Lorentz Force Calculator - 3D Vector Cross Product

A lorentz force calculator evaluates the electromagnetic force F = q (E + v x B) on a particle of charge q moving at velocity v through an electric field E and a magnetic field B, returning the three Cartesian components of the force, the magnitude |F|, and the angle between v and B.

• • Charged particle in a magnetic field: Compute the side push on a proton, electron, or ion in a uniform B field, the basic setup for mass spectrometers and cyclotrons.
• • Cathode ray tube and electron beam steering: Check the deflection of an electron beam in a CRT or electron microscope coil, where the magnetic term dominates.
• • Plasma physics and confinement: Estimate the gyromotion force on a charged particle in a tokamak or Hall-effect thruster where B is the dominant field.
• • Combined E and B field problems: Solve velocity-selector and Wien filter problems where E and B are tuned so the net Lorentz force is zero.

The magnetic part of the force is perpendicular to both v and B, so it bends the particle's path without changing its speed. The electric part is parallel to E and does work on the particle.

When the problem is better framed in terms of net force, mass, and acceleration, the forces-newtons-laws-calculator runs F = m a on the same kind of inputs so the result lines up with a Newtonian read of the same motion.

The calculator computes the cross product v x B component by component, multiplies by the charge, and (in total-Lorentz mode) adds the qE term before reporting the magnitude and the angle between v and B.

• q: Electric charge of the particle in coulombs. Negative values flip the direction.
• v_x, v_y, v_z: Components of the velocity vector v in m/s.
• B_x, B_y, B_z: Components of the magnetic flux density B in tesla.
• E_x, E_y, E_z: Components of the electric field E in V/m. Used only in Total Lorentz mode.
• theta: Angle between v and B. Magnetic force magnitude scales as sin(theta), zero for parallel motion and maximum for perpendicular motion.

The identity |v x B| = |v| |B| sin(theta) is why the magnetic force vanishes for parallel motion and reaches the maximum for perpendicular motion. The angle is reported alongside the components so the result is auditable by hand.

According to Wikipedia - Lorentz force, the Lorentz force on a particle of charge q moving at velocity v in an electric field E and a magnetic field B is F = q (E + v × B), with the magnetic component perpendicular to both v and B.

When the magnetic force bends a charged particle in a circle, the same numbers feed into the centrifugal-force-calculator so the orbital radius and the rotation period come out from the same v and B.

Four ideas from electromagnetism explain why the magnetic force bends a charged particle's path without changing its speed, and why the total Lorentz force does change its speed when an electric field is present.

These four ideas reappear in every charged-particle problem. The right-hand rule fixes the direction, the work-free magnetic term fixes the speed.

When the particle speed approaches a sizable fraction of c, the relativistic correction is handled by the time-dilation-calculator, which returns the Lorentz factor gamma and the dilated time for the same v.

Use the lorentz force calculator in six steps to handle both the magnetic-only and the total-Lorentz cases.

1. 1 Pick the force mode: Choose Magnetic only (F = qv x B) for the standard problem, or Total Lorentz (F = qE + qv x B) when an electric field is also present.
2. 2 Enter the charge: Type the charge in coulombs. Use 1.602176634e-19 for a proton, -1.602176634e-19 for an electron, or zero for a neutral particle.
3. 3 Enter the velocity components: Type v_x, v_y, and v_z in m/s. Zero is fine; the magnetic force just vanishes on the components where v is zero.
4. 4 Enter the magnetic field components: Type B_x, B_y, and B_z in tesla. Most lab values sit between 0.01 T and 1 T.
5. 5 In total mode, enter the electric field: Type E_x, E_y, and E_z in V/m. The qE term is added to the magnetic force in this mode and ignored otherwise.
6. 6 Read the force and the angle: The result panel shows F_x, F_y, F_z, the magnitude |F|, and the angle in degrees between v and B. Negative components mean the force points in the negative direction along that axis.

For a cyclotron check, set Magnetic only mode, q = 1.602e-19 C, v = (1e6, 0, 0) m/s, B = (0, 0, 0.5) T. The page returns F = (0, -8.01e-14, 0) N and angle v-B = 90 deg.

After the force is known, the kinematics-motion-calculator takes the resulting acceleration and returns the velocity, position, and time that the charged particle reaches along its trajectory.

Five practical reasons to compute the Lorentz force with this page instead of by hand.

• • Full 3D vector inputs: v, B, and (optionally) E are all 3D vectors, so the same tool handles textbook 2D problems and arbitrary 3D directions.
• • Both F = qv x B and F = qE + qv x B: Switch between the magnetic-only and the total-Lorentz mode without switching calculators, so velocity-selector and Wien filter problems stay on the same page.
• • Components, magnitude, and angle together: F_x, F_y, F_z, |F|, and the angle between v and B appear in the same panel, so the right-hand rule and the sin(theta) factor are both auditable from one screen.
• • Negative charges supported: Enter -1.602176634e-19 C for an electron and the calculator flips the sign on every component, so the deflection direction matches the lab.
• • Edge-case safe: Parallel v and B, zero charge, zero velocity, and zero field all return a clean 0 N result with a defined angle, so the page does not throw infinities on textbook edge problems.

The same F = q (E + v x B) relation is what every mass spectrometer, cyclotron, and velocity selector is built on, so a single tool that returns the components, magnitude, and angle covers all of them.

When the same problem also asks for the current in a circuit that produces the B field, the ohms-law-calculator returns the voltage, current, and resistance for the coils or electromagnets that generate the field.

Five things that change the result, plus two caveats that limit what the static Lorentz force formula can describe.

• • The calculator uses the static Lorentz force law, so it does not capture radiation reaction, self-fields, or any feedback between the particle and the field.
• • Inputs allow v up to 3e8 m/s, but the displayed result does not include the relativistic gamma factor. For v greater than about 10 percent of c, treat the magnitude as an upper bound.

For most textbook and lab work, the static Lorentz force on a single point charge is the right starting point, and the factors above cover the main effects.

According to OpenStax - University Physics Volume 2, the magnetic force on a moving charge is given by F = q v × B, with the direction set by the right-hand rule perpendicular to both v and B.

According to NIST - SI Special Publication 330, the elementary charge is exactly 1.602176634 × 10⁻¹⁹ C since the 2019 SI redefinition, which fixes the magnitude of the charge on a single electron and proton.

When the E field in the problem is set by a charged capacitor plate, the capacitor-charge-time-calculator returns the charge, voltage, and RC time that produce the E you entered on this page.