This article provides a comprehensive review of relevant studies in the fields of plasma physics, electromagnetism, and space physics. The aim is to demonstrate how the study of the scientific literature can be used to enhance problem-solving abilities and develop innovative solutions in physics. In this paper, we focus on the study of solutions of the specific Grad-Shafranov equation. Two of the new solutions proposed by Yoon and Lui (2005) are used as a basis for the development of a new solution. The new solution presented has singular points similar to the Yoon-Lui-2 solution, but with an inverted configuration, and also presents less rounded double islands compared to the Yoon-Lui-2 solution. Additionally, the new solution does not exhibit the formation of a current ring, a characteristic of the Yoon-Lui-1 solution, and varying its parameters may lead to higher plasma confinement efficiency. In summary, we illustrate how a thorough analysis of literature can serve as a powerful means for generating innovative approaches to resolving theoretical issues in physics.

In part 1 of our physics education series, we introduced a novel solution based on Yoon-Lui’s solutions 1 and 2. Building upon that, this follow-up presents a new solution obtained by combining the generating functions of Yoon-Lui-1 and Yoon-Lui-3, resulting in a new and simplified general solution. We also calculate the singular points and determine their coordinates for various parameter values. A graphical representation of the solution is presented, showing the magnetic field lines and current density distribution. The behavior of the magnetic field and the effect of varying the parameter are discussed. The observed magnetic islands and singular points are relevant in the fields of Plasma Physics and Space Physics, providing insights into magnetic structures in plasmas and their impact on confinement and stability. Furthermore, this study encourages innovation and equips researchers and students with the necessary tools to make meaningful contributions to the field, emphasizing the integration of scientific literature into physics education to promote a comprehensive understanding of physical concepts and their practical applications.

In this article, we work with the basic equation for the fluid velocity in the case of a magnetoconducting fluid in two ways: (i) by Ampère’s law, (ii) taking into account the Maxwell correction. Considering an arbitrary direction between the wave vector and the magnetic field, an expression for the wave’s phase velocity is obtained. The phase velocity study implies three magnetohydrodynamic (MHD) wave modes when we neglect the displacement current. We compared the wave modes with each other, and we also compared the modes obtained with and without the displacement current. The comparison allows us to discuss the variations between these MHD modes in hypothetical cases when the Alfvén speed increases to values close to the speed of light. The article’s main objective is to show the importance of displacement current in the pure and fast MHD waves, in the particular case where the Alfvén speed is higher than the sound wave speed. We find that the displacement current has practically no influence on the value of the wave phase velocity when the Alfvén velocity increases for slow mode.

En este artículo trabajamos con la ecuación de velocidad de un fluido magneto-conductor utilizando dos aproximaciones: (i) la ley de Ampère y (ii) tomando en cuenta la corrección de Maxwell. Considerando una dirección arbitraria entre el vector de onda y el campo magnético se obtiene una expresión para la velocidad de fase de la onda. Existen tres modos de onda magnetohidrodinámicas (MHD) cuando la corriente de desplazamiento (Jd) no es considerada. Primero, se hace una comparación de los modos de onda entre sí, y después se comparan con los modos obtenidos con y sin la Jd. La comparación, permite discutir las variaciones entre estos modos MHD en los casos teóricos, cuando la velocidad de Alfvén aumenta hasta valores cercanos a la velocidad de la luz. El objetivo principal del artículo es mostrar la importancia del efecto de la Jd en los modos puro y rápido de ondas MHD, en el caso particular en donde la velocidad de Alfvén es mayor que la velocidad adiabática de una onda acústica. Encontramos que para el modo lento, la Jd prácticamente no influye en el valor de la velocidad de fase de la onda cuando la velocidad de Alfvén aumenta.

Esse trabalho tem por objetivo apresentar uma formulação teórica na área de física de plasma para obter uma forma simplificada da equação de Grad-Shafranov, em outras palavras, obter a lei de Ampère-Maxwell em função do potencial vetor magnético e da densidade de corrente. Buscamos entender melhor alguns fenômenos que ocorrem no plasma interplanetário. Outro objetivo é apresentar essa pesquisa de forma didática com a maior clareza e concisão possíveis. Para tanto, um desenvolvimento matemático se da como introdução ao trabalho, mostrando as equações que queremos deduzir, a cada seção vamos construindo uma base teórica em física de plasma, usando conceitos da termodinâmica e as leis universais da física, além de matemática, culminando na equação simplificada de Grad-Shafranov. Por fim, apresentamos uma das soluções mais conhecidas para equação de Grad-Shafranov, além de referências para quem quiser se aprofundar na pesquisa.

This article aims to present a theoretical formulation in the area of plasma physics to obtain a specific form of the Grad-Shafranov equation, in other words, to obtain the Ampère-Maxwell law as a function of the magnetic vector potential and current density. We seek to better understand some phenomena that occur in interplanetary plasma. Another objective is to present this theme in a didactic way as clearly and concisely as possible. In particular, a mathematical development takes place as an introduction to the work, showing the equations we want to deduce, with each section we build a theoretical basis in plasma physics, using concepts of thermodynamics and the universal laws of physics, as well as mathematics, culminating in the Grad-Shafranov specific equation. Finally, we present one of the best known solutions for the Grad-Shafranov equation, as well as references for those who want to delve into the subject.