A charge q moving with velocity v feels F = q(E + v × B). The magnetic part is always at right angles to the motion, so it bends the path without changing the speed. On its own it makes the particle circle the field lines at the cyclotron frequency ω = |q|B / (γm) with the Larmor radius r = γm·v⊥ / (|q|B). Any speed along B carries on untouched, so the circle stretches into a helix with pitch p = v∥ · 2π/ω. Positive charges circle left-handed about B and negative charges right-handed — flip the species and watch the spiral reverse.

Add an electric field at right angles to B and the guiding centre slides sideways at the E×B drift velocity v_d = E × B / B² — the same for every charge and mass. That is why plasmas drift as a whole. An electric field parallel to B simply accelerates the particle along the field lines. If E > cB the drift would be faster than light: the magnetic field can no longer trap the particle.

The magnetic mirror slider makes the field stronger away from the centre (B_z grows as 1 + g·z², with the radial component that ∇·B = 0 requires). The magnetic moment μ = m v⊥² / 2B stays almost constant, so v∥ is squeezed into v⊥ until the particle bounces back — the physics of the Van Allen belts and auroras. Particles whose pitch angle is inside the loss cone escape.

The path is integrated with the Boris method, the standard algorithm in plasma codes, which keeps the energy exact in a pure magnetic field. Motion is fully relativistic (it pushes γv), so speeds near c raise γ and slow the gyration. Worked example: a 1 000 km/s electron at right angles to a 1 mT field circles with r = 5.69 mm at f = 28.0 MHz.