Last updated: 2026-06-14

Magnetic Force on a Moving Charge

A magnetic field exerts a force only on charges that are moving. The size of that force is $F = q v B \sin\theta$, where $q$ is the charge, $v$ its speed, $B$ the magnetic flux density and $\theta$ the angle between the velocity and the field. This is the magnetic part of the Lorentz force, and it governs how particle accelerators steer beams, how a mass spectrometer sorts ions, and how the aurora forms as charged particles spiral down Earth's field lines.

This calculator returns the magnitude of that force, along with the maximum force that would act if the motion were perpendicular to the field.

A force that needs motion and angle

Two things switch the magnetic force off. A stationary charge feels nothing, no matter how strong the field — only motion produces the force. And a charge moving exactly along the field lines also feels nothing, because the force depends on $\sin\theta$, which is zero when the velocity and field are parallel. The force is strongest when the charge moves at right angles to the field, where $\sin\theta = 1$. At any angle in between, only the component of velocity perpendicular to the field contributes.

Formula

The direction of the force is perpendicular to both the velocity and the field, set by the right-hand rule. Because it always acts sideways to the motion, the magnetic force does no work — it changes a particle's direction but never its speed, curving it into circles and helices.

Worked example

A charge of 1 mC (0.001 C) moves at 1000 m/s through a 0.5 T field, at right angles to it ($\theta = 90^\circ$). The force is:

If the charge instead moved at 30° to the field, the force would fall to $0.5 \times \sin 30^\circ = 0.25$ N — half as much. The calculator shows both the actual force and this maximum side by side, so you can see how much the angle costs you.

Where it shows up

The magnetic force is the working principle behind a great deal of technology. In a cyclotron or synchrotron, magnets bend charged particles into a ring so they can be accelerated lap after lap. In a mass spectrometer, ions of different mass curve by different amounts, separating into a spectrum. The same force, acting on the electrons drifting through a current-carrying wire, becomes the force that turns every electric motor. And high above the poles, particles from the Sun spiralling along Earth's field lines light up the sky as the aurora.

Limitations

This calculator gives only the magnitude of the magnetic force on a single point charge; the direction requires the right-hand rule and the full vector cross product $F = q\,\vec{v} \times \vec{B}$. It does not include any electric force that may also be present — the complete Lorentz force is $F = q(\vec{E} + \vec{v} \times \vec{B})$. For a current-carrying wire rather than a single charge, the equivalent expression is $F = B I L \sin\theta$, where $I$ is the current and $L$ the length of wire in the field.