Cyclotron motion and magnetic moment

Let $\mathbf B=B\hat{\mathbf z}$ with $B>0$ and no electric field. Cartesian Lorentz-force components give $\dot v_x=\Omega_c v_y$ and $\dot v_y=-\Omega_c v_x$, where $\Omega_c=qB/m$ is signed. With initial perpendicular velocity along positive x, a positive charge initially accelerates toward negative y. Viewed from positive z toward the origin, its orbit is clockwise; changing the charge sign reverses that direction without changing the speed.

The positive gyration rate is $|\Omega_c|$, its period is $2\pi/|\Omega_c|$, and the Larmor radius uses the velocity perpendicular to the field. Parallel velocity instead sets the helix pitch per turn. The magnetic force is always orthogonal to the instantaneous total velocity, so kinetic energy is exactly constant in a static magnetic field, even when that field varies in space. That exact statement does not by itself establish a circular orbit in a nonuniform field.

Define the scalar magnetic moment $\mu=mv_\perp^2/(2B)$ using the fast gyration relative to the guiding centre. This positive quantity is perpendicular kinetic energy divided by field magnitude. It is independent of the sign of charge for given mass, speed, and field. The associated orbital magnetic dipole vector points opposite to $\mathbf B$ for either charge sign: reversing the orbit and reversing charge cancel in the current-loop orientation. Do not replace this scalar by signed charge.

In a slowly varying field, the gyro-averaged scalar is approximately conserved. Require $\rho_L/L_B\ll1$ and a fractional field change during a gyrotime much smaller than unity, with the latter measured along the particle's motion as well as at a fixed location. Collisions and resonant fluctuations can invalidate the averaging. A small laboratory time derivative alone is insufficient if the particle traverses a sharp spatial gradient during one turn.

In a static nonuniform magnetic field with negligible electric field, total kinetic energy stays fixed while conserved $\mu$ makes perpendicular energy increase as the particle reaches stronger field. Parallel energy must decrease by the same amount. The resulting mirror force is an averaged consequence of the Lorentz dynamics, not work by the magnetic force. This explains how a particle can slow along a field even though its total speed remains constant.

If the field is instead increased slowly in time, Faraday's law generally supplies an induced electric field. That field can do work on the gyromotion, increasing perpendicular energy while approximately preserving $\mu$. Confusing these two situations leads to an apparent contradiction about magnetic work. Always distinguish spatial travel through a static field from temporal field compression, and check whether electric forces, collisions, or a vanishing field destroy the assumptions before using the invariant.