Moving Charges and Magnetism

Always perpendicular to $v$, so it does zero work: speed never changes, only direction. Direction from the right-hand rule.

1. r: directly proportional to v — faster particle, bigger circle.
2. T and f: no v (or r) anywhere — completely independent of speed, depend only on q, m, B.
3. Why it works: bigger circle ($\propto v$) covered at proportionally higher speed — the two effects cancel exactly, leaving lap time fixed.

This is the cyclotron’s whole principle: a fixed-frequency accelerating voltage stays in sync with the particle no matter how fast it gets. See it live: interactive orbit simulation →

Maximum when the wire is perpendicular to B, zero when current flows along B.

1. Same direction: attractive.
2. Opposite direction: repulsive (reverse of like/unlike charges).

Defines the ampere: 1 A in each of two wires, 1 m apart, gives 2×10⁻⁷ N per metre of force.

Uniform deep inside, along the axis; n = turns per unit length.

Same form as torque on any magnetic dipole, $\tau=m\times B$. A moving-coil galvanometer balances this against a spring, so deflection becomes proportional to current.