Lorentz Force Calculator

The Lorentz force describes how electric and magnetic fields act on a charged particle. In vector form, force equals charge multiplied by the sum of electric field and velocity crossed with magnetic field. The electric contribution can accelerate a particle whether it is moving or stationary. The magnetic contribution exists only when the particle moves relative to the field and acts perpendicular to both velocity and magnetic field. This calculator evaluates the complete three-dimensional vector and its magnitude in SI units. Cartesian components make direction explicit. The cross product velocity crossed with magnetic field has an X component equal to velocity Y times magnetic Z minus velocity Z times magnetic Y. Its Y and Z components follow the same cyclic pattern. The calculator adds the corresponding electric-field component and multiplies each sum by charge. A negative charge reverses every force component relative to the direction predicted for a positive charge. The magnitude is the square root of the sum of the three squared force components and is always nonnegative. Units must be consistent. Enter charge in coulombs, electric field in newtons per coulomb, velocity in meters per second, and magnetic flux density in teslas. The resulting components and magnitude are newtons. Elementary-particle charges and microscopic forces are often extremely small, so scientific notation such as 1.6e-19 is accepted and small outputs are displayed in exponential form. Use signed values to represent vector direction along each selected axis, and enter zero where a vector has no component. Several useful special cases provide checks. With no magnetic field, force reduces to charge times electric field. With no electric field, magnetic-force magnitude is the absolute charge times speed times magnetic field times the sine of the angle between them. Parallel velocity and magnetic-field vectors have a zero cross product, while perpendicular vectors maximize magnetic force. Because magnetic force is perpendicular to instantaneous velocity, it changes direction but does no work by itself; electric force can change kinetic energy. The nonrelativistic equation is fundamental in mass spectrometers, particle accelerators, plasma physics, electron optics, Hall-effect devices, and introductory electromagnetism. At speeds approaching the speed of light, trajectory and momentum analysis must use relativistic mechanics even though the force law retains its vector form. Real trajectories also require particle mass, initial position, spatially varying fields, and numerical integration over time. This calculator gives the instantaneous force for the entered local conditions and coordinate system.