We develop a long-wavelength approximation in order to describe the low-energy states of carbon nanotubes in a transverse magnetic field. We show that in the limit where the square of the magnetic length l = ͱ បc / eB is much larger than the C-C distance times the nanotube radius R, the low-energy theory is given by the linear coupling of a two-component Dirac spinor to the corresponding vector potential. We investigate in this regime the evolution of the band structure of zigzag nanotubes for values of R / l Ͼ 1, showing that for radius R Ϸ 20 nm a clear pattern of Landau levels starts to develop for magnetic field strength B տ 10 T. The levels tend to be fourfold degenerate, and we clarify the transition to the typical twofold degeneracy of graphene as the nanotube is unrolled to form a curved strip. We show that the dynamics of the Dirac fermions leads to states which are localized at the flanks of the nanotube and that carry chiral currents in the longitudinal direction. We discuss the possibility of observing the quantization of the Hall conductivity in thick carbon nanotubes, which should display steps at even multiples of 2e 2 / h, with values doubled with respect to those in the odd-integer quantization of graphene.

We address some issues in colour theory, which are of relevance from an educational perspective. Spectra of emitted RGB and of transmitted CMYK colours are quantitatively processed and analysed with quite inexpensive homemade instruments, making use of smartphones as affordable digital cameras. LCD monitors and paper sheets with pigments coming from a laser printer are used to point out the basic differences between additive and subtractive colour formation. As an especially relevant aspect, we point out how it is possible to construct a simple model to explain the subtractive mixing process in terms of convolution of primary colour filters. The analysis presented in this work is particularly suited for enhancing the need for a proper understanding of the physiology of human eye–brain action in light acquisition and perception of colours.

We designed a teaching-learning sequence on rolling motion, rooted in previous research about student conceptions, and proposing an educational reconstruction strongly centred on the role of friction in different cases of rolling. A series of experiments based on video analysis is used to highlight selected key concepts and to motivate students in their exploration of the topic; and interactive simulations, which can be modified on the fly by students to model different physical situations, are used to stimulate autonomous investigation in inquiry activities. The activity sequence was designed for students on introductory physics courses and was tested with a group of student teachers. Comparisons between pre-and post-tests, and between our results and those reported in the literature, indicate that students' understanding of rolling motion improved markedly and some typical difficulties were overcome.