// interactive reference · module 07
Charged Particles
in Magnetic Fields
Circular motion, the Lorentz force, and magnetic confinement — built from the ground up.
Vectors live. Physics breathes.
The Lorentz Force
\[ \textbf{F} = q\,\textbf{v} \times \textbf{B} \]
Force \(\textbf{F}\) is always perpendicular to velocity \(\textbf{v}\) — so it does no work, only curves the path. \(q\) is charge, \(\textbf{B}\) is the magnetic field vector.
Radius of Circular Motion
\[ r = \frac{mv}{|q|B} \]
Set \(|F_\text{Lorentz}| = |F_\text{centripetal}|\). Mass \(m\) and speed \(v\) want to fling the particle out; charge \(|q|\) and field \(B\) pull it back in. Balance = circle.
Cyclotron Frequency
\[ \omega_c = \frac{|q|B}{m}, \quad f_c = \frac{\omega_c}{2\pi} \]
The angular frequency of the circular orbit — independent of \(v\)! This is the isochronous property that makes cyclotrons tick.
Magnetic Mirror Force
\[ F_\parallel = -\mu\,\frac{\partial B}{\partial z} \]
In a magnetic bottle, the gradient of \(B\) along the axis creates a mirror force \(F_\parallel\) that pushes particles back toward weaker-field regions — the trapping mechanism.
// simulation 01
Circular Motion in Uniform B
A positive charge moves in the xy-plane. \(\textbf{B}\) points into or out of the page (z-direction).
Watch the Lorentz force (rose) always point inward — centripetal, never tangential.
// simulation 02
Magnetic Bottle — 3D
Two coils create a region of weak \(B\) in the center and strong \(B\) at the ends.
The gradient force \(F_\parallel = -\mu\,\partial B/\partial z\) acts as a mirror, trapping the particle.
Drag to orbit · scroll to zoom.
⟳ DRAG TO ROTATE
⊕ SCROLL TO ZOOM
REAL-TIME VECTORS