We describe how a classic electrostatics experiment can be modified to be a four-point probe lab experiment. Students use the four-point probe technique to investigate how the measured resistance varies as a function of the position of the electrodes with respect to the edge of the sample. By using elementary electromagnetism concepts such as the superposition principle, the continuity equation, the relation between electric field and electric potential, and Ohm's law, a simple model is derived to describe the four-point probe technique. Although the lab introduces the students to the ideas behind the Laplace equation and the methods of images, advanced mathematics is avoided so that the experiment can be done in trigonometry and algebra based physics courses. In addition, the experiment introduces the students to a standard measurement technique that is widely used in industry and thus provides them with useful hands-on experience.
The unique advantage of this system is to allow rapid blood sampling after a puff of cigarette smoke, with the benefits of reproducibility, reduction in labor intensity, and high temporal resolution.
Abstract. We prove that every hereditarily indecomposable continuum is the image under an open, monotone map of a one-dimensional hereditarily indecomposable continuum. Thus there exists a one-dimensional hereditarily indecomposable continuum with infinite dimensional hyperspace.Eberhart and Nadler have shown that every hereditarily indecomposable continuum has a hyperspace of dimension either two or infinite. Every planar hereditarily indecomposable continuum (as well as every other standard one-dimensional example) has a two-dimensional hyperspace. Every hereditarily indecomposable continuum of dimension at least two has an infinite-dimensional hyperspace.Lau [L-l] has shown that an hereditarily indecomposable continuum A has an infinite-dimensional hyperspace if and only if there exists a monotone map f: X -> Y for some Y of dimension greater than 1. In this paper, we prove that every hereditarily indecomposable continuum is the image under an open, monotone map of a one-dimensional hereditarily indecomposable continuum. Thus there exist one-dimensional hereditarily indecomposable continua with infinite-dimensional hyperspaces.
Preliminaries. A continuum is a nondegenerate compact connected metricspace.An arc is e-crooked if for each pair of its points p and q there are points r and s between p and q such that r lies between p and s, dist(/>, s) < e and dist(r, q) < e [B-l].We can assume that the target space C" is embedded in Euclidean space E2n+X (or in the Hilbert Cube Q if n is infinite). Since C" is hereditarily indecomposable, if U is a cover of C" (by sets open in E2n+' or Q) and e > 0, there exists a cover (by sets open in E2n+X or Q) V of C", closure refining U, of mesh less than e, so that every arc in V* is c-crooked.From now on every open set will be open in the appropriate Euclidean space.If A is a complex, K' will be the one-skeleton of the first barycentric subdivision of K.
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