶) × (0.5) × 1
F = 2.403 × 10⁻¹³ N
This tiny force is more than sufficient to curve the proton's path into a tight circle, as protons have a very small mass of 1.673 × 10⁻²⁷ kg. The resulting centripetal acceleration is enormous — approximately 1.44 × 10¹⁴ m/s², demonstrating how effectively magnetic fields control particle beams in accelerators.
Now consider the same proton but traveling at an angle of θ = 30° to the field instead. The force becomes:
F = (1.602 × 10⁻¹⁹) × (3.0 × 10⁶) × (0.5) × sin(30°) = 2.403 × 10⁻¹³ × 0.5 = 1.201 × 10⁻¹³ N — exactly half the perpendicular case, as expected from the sine function.
Limitations & notes
This calculator computes the magnitude of the magnetic force using the scalar form of the Lorentz law. It does not resolve the three-dimensional direction of the force, which requires a full vector cross product (F = qv × B) to determine. The result is always perpendicular to both the velocity and the field, but the specific direction depends on the orientation of those vectors in 3D space and on the sign of the charge — information that must be determined separately using the right-hand rule or left-hand rule for negative charges. Additionally, this formula assumes a uniform, static magnetic field; in non-uniform or time-varying fields, the analysis becomes significantly more complex. Relativistic effects should be accounted for when particle speeds approach a significant fraction of the speed of light (typically above ~10% of c, or ~3 × 10⁷ m/s), as the relativistic mass increase alters the effective force response. The calculator also does not include any simultaneous electric field contributions — the full Lorentz force including electric effects is F = q(E + v × B).
Frequently asked questions
What is the Lorentz force law?
The Lorentz force law describes the total electromagnetic force on a charged particle: F = q(E + v × B), where E is the electric field and B is the magnetic field. When only a magnetic field is present, this simplifies to F = qv × B, with magnitude F = qvB sin(θ). It is one of the foundational equations of classical electromagnetism.
Why is the magnetic force zero when a particle moves parallel to the field?
When the velocity vector is parallel to the magnetic field (θ = 0° or 180°), the cross product v × B equals zero because two parallel vectors have no perpendicular component relative to each other. Physically, the particle moves along the field lines and experiences no sideways deflection, so the magnetic force is zero. Only the component of velocity perpendicular to B contributes to the force.
Does the magnetic force speed up a charged particle?
No. The magnetic force is always perpendicular to the velocity of the particle. Since work is only done by a force component parallel to the displacement, the magnetic force does zero net work and cannot change the particle's kinetic energy or speed. It only changes the direction of motion, which is why charged particles in uniform magnetic fields orbit in circles at constant speed.
What units are used for magnetic field strength, and how do Tesla and Gauss relate?
Magnetic field strength B is measured in Tesla (T) in the SI system. One Tesla equals 10,000 Gauss (G), where Gauss is the older CGS unit. Earth's magnetic field is roughly 25–65 microtesla (0.25–0.65 Gauss), while MRI machines typically operate at 1.5 to 3 Tesla. Always convert to Tesla before using this calculator.
How is this formula used in a mass spectrometer?
In a mass spectrometer, ions are accelerated through an electric field and then enter a magnetic field region. The magnetic force curves their path into a circular arc. By measuring the radius of curvature and knowing the charge and velocity, the mass of the ion can be determined using the relation r = mv/(qB). Different isotopes have different masses and therefore follow arcs of different radii, allowing separation and identification.
Last updated: 2025-01-15 · Formula verified against primary sources.