Four errors students make with inverse-square law vectors

In this paper, we discuss four errors introductory physics students make when attempting to add two inverse-square law vectors. We observe multiple instances in which students 1) add vectors as if they were scalars, 2) project the $r$ (or $r^2$) in the denominator, instead of the entire vector, when attempting to find the vector's components, 3) incorrectly apply the Pythagorean theorem when attempting to calculate the magnitude of the resultant vector, and 4) incorrectly relate the signs of the components of an electric field (or force) to the signs of the electric charges. While these are not the only errors students make, they are the most frequently occurring based on our analysis of 678 exams taken by students in either introductory mechanics or electricity and magnetism (E&M). We then show how these errors can be encoded into a new type of activity or assessment question which we call a ``student error task." Introductory physics instructors can use the student error task in this paper as a way to engage or assess their students' understandings of how to add two inverse-square law vectors.Comment: 22 pages, 7 figures, submitted to the European Journal of Physic

Entropy creation inside black holes points to observer complementarity

Heating processes inside large black holes can produce tremendous amounts of entropy. Locality requires that this entropy adds on space-like surfaces, but the resulting entropy (10^10 times the Bekenstein-Hawking entropy in an example presented in the companion paper) exceeds the maximum entropy that can be accommodated by the black hole's degrees of freedom. Observer complementarity, which proposes a proliferation of non-local identifications inside the black hole, allows the entropy to be accommodated as long as individual observers inside the black hole see less than the Bekenstein-Hawking entropy. In the specific model considered with huge entropy production, we show that individual observers do see less than the Bekenstein-Hawking entropy, offering strong support for observer complementarity.Comment: 13 pages. This is a companion paper to arXiv:0801.4415; Added reference

Clebsch-Gordan Construction of Lattice Interpolating Fields for Excited Baryons

Large sets of baryon interpolating field operators are developed for use in lattice QCD studies of baryons with zero momentum. Operators are classified according to the double-valued irreducible representations of the octahedral group. At first, three-quark smeared, local operators are constructed for each isospin and strangeness and they are classified according to their symmetry with respect to exchange of Dirac indices. Nonlocal baryon operators are formulated in a second step as direct products of the spinor structures of smeared, local operators together with gauge-covariant lattice displacements of one or more of the smeared quark fields. Linear combinations of direct products of spinorial and spatial irreducible representations are then formed with appropriate Clebsch-Gordan coefficients of the octahedral group. The construction attempts to maintain maximal overlap with the continuum SU(2) group in order to provide a physically interpretable basis. Nonlocal operators provide direct couplings to states that have nonzero orbital angular momentum.Comment: This manuscript provides an anlytical construction of operators and is related to hep-lat/0506029, which provides a computational construction. This e-print version contains a full set of Clebsch-Gordan coefficients for the octahedral grou

Using graphical and pictorial representations to teach introductory astronomy students about the detection of extrasolar planets via gravitational microlensing

The detection and study of extrasolar planets is an exciting and thriving field in modern astrophysics, and an increasingly popular topic in introductory astronomy courses. One detection method relies on searching for stars whose light has been gravitationally microlensed by an extrasolar planet. In order to facilitate instructors' abilities to bring this interesting mix of general relativity and extrasolar planet detection into the introductory astronomy classroom, we have developed a new Lecture-Tutorial, "Detecting Exoplanets with Gravitational Microlensing." In this paper, we describe how this new Lecture-Tutorial's representations of astrophysical phenomena, which we selected and created based on theoretically motivated considerations of their pedagogical affordances, are used to help introductory astronomy students develop more expert-like reasoning abilities.Comment: 10 pages, 10 figures, accepted for publication in the American Journal of Physic