Description of the Dynamics of Charged Particles in Electric Fields: An Approach Using Fractional Calculus

The Free Electron Lasers (FEL) uses a beam of electrons accelerated to relativistic velocities as the active medium to laser generation; these electrons are bound to atoms, but move freely in a magnetic field. The efficiency of FEL depends on several parameters such as relaxation time, longitudinal effects and transverse variations of the optical field. Moreover, the electron dynamics in a magnetic field undulator serves as a means of radiation source for coupling to the electric field. The transverse motion of the electrons leads to either a gain or loss energy from or to the field; this depends on the position of the particle regarding the phase of the external radiation field. On the other hand, optical tweezers are noninvasive tools that use a laser beam to generate powerful forces enough to manipulate microscopic matter by using electric and magnetic fields. In this work, we described the fractional dynamics of charged particles in electric fields to knowtheir displacement. Fractional Newton’s second law is considered and the order of the fractional differential equation is 0 < ɣ ≤ 1. We use the Laplace transform of the fractional derivative in Caputo sense. Dissipative effects are observed in the study cases of the particle dynamics due to the order of the derivative, and the standard electrodynamics is recovered by taking the limit when ɣ = 1.