The motion of charged particles with charge \(q\) and mass \(m\) in a magnetic field, is described by the Lorenz equation of motion:

\begin{equation}\label{eq:lorenz}

\mathbf{F}=\frac{d\mathbf{p}}{dt}=q \cdot (\mathbf{v} \times \mathbf{B})

\end{equation}

This equation describes the force exerted on a charged particle at a position \(\mathbf{r}\) by the external magnetic field \(\mathbf{B}\). For particles in non-idealized magnetic fields, it is usually impossible to solve this equation analytically to get the full solution, and trajectories can be very complicated. Under the assumption that the radius of gyration of the particle is smallwith respect to the magnetic term. the particle movement can be described by the so called \(\it{guiding \quad center}\) approximation, that leat to the adiabatic approach for particle motion.

The trajectory, in relation to the structure of the field, can result in magnetic trapping along the field lines, and the trapping, and this phenomena is the superposition of three different motions: spiral motion around the field lines, a drift motion due to a gradient in the field, and finally a bounce motion for particular field configurations.

The magnetic force is much stronger near the Earth than far away, and on any field line it is greatest at the ends, where the line enters the atmosphere. Thus electrons and ions can remain **trapped** for a long time, **bouncing back and forth** from one hemisphere to the other

Charged particles, when trapped in the Earth's magnetic field, can perturb the field. Positive trapped particles experience the westward drift, under the influence of the inhomogeneous, curving dipole field, while negative particles drift eastward, and the motion of electric charges is equivalent to an **electric current** circling the Earth clockwise.

That is the so-called **ring current**, whose magnetic field slightly weakens the field observed over most of the Earth's surface. During magnetic storms the ring current receives many additional ions and electrons from the nightside "tail" of the magnetosphere and its effect increases, though at the Earth's surface it is always very small, only rarely exceeding 1% of the total magnetic field intensity.

In the following sections we will discuss in more details the Adiabatic Invariants and the Radiation Belts.