Magnetic Force on a Moving Charge

Magnetic fields are choosy: they push only on charges that move, and only on the component of motion that cuts across the field lines. The force is greatest when velocity and field are perpendicular (θ = 90°), and vanishes entirely for a charge coasting along the field. Because the push is always sideways — perpendicular to both v and B — it does no work; it bends paths into circles and spirals instead of speeding particles up. That steering is the working principle of particle accelerators, mass spectrometers, and the aurora, where solar particles spiral down Earth's field lines to the poles.

Worked example: a proton (q = 1.602×10⁻¹⁹ C) crossing a 0.5 T field at 10⁶ m/s and 90° feels F = 8×10⁻¹⁴ N — tiny, yet enough to whirl it in a tight circle. Note that θ itself is not solvable here: arcsin cannot tell θ from 180° − θ, so the inversion is ambiguous.