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Unformatted text preview: Chapter 30 The Electric Field Due to a Continuous Distribution of Charge on a Line 267 30 The Electric Field Due to a Continuous Distribution of Charge on a Line Every integral must include a differential (such as dx, dt, dq, etc.). An integral is an infinite sum of terms. The differential is necessary to make each term infinitesimal (vanishingly small). ∫ x d x f ) ( is okay, ∫ dy y g ) ( is okay, and ∫ t d t h ) ( is okay, but never write ∫ ) ( x f , never write ∫ ) ( y g and never write ∫ ) ( t h . Here we revisit Coulomb’s Law for the Electric Field. Recall that Coulomb’s Law for the Electric Field gives an expression for the electric field, at an empty point in space, due to a charged particle. You have had practice at finding the electric field at an empty point in space due to a single charged particle and due to several charged particles. In the latter case, you simply calculated the contribution to the electric field at the one empty point in space due to each charged particle, and then added the individual contributions. You were careful to keep in mind that each contribution to the electric field at the empty point in space was an electric field vector, a vector rather than a scalar, hence the individual contributions had to be added like vectors. A Review Problem for the Electric Field due to a Discrete 1 Distribution of Charge Let’s kick this chapter off by doing a review problem. The following example is one of the sort that you learned how to do when you first encountered Coulomb’s Law for the Electric Field. You are given a discrete distribution of source charges and asked to find the electric field (in the case at hand, just the x component of the electric field) at an empty point in space. The example is presented on the next page. Here, a word about one piece of notation used in the solution. The symbol P is used to identify a point in space so that the writer can refer to that point, unambiguously, as “point P .” The symbol P in this context does not stand for a variable or a constant. It is just an identification tag. It has no value. It cannot be assigned a value. It does not represent a distance. It just labels a point. 1 The charge distribution under consideration here is called a discrete distribution as opposed to a continuous distribution because it consists of several individual particles that are separated from each other by some space. A continuous charge distribution is one in which some charge is “smeared out” along some line or over some region of space. Chapter 30 The Electric Field Due to a Continuous Distribution of Charge on a Line 268 Example 30-1 (A Review Problem) There are two charged particles on the x-axis of a Cartesian coordinate system, q 1 at x = x 1 and q 2 at x = x 2 where x 2 > x 1 . Find the x component of the electric field, due to this pair of particles, valid for all points on the x- y plane for which x > x 2 ....
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