Lorentz Force Calculator finds the electric force, magnetic force, net force, drift velocity, and field energy density for a charge moving through electric and magnetic fields.
Find the Net Force on a Charged Particle in Combined Electric and Magnetic Fields
This Lorentz force calculator finds the electric force, magnetic force, and combined net force acting on a moving charged particle, along with drift velocity and field energy density. Physicists, plasma researchers, and particle-accelerator or beam-optics engineers use it to work out how a charge behaves as it moves through overlapping electric and magnetic fields.
How to Use the Lorentz Force Calculator
Enter charge $q$ in coulombs, electric field $E$ in V/m, particle velocity $v$ in m/s, magnetic field $B$ in tesla, and angle $\theta$ in degrees between velocity and the magnetic field. The calculator returns net force, its electric and magnetic components, drift velocity, and field energy density.
Lorentz Force Formula: Electric and Magnetic Force Components
The Lorentz force law describes the total electromagnetic force on a moving charge, per Griffiths’ Introduction to Electrodynamics, the standard electromagnetism textbook:
$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$
This splits into an electric component and a magnetic component:
$$F_E = qE \qquad F_B = qvB\sin\theta$$
where $\theta$ is the angle between velocity and the magnetic field — the electric force has no angle dependence at all. The magnetic force is always perpendicular to velocity, so per Griffiths it can never do work on the particle, regardless of field strength:
$$W_B = 0$$
This calculator treats $F_E$ and $F_B$ as perpendicular vectors and combines them with the Pythagorean sum, not simple addition:
$$F_{net} = \sqrt{F_E^2 + F_B^2}$$
Field energy densities use the vacuum permittivity and permeability recommended by NIST CODATA 2022 (NIST SP 961):
$$u_E = \frac{1}{2}\varepsilon_0 E^2 \qquad u_B = \frac{B^2}{2\mu_0}$$
with $\varepsilon_0 = 8.8541878188 \times 10^{-12}\ F/m$ and $\mu_0 = 1.25663706127 \times 10^{-6}\ N/A^2$.
Common input mistake: $\theta$ is the angle between velocity and the magnetic field only — not between velocity and the electric field, and not between $E$ and $B$. Changing $\theta$ shifts $F_B$ alone; $F_E$ stays fixed at $qE$ no matter what angle you enter.
A nuance most calculators don’t explain: the “drift” value shown ($E/B$) is not derived from the velocity you entered at all. It’s the velocity-selector condition — the one specific speed at which the electric and magnetic forces exactly cancel for a particle moving perpendicular to both fields, used in Wien filters and mass spectrometers to filter particles by speed regardless of their charge or mass. Change your input velocity from 5 m/s to 500 m/s and this number won’t move, because $E/B$ never depended on it in the first place.
Lorentz Force Diagram: Electric and Magnetic Components Combining Into Net Force
Common Questions About the Lorentz Force Calculator
Why does the calculator always show 0 W for deflection work?
The magnetic force acts perpendicular to velocity at every instant, so it can never add or remove kinetic energy from the particle. This holds for any $B$, $v$, or $\theta$ — it’s not specific to your entered values.
What does the “drift” value actually represent?
It’s $E/B$, the velocity-selector speed at which electric and magnetic forces cancel for a particle moving perpendicular to both fields. It’s independent of the velocity you entered — used in Wien filters and mass spectrometers.
Why does net force use a square root instead of simple addition?
The calculator treats $F_E$ and $F_B$ as perpendicular vectors. Combining perpendicular magnitudes requires $\sqrt{F_E^2+F_B^2}$; adding them directly would overstate the true resultant force.
Does the angle θ change the electric force?
No. $\theta$ only appears in the magnetic force term, $qvB\sin\theta$. The electric force, $qE$, has no angle dependence and stays constant regardless of the particle’s direction of travel.
Why are the electric and magnetic field energy densities so different in magnitude?
$\varepsilon_0$ is roughly $10^{22}$ times smaller than $1/\mu_0$. So even with comparable-looking $E$ and $B$ inputs, magnetic field energy density (MJ/m³) vastly outweighs electric field energy density (pJ/m³), per the NIST CODATA vacuum constants.