Fractional solution of helical motion of a charged particle under the influence of Lorentz force

Abstract

In this study, a generalized solution for the helical motion of a charged particle in uniform electric and magnetic fields is obtained using a powerful fractional derivative approach. Using this approach, the differential equations that describe the helical motion of a charged particle in the fields were obtained. The solution for the fractional differential equations is presented in great detail in terms of a series solution using the Mittag-Leffler function. The Laplace transform technique was used to solve the differential equations in the regular form and in the fractional form (with fractional parameter γ). Two and three-dimensional plots were presented for the trajectory of the particle before and after introducing the fractional operator for different values of γ. Features of delay in the motion and dissipation in the medium have been observed in the fractional solution too. The importance of our work stems from the two- and three-dimensional visualization of the obtained generalized helical trajectories that can be applied to similar types of motions in nature and the universe.