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Unformatted text preview: Chapter 2 Maxwell’s Equations This chapter introduces Maxwell’s Equations. These equations describe the time- independent (static) and time-dependent (dynamic) behavior of electric and mag- netic fields and their relationships to charges and currents. We simply state and interpret these equations here, leaving the proof of their validity to your physics courses. Maxwell’s Equations are written in terms of vector calculus, so we will start with a review of scalar and vector functions in the three common coordinate systems used for three-dimensional spaces. Application of Maxwell’s Equations to engineering problems will then occupy us for the rest of this book. August 25, 2009 2.1 Coordinate Systems The fundamental parameters we deal with in this course are the electric and magnetic fields that describe electromagnetic phenomena. These parameters are vector fields which, in general, depend upon both space and time, where the spatial quantities are expressed in terms of orthogonal coordinate systems. You are most likely familiar with the cartesian system illustrated below. y x z y x z z ( ) z y x , , y P r & Figure 2.1: Cartesian coordinate sys- tem. y x z y x z z y z ˆ x ˆ y ˆ Figure 2.2: Local unit vectors in the cartesian coordinate system. 2.1 2.2 CHAPTER 2. MAXWELL’S EQUATIONS If we use a cartesian coordinate system, the electric field, for example, can be ex- pressed in the form ~ E( x,y,z,t ) where the arrow symbol implies a vector. A general vector field, ~ A( x,y,z,t ), can be expressed in at least three equivalent ways, all of which we shall use in this text. The location of a given point in three space is de- scribed by the point ( x,y,z ), which is written with the shorthand notation ~ r and called the position vector of point P. A vector field can then be expressed as ~ A( ~ r ,t ). There are many possible representations of a vector involving various ways of ex- pressing the unit vectors. An example of the form we will use in this text is given by ~ A( ~ r ,t ) = A x ˆx + A y ˆy + A z ˆ z. In this system, it is important to note that no matter where a vector is to be defined in three dimensional space, the local unit vector system is simply a translation of the unit vectors from the origin. This is not the case in the other two systems we use. Depending on the symmetry of the physical phenomenon of interest, other coordinate systems may be more appropriate. The two most important coordinate systems in addition to cartesian are illustrated next, along with the representations of ~ A( ~ r ,t ) in each. In these systems, the direction of the local unit vectors (ˆ r , ˆ φ, ˆ z) and (ˆ r , ˆ θ, ˆ φ ) in general are not parallel to the set at the origin....
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This note was uploaded on 01/16/2010 for the course ECE 3030 at Cornell.