# On the validity of the guiding-center approximation in the presence of strong magnetic gradients

## TL;DR

This paper investigates the accuracy of the guiding-center approximation for charged particle motion in nonuniform magnetic fields with gradients, demonstrating its validity even beyond traditional small-parameter limits through exact solutions.

## Contribution

It provides an exact solution for particle motion in a magnetic field with a gradient and compares it to guiding-center theory, showing the approximation's robustness beyond standard assumptions.

## Key findings

- Guiding-center theory agrees well with exact solutions in this scenario.
- The approximation remains valid even when the small parameter condition is relaxed.
- The work bridges the gap between exact particle motion and approximate guiding-center predictions.

## Abstract

The motion of a charged particle in a nonuniform straight magnetic field with a uniform magnetic-field gradient is solved exactly in terms of elliptic functions. The connection between this problem and the guiding-center approximation is discussed. It is shown that, for this problem, the predictions of guiding-center theory agree very well with the orbit-averaged particle motion and hold well beyond the standard guiding-center limit $\epsilon \equiv \rho/L \ll 1$, where $\rho$ is the gyromotion length scale and $L$ is the magnetic-field gradient length scale.