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Unformatted text preview: Chapter 3 Electrostatic Fields and Potentials Stationary charges produce vector electric fields that emanate outward from positive charges and terminate on negative charges. The study of the fields produced by sta- tionary charges is called electrostatics . We use electrostatics to explain the workings of a variety of common circuit devices, most notably the capacitor. But electrostat- ics also provides the mathematical framework for understanding elements of two of the most spectacular natural light shows: lightning and aurora. In this chapter, we first apply Maxwells Equations to isolated charges and then to charge distributions with symmetric properties (i.e., invariant in one or more directions) that keep the vector mathematics simple enough for us to directly solve for the electric field. For the more complicated (and more interesting) charge distributions, we introduce the electrostatic potential, which can be determined from a scalar equation. The gradient operating on the potential will then yield the vector electric field. August 25, 2009 3.1 Applications of Gauss Law to Symmetric Charge Distributions The electric field from some distributions of charge can be determined using the integral version of Gauss Law. Con- sider the simplest case of a single point charge of value q . By symmetry, the associated electric field can only point outward from or inward toward the charge, so a spherical coordinate system centered on the charge is appropriate. If we surround the charge with an imaginary spherical surface of radius r , as shown to the right, then the surface integral in (2.7a) becomes 4 r 2 E and the volume integral becomes q and ~ E = q 4 r 2 r (3.1) r q Figure 3.1: Gaussian surface surrounding a point charge. 3.1 3.2 CHAPTER 3. ELECTROSTATIC FIELDS AND POTENTIALS This sort of imaginary surface is called a Gaussian surface in this context. Gaussian surfaces give us a way to visualize symmetries in the fields and can be used to help solve a variety of problems. In circuit courses, superposition is used to investigate the role of one source when the others are suppressed. This technique works when the equations are linear, as is the case here. We thus can add the separate solutions for each charge to obtain the complete solution. For example, suppose there are two charges, q 1 and q 2 , as shown in Figure 3.2 with the solu- tions ~ E 1 and ~ E 2 as given in (3.1). Then by superposition, ~ E TOTAL = ~ E 1 + ~ E 2 where ~ E 1 = q 1 4 r 2 1 r 1 and ~ E 2 = q 2 4 r 2 2 r 2 where r 1 and r 2 are unit vectors in the direc- tions of ~ r 1 and ~ r 2 , respectively, and the solu- tions add in vector form. 1 q 2 q 1 r & 2 r & 1 E & 2 E & TOTAL E & Figure 3.2: Electric field from two point charges....
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