chapt05 - CHAPTER 5 CONDUCTORS, DIELECTRICS, AND...

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CHAPTER 5 CONDUCTORS, DIELECTRICS, AND CAPACITANCE In this chapter we intend to apply the laws and methods of the previous chapters to some of the materials with which an engineer must work. After defining current and current density and developing the fundamental continuity equation, we shall consider a conducting material and present Ohm's law in both its microscopic and macroscopic forms. With these results we may calculate resis- tance values for a few of the simpler geometrical forms that resistors may assume. Conditions which must be met at conductor boundaries are next obtained, and this knowledge enables us to introduce the use of images. After a brief consideration of a general semiconductor, we shall investigate the polarization of dielectric materials and define relative permittivity, or the dielectric constant, an important engineering parameter. Having both conduc- tors and dielectrics, we may then put them together to form capacitors. Most of the work of the previous chapters will be required to determine the capacitance of the several capacitors which we shall construct. The fundamental electromagnetic principles on which resistors and capaci- tors depend are really the subject of this chapter; the inductor will not be intro- duced until Chap. 9. 119
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5.1 CURRENT AND CURRENT DENSITY Electric charges in motion constitute a current . The unit of current is the ampere (A), defined as a rate of movement of charge passing a given reference point (or crossing a given reference plane) of one coulomb per second. Current is symbo- lized by I , and therefore I ± dQ dt ² 1 ³ Current is thus defined as the motion of positive charges, even though conduc- tion in metals takes place through the motion of electrons, as we shall see shortly. In field theory we are usually interested in events occurring at a point rather than within some large region, and we shall find the concept of current density , measured in amperes per square meter (A/m 2 ), more useful. Current density is a vector 1 represented by J : The increment of current ± I crossing an incremental surface ± S normal to the current density is ± I ± J N ± S and in the case where the current density is not perpendicular to the surface, ± I ± J Á ± S Total current is obtained by integrating, I ± ± S J Á d S ² 2 ³ Current density may be related to the velocity of volume charge density at a point. Consider the element of charge ± Q ± ± v ± v ± ± v ± S ± L , as shown in Fig. 5 : 1 a . To simplify the explanation, let us assume that the charge element is oriented with its edges parallel to the coordinate axes, and that it possesses only an x component of velocity. In the time interval ± t , the element of charge has moved a distance ± x , as indicated in Fig. 5 : 1 b . We have therefore moved a charge ± Q ± ± v ± S ± x through a reference plane perpendicular to the direction of motion in a time increment ± t , and the resultant current is ± I ± ± Q ± t ± ± v ± S ± x ± t As we take the limit with respect to time, we have ± I ±
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This note was uploaded on 03/30/2010 for the course EE 2317 taught by Professor Wilton during the Spring '10 term at University of Houston.

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chapt05 - CHAPTER 5 CONDUCTORS, DIELECTRICS, AND...

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