F = qvB·sinθ — Magnetic Force on a Moving Charge

Magnetic force on a moving charge — the Lorentz force law — with a worked electron example.

F = qvB\sin\theta

Magnetic force equals charge times speed times field strength times the sine of the angle.

What it means

A magnetic field pushes on moving charges — stationary charges feel nothing. The force is perpendicular to both the velocity and the field, so it curves trajectories into circles and spirals instead of speeding particles up. This is how mass spectrometers sort ions and auroras paint the sky.

Variables

Symbol	Name	Unit	Meaning
$F$	Magnetic force	N	Force on the charge
$q$	Charge	C	Moving charge (sign gives direction)
$v$	Velocity	m/s	Charge's velocity
$B$	Magnetic field	T	Magnetic flux density
$θ$	Angle	degrees or rad	Angle between velocity and field

Assumptions

Charge moves with velocity v in a uniform field B
θ measured between v and B
Non-relativistic speeds

Worked example

Problem

An electron moves at 2×10⁶ m/s perpendicular to a 0.5 T magnetic field. What magnetic force acts on it?

Given:

q = 1.6×10⁻¹⁹ C (electron charge magnitude)
v = 2×10⁶ m/s
B = 0.5 T
θ = 90° (perpendicular)

Write the formula: $F = qvB\sin\theta$
Substitute: $F = 1.6\times10^{-19} \times 2\times10^6 \times 0.5 \times \sin 90°$
Calculate: $F = 1.6\times10^{-19} \times 10^6 = 1.6\times10^{-13}\ \mathrm{N}$

1.6×10⁻¹³ N

The electron feels 1.6×10⁻¹³ N — tiny in absolute terms, but enormous for a particle with mass 9.11×10⁻³¹ kg (acceleration ≈ 1.8×10¹⁷ m/s²). Because the force is always perpendicular to velocity, it bends the path into a circle without changing speed.