Lorentz Force Formula
The Lorentz force formula F = qE + qv × B calculates the force on a charged particle in electric and magnetic fields.
With worked examples.
The Formula
The Lorentz force describes the total electromagnetic force acting on a charged particle. It combines the electric force (qE) and the magnetic force (qv × B) into a single expression.
Named after the Dutch physicist Hendrik Lorentz, this formula is fundamental to understanding how charged particles behave in electromagnetic fields. It governs everything from particle accelerators to the aurora borealis.
The magnetic component is special because it depends on the particle's velocity. A stationary charge feels no magnetic force — only a moving charge interacts with a magnetic field.
Variables
| Symbol | Meaning |
|---|---|
| F | Total force on the particle (measured in newtons, N) |
| q | Electric charge of the particle (measured in coulombs, C) |
| E | Electric field strength (measured in newtons per coulomb, N/C, or volts per meter, V/m) |
| v | Velocity of the charged particle (measured in meters per second, m/s) |
| B | Magnetic field strength (measured in tesla, T) |
Special Cases
- Electric field only (B = 0): F = qE — the force is in the direction of the electric field
- Magnetic field only (E = 0): F = qvB sin θ — the force is perpendicular to both velocity and field
- Particle at rest (v = 0): F = qE — only the electric force acts
Example 1
An electron (q = 1.6 × 10⁻¹⁹ C) moves at 5.0 × 10⁶ m/s perpendicular to a magnetic field of 0.2 T. There is no electric field. What is the magnetic force?
Since E = 0 and the velocity is perpendicular to B (θ = 90°, sin 90° = 1):
F = qvB = 1.6 × 10⁻¹⁹ × 5.0 × 10⁶ × 0.2
F = 1.6 × 10⁻¹⁹ × 1.0 × 10⁶
F = 1.6 × 10⁻¹³ N
Example 2
A proton (q = 1.6 × 10⁻¹⁹ C) is in an electric field of 500 N/C and also moves at 3.0 × 10⁴ m/s perpendicular to a magnetic field of 0.1 T. What is the total force?
Electric force: F_E = qE = 1.6 × 10⁻¹⁹ × 500 = 8.0 × 10⁻¹⁷ N
Magnetic force: F_B = qvB = 1.6 × 10⁻¹⁹ × 3.0 × 10⁴ × 0.1 = 4.8 × 10⁻¹⁶ N
The total depends on the relative directions of E and v × B.
If both forces are in the same direction: F = 8.0 × 10⁻¹⁷ + 4.8 × 10⁻¹⁶
F = 5.6 × 10⁻¹⁶ N (when forces align)
When to Use It
Use the Lorentz force formula when dealing with charged particles in electromagnetic fields.
- Calculating the trajectory of charged particles in particle accelerators
- Understanding how mass spectrometers separate ions by mass
- Designing velocity selectors that filter particles by speed
- Analyzing the motion of electrons in cathode ray tubes
- Studying plasma behavior in fusion reactors