Abstract
In this project, we study the general 3D mechanics of point charges moving under specific electric and magnetic fields. First, we consider the motion in the absence of any damping forces. In this category, we break down our study into two subcategories of motion under time-independent fields as well as under time-dependent fields (for simplicity of derivations, we assume that the magnitude of time dependencies are small compared with the constant fields, therefore, the electromagnetic induction can be neglected). In each subcategory, we study different situations where electric and magnetic fields are present individually, as well as when combined together. We apply the laws of mechanics and electromagnetism to derive the differential equations of motion, and then work out all details of their solutions step by step using analytical methods as well as numerical and computational techniques by developing codes in Python. For better visualization of the motion, we also develop simulations in which the motion of the charged particles can be visualized in real time. The second category of our investigation involves inclusion of damping forces (both linear as well as nonlinear damping) in which we examine the effects of damping on deformation of the paths that we derived in the first part of our study. In this category, too, we give all details of derivation of the differential equations of motion from the laws of mechanics and electromagnetism, and then work out their solutions step by step. The mathematical complexities, however, are significantly higher and include difficulties of solving a system of nonlinear coupled ordinary differential equations. Therefore, the importance of computational techniques is even more crucial in this category of our investigation than when the damping is absent.
