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Abstract The calculation of the gravitational self-energy of a disk with constant surface mass density is crucial because it quantifies the energy required to assemble the disk from individual particles considering their mutual gravitational attraction. This concept is important in astrophysics and planetary science, especially when analyzing the formation and stability of structures like galaxies or accretion disks around black holes. In this work, we show how to calculate the gravitational self-energy of a disk with constant surface mass density by using Fourier transform techniques. From a pedagogical perspective, finding the gravitational self-energy of a disk using the Fourier transform method helps undergraduate students learn how to use powerful mathematical tools to solve physics problems more easily. It also shows how symmetry in combination with the Fourier transform method can simplify the calculation of complicated multi-dimensional integrals that arise in common topics in classical mechanics. 

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. 

Abstract The calculation of the electrostatic potential and/or electrostatic field due to a continuous distribution of charge is a well-covered topic in all calculus-based undergraduate physics courses. The most common approach is to consider bodies with uniform charge distribution and obtain the quantity of interest by integrating over the contributions from all the differential charges. The examples of a uniformly charged disk and ring are prominent in many physics textbooks since they illustrate well this technique at least for special points or directions of symmetry where the calculations are relatively simple. Surprisingly, the case of a uniformly charged annulus, namely, an annular disk, is largely absent from the literature. One might speculate that a uniformly charged annulus is not extremely interesting since after all, it is a uniformly charged disk with a central circular hole. However, we show in this work that the electrostatic potential created by a uniformly charged annulus has features that are much more interesting than one might have expected. A uniformly charged annulus interpolates between a uniformly charged disk and ring. However, the results of this work suggest that a uniformly charged annulus has such electrostatic features that may be essentially viewed as ring-like. The ring-like characteristics of the electrostatic potential of a uniformly charged annulus are evident as soon as a hole is present no matter how small the hole might be. The solution of this problem allows us to draw attention to the pedagogical aspects of this overlooked, but very interesting case study in electrostatics. In our opinion, the problem of a uniformly charged annulus and its electrostatic properties deserves to be treated at more depth in all calculus-based undergraduate physics courses covering electricity and magnetism. 

Abstract It is common in mesoscopic systems to find instances where several charges interact among themselves. These particles are usually confined by an external potential that shapes the symmetry of the equilibrium charge configuration. In the case of classical charges moving on a plane and repelling each other via the Coulomb potential, they possess a ground state à la Thomson or Wigner crystal. As the number of particles N increases, the number of local minima grows exponentially and direct or heuristic optimization methods become prohibitively costly. Therefore the only feasible approximation to the problem is to treat the system in the continuum limit. Since the underlying framework is provided by potential theory, we shall by‐pass the corresponding mathematical formalism and list the most common cases found in the literature. Then we prove a (albeit known) mathematical correspondence that will enable us to re‐discover analytical results in electrostatics. In doing so, we shall provide different methods for finding the equilibrium surface density of charges, analytical and numerical. Additionally, new systems of confined charges in three‐dimensional surfaces will be under scrutiny. Finally, we shall highlight exact results regarding a modified power‐law Coulomb potential in thed‐dimensional ball, thus generalizing the existing literature. 

Abstract We present a theory of spontaneous Fermi surface deformations for half-filled Landau levels (filling factors of the form$$\nu =2 \, n+1/2$$ $\nu =2n+1/2$). We assume the half-filled level to be in a compressible, Fermi liquid state with a circular Fermi surface. The Landau level projection is incorporated via a modified effective electron-electron interaction and the resulting band structure is described within the Hartree-Fock approximation. We regulate the infrared divergences in the theory and probe the intrinsic tendency of the Fermi surface to deform through Pomeranchuk instabilities. We find that the corresponding susceptibility never diverges, though the system is asymptotically unstable in the$$n \rightarrow \infty $$ $n\to \infty$limit. 

Abstract We consider the stability of the circular Fermi surface of a two-dimensional electron gas system against an elliptical deformation induced by an anisotropic Coulomb interaction potential. We use the jellium approximation for the neutralizing background and treat the electrons as fully spin-polarized (spinless) particles with a constant isotropic (effective) mass. The anisotropic Coulomb interaction potential considered in this work is inspired from studies of two-dimensional electron gas systems in the quantum Hall regime. We use a Hartree–Fock procedure to obtain analytical results for two special Fermi liquid quantum electronic phases. The first one corresponds to a system with circular Fermi surface while the second one corresponds to a liquid anisotropic phase with a specific elliptical deformation of the Fermi surface that gives rise to the lowest possible potential energy of the system. The results obtained suggest that, for the most general situations, neither of these two Fermi liquid phases represent the lowest energy state of the system within the framework of the family of states considered in this work. The lowest energy phase is one with an optimal elliptical deformation whose specific value is determined by a complex interplay of many factors including the density of the system.