Moving Charges and Magnetism

Electric field, Magnetic field

The Lorentz force F = q(E + v × B) includes contributions from both electric field E and magnetic field B. The electric force qE can do work on the particle, while the magnetic force q(v × B) is always perpendicular to velocity and does no work.

μ₀I/2R

At the center of a circular loop carrying current I with radius R, the magnetic field is B = μ₀I/2R. This formula comes from integrating the Biot-Savart law around the circular loop, where all current elements contribute in the same direction at the center.

Speed of the particle, Kinetic energy of the particle, Angular frequency, Radius of circular path

Since magnetic force does no work (being perpendicular to velocity), the speed and kinetic energy remain constant. The radius r = mv/qB and angular frequency ω = qB/m also remain constant for uniform B. Only the direction of velocity changes continuously.

Charge and mass of particle, and magnetic field strength

The cyclotron frequency ν = qB/2πm depends only on the charge q, mass m of the particle, and magnetic field strength B. It is independent of the particle's speed and radius, which is the key principle used in cyclotron accelerators.