Exact solution for a charged particle in an inductive time-dependent increasing magnetic field

We study the classical motion of a charged particle in presence of an inductively increasing time-dependent magnetic field as the one created inside a resistor-inductor series circuit driven by a voltage source. The inductor is treated as an infinite solenoid. In such a scenario, the expression for the time-dependent magnetic field generated when the circuit is turned on can be easily derived. We consider the case study of two-dimensional motion since the generalization to three-dimensions is elementary. The resulting differential equations for the two-dimensional motion of the charged particle are solved by using a particular method which relies in deployment of complex variables. The ensuing motion has interesting features that highlight the challenges faced in studies of charged particles in a time-dependent magnetic field.This study has applications in magnetic plasma confinement, where understanding charged particle dynamics in time-varying magnetic fields helps optimize stability and energy retention in fusion devices.