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The problem of the two-dimensional motion of a charged particle with constant mass in the presence of a uniform constant perpendicular magnetic field features in several undergraduate and graduate quantum physics textbooks. This problem is very important to studies of two-dimensional materials that manifest quantum Hall behavior, as evidenced by several major discoveries over the last few years. Many real experimental samples are more complicated due to the anisotropic mass of the electrons. In this work, we provide the exact solution to this problem by means of a clever scaling of coordinates. Calculations are done for a symmetric gauge of the magnetic field. This study allows a broad audience of students and teachers to understand the mathematical techniques that lead to the solution of this quantum problem.
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Exact solution for a charged particle in an inductive time-dependent increasing magnetic field
Abstract 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.
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- Award ID(s):
- 2001980
- PAR ID:
- 10629950
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Physica Scripta
- Volume:
- 100
- Issue:
- 8
- ISSN:
- 0031-8949
- Format(s):
- Medium: X Size: Article No. 085991
- Size(s):
- Article No. 085991
- Sponsoring Org:
- National Science Foundation
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