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Lab02-PreLab - PHYS 2212 Lab Exercise 02 Charge...

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PHYS 2212 Lab Exercise 02: Charge Distributions & Integration PRELIMINARY MATERIAL TO BE READ BEFORE LAB PERIOD I. Density Functions: When we consider a charge distribution consisting of very many point charges, it is convenient to introduce the idea of a charge density , which will allow us to treat the distribution as being continuous . A “purist” might object to this, pointing out that ultimately, at the atomic scale, all distributions are composed of individual point charges (protons and electrons). However, if we view the distribution at a macroscopic scale, even a region that we would consider to be pointlike—e.g., a box of length 0.1 mm on a side—is of vast size in comparison to atomic scales, and would consist of tens of millions of individual point charges. In that context, treating a general charge distribution as being continuous is not all that unrealistic. A continuous charge density function , then, is a means of describing how a certain amount of charge is spread out over some particular region. Depending upon the type of region that is being spread over, we have three different “categories” of density functions: If an amount of charge is spread throughout a three-dimension volume, we have a volume charge density , conventionally denoted by the symbol “ ρ ”. (This is, perhaps, suggestive of the familiar mass density that describes distributions of matter.) A volume charge density describes the amount of charge found charge per unit volume examined . Thus, if you were to state that the volume density at some point were 15 μ C/mm 3 , you would essentially be asserting that a tiny box around that point (again, lets say, of dimension 0.1 mm on a side) would contain a total charge of: (15 x 10 –6 C/mm 3 ) × (0.1 mm) 3 = 15 nC. If an amount of charge is spread over some two-dimensional surface , we have a surface charge density , and will use the symbol η to represent such a distribution. It is important to recognize that we are treating such a distribution as having “zero thickness”—even though, technically, there is some non-zero thickness to a typical “ η ”. Provided, however, that the thickness “ t” is very small compared to the area of the surface, we can neglect t and treat the distribution as if it were truly “2D”. Surface charge density is then calculated in a manner comparable to volume charge density. In this case, however, we measure the amount of charge found per unit area of the surface examined. Keep in mind, as well, that there is no a priori reason why the surface in question has to be flat —it is completely possible to deal with a situation where charge is spread over the curved surface of a sphere or cylinder.
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