Magnetic Field & Forces

A moving charge creates a magnetic field and experiences a force in an external magnetic field.

The Lorentz force on a charge: F = qv × B. The force is always perpendicular to both v and B.

Since the force is perpendicular to velocity, the magnetic force does NO work on the charge (speed stays constant).

A charged particle in a uniform magnetic field moves in a circle if v ⊥ B, or a helix if v has a component along B.

Radius of circular motion: r = mv/(qB). Period: T = 2πm/(qB) — independent of velocity!

The cyclotron uses this principle: T is constant, so particles can be accelerated with a fixed-frequency oscillating field.

Biot-Savart Law: dB = (μ₀/4π)(Idl × r̂)/r² gives the magnetic field due to a current element.

Ampere's Circuital Law: ∮B·dl = μ₀I_enclosed — useful for high-symmetry situations.