Neste trabalho calculamos o potencial elétrico para algumas distribuições infinitas de cargas puntiformes. Para cada distribuição discutimos os critérios de convergência das séries obtidas. Calculamos também a densidade de carga superficial induzida em um plano condutor infinito "aterrado" e em uma esfera condutora "aterrada" de raio a por uma distribuição infinita de cargas puntiformes com sinais alternados e magnitude que diminui harmonicamente com n (∼ q/n). Palavras-chave: potencial eletrostático, séries infinitas e convergência de séries.In this work we obtain the electric potential for some infinite distributions of point charges. The convergence criteria for the resulting series are discussed. We also calculate the surface charge density induced on an infinite grounded conducting plane and on a grounded conducting sphere of radius a by an infinite distribution of point charges with alternate signs and harmonically decreasing magnitude (∼ q/n).

In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The exact wave functions are obtained by solving the respective Milne-Pinney equation for each system. We obtain the solutions for the system with m1 = m2 = m0e γt , ω1 = ω01e −γt/2 , ω2 = ω02e −γt/2 and k = k0.

Since the late twentieth century, there has been discussion about the insertion of Modern and Contemporary Physics (MCP) in Basic Education. Considering the current panorama of discussions about the need for these subjects to be present in everyday school life, more precisely in high school (HS), this research aims to investigate the conceptions of physics teachers of Sobral public schools about the teaching of MCP in your HS classes. For this, a semi-structured questionnaire was applied to all physics teachers of the Sobral public schools who completed their graduation after 2002. It was observed that the time, the curriculum of this level and the absence of these subjects in the entrance exams are the main obstacles pointed out by the students. teachers to teach them and that 67% claim to insert MCP in their classes, as well as draw attention to the need for new methodologies for teaching these contents in HS and 47% had difficulties in these subjects during their training.

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Considering the growing discussions about the insertion of topics of Modern and Contemporary Physics (MCP) in Basic Education and the research that deals with the formation of Physics teachers in these contents, a qualitative and quantitative research was developed, using a semi-structured questionnaire as a collection instrument of data, with all physics teachers in all public high schools (HS) in the city of Sobral - CE, who graduated after the year 2002, totaling 15 teachers in 13 schools, aiming to discuss and analyze the aspects related to the teaching of these themes during the graduation of teachers, as well as their implications for HS. It was observed that: the difficulties they had in the study of these subjects during graduation are related to Mathematics; the most recurrent assessment tool was evidence; and that the mathematical aspect of MCP was the one most worked on by the teacher teachers, neglecting the conceptual, historical and experimental aspects of these subjects, for example. Regarding teaching, it was noticed that the class time that teachers have in the HS and the curriculum are the main obstacles faced by teacher.