Lorentz Force

Total force on a charge in combined electric and magnetic fields. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The Lorentz force is the fundamental force experienced by any charged particle in the presence of electromagnetic fields. It is the bridge between electrostatics (forces on stationary charges) and electrodynamics (forces on moving charges): it unifies in a single expression the electric force and the magnetic force.

In Phyxio's physics tree, the Lorentz force is placed after electric charge, the electric field and the magnetic field, because it is the synthesis of all of them. Understanding the Lorentz force means understanding how particles respond to the fields surrounding them: it is the equation that closes the cycle between "creating fields" and "interacting with them".

The Lorentz force also underpins countless devices: the cathode-ray tube, the mass spectrometer, the cyclotron, the electric motor, the loudspeaker, auroras boreales and particle accelerators all operate thanks to this single law.

🟢 Essential level

A particle with charge $q$ placed in a field experiences an action on it. When the source is an electrostatic origin, this action points along the field direction for positive $q$ and opposite for negative $q$; it acts on the particle whether it is at rest or moving.

If the particle is also moving, the field $B$ exerts an additional action perpendicular to both the velocity $v$ and the field itself. This action cannot accelerate or decelerate the particle: it only curves its trajectory.

The Lorentz force $F_L$ is the sum of both actions. For positive $q$ moving perpendicular to $B$, the action due to $B$ points in the third orthogonal direction (right-hand rule). For negative $q$, the direction reverses.

The most important point: the action of $B$ on the particle never does work because it is always perpendicular to the motion. The field $B$ can only change the direction of $v$, never its magnitude.

A quick self-check before calculating: decide first whether the particle is at rest (only the electric term survives) or moving (both terms may contribute), since that single check determines which part of the formula actually applies.

🔵 Formal level

Basic formulas

The complete expression of the Lorentz force combines both components. For the magnitude of the magnetic component:

Magnitude of the magnetic Lorentz force

Here $F_m$ is the magnetic force magnitude, $q$ is the particle charge, $v$ is its speed, $B$ is the magnetic field magnitude, and $\sin\theta$ is the sine of the angle between the velocity vector and the magnetic field vector.

Real student questions

Why does the magnetic force do no work?

Work is the integral of force in the direction of displacement. The magnetic force $F_m$ is always perpendicular to $v$, which is the direction of instantaneous displacement. The dot product of $F_m$ and the displacement is always zero: the magnetic force cannot transfer energy to the particle.

How do I find the direction of the force if the cross product is hard to visualize?

The right-hand rule: point your fingers in the direction of $v$, curl toward $B$; your thumb points in the direction of $F_m$ for positive charge. For negative charge, the force is opposite. With practice, this rule is fast and reliable.

What happens if the particle is initially at rest?

The magnetic component is zero (it depends on $v$). The only force is the electric $F_E$ if there is an electric field. The magnetic field has no influence on a stationary charge.

Final synthesis

The Lorentz force unifies the electric and magnetic forces in a single expression. The electric component acts on charges at rest or in motion; the magnetic one acts only on moving charges and never does work. Full mastery requires controlling the geometric factor $\sin\theta$ and the vector geometry of the cross product.