Analysis of the behavior of charged particles in electrical and magnetic fields by prospective physics teachers

Abstract

This study investigates how prospective teachers analyzed the behavior of charged particles in electrical and magnetic fields. A test consisting of eight questions each related to five sub-concepts and involving the analysis of the path followed by a charged particle moving into a uniform electrical or magnetic field was prepared. Three different categories of assessment were applied to the data obtained from the test: detailed, Combined, and Associated. The study was carried out with 80 prospective physics teachers. It was found in the study that difficulties begin with insufficient understanding relating to the nature of vector quantities and the structure of the Lorentz force. Some prospective teachers overlooked the relationship between the direction of the force and the type of the charge in the electrical field. Thus, there were significant errors related to the direction of the force. Further, prospective teachers failed to determine that magnetic force would cause the charged particle to follow a circular path in the magnetic field. Difficulties experienced by the prospective teachers in relation to force were reflected in acceleration and velocity, as well. The rest of the difficulties were related to Newton’s second law. The prospective teachers either did not understand Newton’s second law or could not apply it in other areas of physics. Difficulties relating to mechanics were observed to continue in electricity and magnetism as well. One of the surprising findings of the study was that the prospective teachers had significantly more success in questions relating to the concept of work. According to the combined results, the prospective teachers were found to follow the conceptual order force, acceleration, velocity, and path when answering questions.

Detailed solution of the problem of Landau states in a symmetric gauge

“…Based on the quantum constraints of dealing with only square integrable wave functions over an infinite 2D space (in this case), one expects that both x R( ) and its first derivative should go smoothly to zero in the x  ¥ limit. In such a limit, the first term in equation (31) dominates over the second term. By similar reasoning, one concludes that the term proportional to x 4 dominates among the remaining last three terms in equation (31) in the x  ¥ limit.…”

“…. Another way to frame the discussion is to argue that terms in equation (31) that contain derivatives should eliminate the x 1 divergence in the x  0 limit. A power function is the simplest choice to achieve such an objective.…”

“…A power function is the simplest choice to achieve such an objective. Based on these hints, one looks for a solution to equation (31) as a valid quantum solution in the x  0 limit.…”

“…where x w ( ) is a new function to be determined. After substituting equation (33) into equation (31), we arrive at the following equation for x w ( ):…”

“…A step-by-step guide of the solution of the resulting differential equations is rarely provided in typical quantum physics textbooks [22][23][24][25][26][27]. Studies indicate that even prospective physics teachers have a lot of difficulty to analyze the behavior of charged particles in electrical and magnetic fields [31]. For these reasons, the motivation of this work is to provide an easy to follow and detailed mathematical solution of the quantum problem of the 2D motion of a free charged particle in an external perpendicular magnetic field.…”