We discuss common difficulties in upper-division electricity and magnetism (E&M) in the areas of Gauss's law, vector calculus, and electric potential using both quantitative and qualitative evidence. We also show that many of these topical difficulties may be tied to student difficulties with mathematics. At the junior level, some students struggle to combine mathematical calculations and physics ideas, to account for the underlying spatial situation when doing a mathematical calculation, and to access appropriate mathematical tools. We discuss the implications of our findings for E&M instruction at the junior level.
The flagellated protozoan Salpingoeca rosetta is one of the closest relatives of multicellular animals. Unicellular S. rosetta can be induced to form multicellular colonies, but colonies swim more slowly than individual cells so the advantages conferred by colony formation are uncertain. Here we use theoretical models to show that hydrodynamic cooperation between cells can increase the fluid supply to the colony, an important predictor of feeding rate. Our results suggest that hydrodynamic benefits may have been an important selective factor in the evolution of early multicellular animals.
We study systematically the effect of substrate compliance on the threshold for splashing of a liquid drop using an elastic membrane under variable tension. We find that the splashing behavior is strongly affected by the tension in the membrane and splashing can be suppressed by reducing this tension. The deflection of the membrane upon droplet impact is measured using a laser sheet, and the results allow us to estimate the energy absorbed by the film upon drop impact. Measurements of the velocity and acceleration of the spreading drop after impact indicate that the splashing behavior is set at very early times after, or possibly just before, impact, far before the actual splash occurs. We also provide a model for the tension dependence of the splashing threshold based on the pressure in the drop upon impact that takes into account the interplay between membrane tension and drop parameters.
Microscopic sessile suspension feeders are a critical component in aquatic ecosystems, acting as an intermediate trophic stage between bacteria and higher eukaryotic taxa. Because they live attached to boundaries, it has long been thought that recirculation of the feeding currents produced by sessile suspension feeders inhibits their ability to access fresh fluid. However, previous models for the feeding flows of these organisms assume that they feed by pushing fluid perpendicular to surfaces they live upon, whereas we observe that sessile suspension feeders often feed at an angle to these boundaries. Using experiments and calculations, we show that living suspension feeders (Vorticella) likely actively regulate the angle that they feed relative to a substratum. We then use theory and simulations to show that angled feeding increases nutrient and particle uptake by reducing the reprocessing of depleted water. This work resolves an open question of how a key class of suspension-feeding organisms escapes physical limitations associated with their sessile lifestyle.
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