Magnetic Force Calculator (F = qvB and BIL)

The magnetic force is the component of the electromagnetic force that acts on a moving electric charge or current-carrying conductor when it is in a magnetic field. This force is described by the Lorentz force law and is fundamental to electric motors, generators, particle accelerators, and many other technologies.

Force on a Moving Charge

When a charged particle moves through a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field:

F = q × v × B × sin(θ)

Where:

• F = magnetic force (Newtons, N)
• q = electric charge (Coulombs, C)
• v = velocity of the particle (meters per second, m/s)
• B = magnetic field strength (Tesla, T)
• θ = angle between velocity vector and magnetic field vector

The force is maximum (θ = 90°) when velocity is perpendicular to the field, and zero (θ = 0°) when velocity is parallel to the field.

Force on a Current-Carrying Wire

A wire carrying current in a magnetic field experiences a force:

F = I × L × B × sin(θ)

Where:

• I = current (Amperes, A)
• L = length of the wire in the field (meters, m)
• B = magnetic field strength (Tesla, T)
• θ = angle between wire and field

This is the principle behind electric motors: current in the armature windings interacts with the motor’s magnetic field to produce rotational force (torque).

Magnetic Field Strength Reference

Direction of the Force

The direction of the magnetic force is given by the right-hand rule: point fingers in the direction of velocity (or current), curl toward the magnetic field, and the thumb points in the direction of force (for positive charges).

The magnetic force does no work. Because F is always perpendicular to v, the dot product F·v is zero. That means a magnetic field can change the direction of a charged particle’s motion but never its speed or kinetic energy. This is the deep reason why magnetic fields are used for steering charged beams (mass spectrometers, particle accelerators, electron microscopes) but never for accelerating them, which requires an electric field.

Circular motion in a uniform field. A charged particle moving perpendicular to a uniform magnetic field travels in a circle. Setting the magnetic force equal to the centripetal force gives:

r = mv / (qB)

where r is the radius, m is the particle mass, v is the speed, q is the charge magnitude, and B is the field strength. This relationship is what makes mass spectrometers work: ions of different mass-to-charge ratios trace different radii and land at different detector positions. Cyclotrons and synchrotrons exploit the same principle on a much larger scale. Enter a particle mass and the result panel works out the radius for you, along with the time to complete one orbit.

The cyclotron period is the surprising part. That time, 2πm/(qB), contains no velocity term at all. A slow particle traces a tight circle, a fast one traces a wide circle, and both finish a lap in the same interval. Lawrence spotted this in 1929, and it is the reason a cyclotron can drive its accelerating gap at one fixed frequency while the beam spirals outward. The scheme only breaks when the particles get fast enough for relativistic mass increase to matter, which is what the synchrocyclotron was built to fix.

Adding an electric field: the full Lorentz force

Magnetic force is one half of the Lorentz force, F = q(E + v × B). Give the calculator an electric field as well and it will combine the two, but it has to ask you one thing first: which way the electric force points relative to the magnetic one. That is not pedantry. Two forces of 5 kN each can total anything from 0 to 10 kN depending on their geometry, and getting it wrong is the single easiest way to produce a confidently wrong answer here.

The arrangement worth knowing is the velocity selector, or Wien filter. Set E and B so the electric force exactly cancels the magnetic one, and only particles travelling at v = E/B get through undeflected. Slower ones feel more electric force and bend one way; faster ones feel more magnetic force and bend the other. Note what is missing from v = E/B: charge and mass. The filter sorts purely on speed, which is why a mass spectrometer puts one ahead of its analyser to deliver a clean single-speed beam. Thomson used exactly this trick in 1897 to pin down the charge-to-mass ratio of the electron.