Lorentz Force

Force on charges and conductors in magnetic fields

Calculation
Quick Introduction

Lorentz force describes the force exerted by magnetic fields on moving charges and current-carrying conductors. The force is maximum at 90° between motion/current direction and magnetic field.

The angle term \(\sin\alpha\) is critical: at 0° or 180° the force is zero, while at 90° it is maximal.

Sizing schema:
1) choose charge or conductor case
2) set field level, current/velocity, geometry
3) compute force and evaluate mechanical impact
Formulas (MathJax)
\[F=q\,v\,B\,\sin(\alpha)\]
\[F=B\,I\,l\,\sin(\alpha)\]
\[B=\frac{F}{I\,l\,\sin(\alpha)}\]
\[\sin(0^\circ)=0,\;\sin(90^\circ)=1\]
Symbol Legend
  • \(F\): Lorentz force [N]
  • \(q\): charge [C]
  • \(v\): velocity [m/s]
  • \(B\): magnetic flux density [T]
  • \(I\): current [A]
  • \(l\): active conductor length in field [m]
  • \(\alpha\): angle between current/motion and B


Examples
Example 1 (conductor): \(B=0.8T\), \(I=12A\), \(l=0.2m\), \(\alpha=90^\circ\) ⇒ \(F=1.92N\).
Example 2 (charge): \(q=12\,\mu C\), \(v=250000m/s\), \(B=0.35T\), \(\alpha=90^\circ\) ⇒ \(F\approx1.05\times10^{-3}N\).
Example 3: at \(\alpha=30^\circ\), force is half of the 90° case (\(\sin30^\circ=0.5\)).
Detailed Documentation & Summary

Lorentz force is a fundamental concept of electrodynamics. It explains particle motion in magnetic fields and force generation in conductors used in machines and actuators. In electric motors, electromagnetic torque emerges from distributed conductor forces in the air-gap field.

This calculator computes magnitudes. Actual direction follows vector rules (right-hand rule for positive charge/current). For electrons, direction reverses due to negative charge. Real applications may require accounting for field non-uniformity, thermal effects, and time-varying current waveforms.

For practical design, the angle between field and current/motion may vary along the path. Then integration or an effective average angle factor is needed. This tool provides robust first-order estimates for concept and predesign stages.

Summary
  • Computes Lorentz force for charge and conductor cases, plus inverse B mode
  • Includes angle dependency via \(\sin\alpha\)
  • Useful for motor, actuator, sensor, and magnetic system predesign

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