MIT OpenCourseWare http ocw mit edu Electromagnetic Field Theory A Problem Solving Approach For any use or distribution of this textbook please cite as follows Markus Zahn Electromagnetic Field Theory A Problem Solving Approach Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms chapter 1 review of vector analysis 2 Review of Vector Analysis Electromagnetic field theory is the study of forces between charged particles resulting in energy conversion or signal transmission and reception These forces vary in magnitude and direction with time and throughout space so that the theory is a heavy user of vector differential and integral calculus This chapter presents a brief review that highlights the essential mathematical tools needed throughout the text We isolate the mathematical details here so that in later chapters most of our attention can be devoted to the applications of the mathematics rather than to its development Additional mathematical material will be presented as needed throughout the text 1 1 COORDINATE SYSTEMS A coordinate system is a way of uniquely specifying the location of any position in space with respect to a reference origin Any point is defined by the intersection of three mutually perpendicular surfaces The coordinate axes are then defined by the normals to these surfaces at the point Of course the solution to any problem is always independent of the choice of coordinate system used but by taking advantage of symmetry computation can often be simplified by proper choice of coordinate description In this text we only use the familiar rectangular Cartesian circular cylindrical and spherical coordinate systems 1 1 1 Rectangular Cartesian Coordinates The most common and often preferred coordinate system is defined by the intersection of three mutually perpendicular planes as shown in Figure 1 la Lines parallel to the lines of intersection between planes define the coordinate axes x y z where the x axis lies perpendicular to the plane of constant x the y axis is perpendicular to the plane of constant y and the z axis is perpendicular to the plane of constant z Once an origin is selected with coordinate 0 0 0 any other point in the plane is found by specifying its x directed ydirected and z directed distances from this origin as shown for the coordinate points located in Figure 1 lb I CoordinateSystems 3 T 2 2 3 I 3 3 2 1 2 2 I I I I i b1 2 3 4 b xdz dS Figure 1 1 Cartesian coordinate system a Intersection of three mutually perpendicular planes defines the Cartesian coordinates x y z b A point is located in space by specifying its x y and z directed distances from the origin c Differential volume and surface area elements By convention a right handed coordinate system is always used whereby one curls the fingers of his or her right hand in the direction from x to y so that the forefinger is in the x direction and the middle finger is in the y direction The thumb then points in the z direction This convention is necessary to remove directional ambiguities in theorems to be derived later Coordinate directions are represented by unit vectors i i and i2 each of which has a unit length and points in the direction along one of the coordinate axes Rectangular coordinates are often the simplest to use because the unit vectors always point in the same direction and do not change direction from point to point A rectangular differential volume is formed when one moves from a point x y z by an incremental distance dx dy and dz in each of the three coordinate directions as shown in 4 Review of VectorAnalysis Figure 1 Ic To distinguish surface elements we subscript the area element of each face with the coordinate perpendicular to the surface 1 1 2 CircularCylindrical Coordinates The cylindrical coordinate system is convenient to use when there is a line of symmetry that is defined as the z axis As shown in Figure 1 2a any point in space is defined by the intersection of the three perpendicular surfaces of a circular cylinder of radius r a plane at constant z and a plane at constant angle 4 from the x axis The unit vectors i i6 and iz are perpendicular to each of these surfaces The direction of iz is independent of position but unlike the rectangular unit vectors the direction of i and i6 change with the angle 0 as illustrated in Figure 1 2b For instance when 0 0 then i i and i i while if ir 2 then i i and i ix By convention the triplet r 4 z must form a righthanded coordinate system so that curling the fingers of the right hand from i to i4 puts the thumb in the z direction A section of differential size cylindrical volume shown in Figure 1 2c is formed when one moves from a point at coordinate r 0 z by an incremental distance dr r d4 and dz in each of the three coordinate directions The differential volume and surface areas now depend on the coordinate r as summarized in Table 1 1 Table 1 1 Differential lengths surface area and volume elements for each geometry The surface element is subscripted by the coordinate perpendicular to the surface CARTESIAN CYLINDRICAL SPHERICAL dl dx i dy i dz i dl dri r d0 i dz i dl dri rdOis r sin 0 do i dS r 9 sin 0 dO d4 dSr r dO dz dS dy dz dS r sin Odr d4 dS drdz dS dx dz dS dx dy dS r dr do dS rdrdO dV r 2 sin drdO d dV r dr d4 dz dV dxdydz 1 1 3 Spherical Coordinates A spherical coordinate system is useful when there is a point of symmetry that is taken as the origin In Figure 1 3a we see that the spherical coordinate r 0 0 is obtained by the intersection of a sphere with radius r a plane at constant CoordinateSystems 5 b V rdrdodz c Figure 1 2 Circular cylindrical coordinate system a Intersection of planes of constant z and 4 with a cylinder of constant radius r defines the coordinates r 4 z b The direction of the unit vectors i and i vary with the angle 46 c Differential volume and surface area elements angle 4 from the x axis as defined for the cylindrical coordinate system and a cone at angle 0 from the z axis The unit vectors i is and i are perpendicular to each of these surfaces and change direction from point to point The triplet r 0 4 must form a right handed set of coordinates The differential size spherical volume element formed by considering incremental displacements dr rdO r sin 0 d4 6 Al I Review of Vector Analysis ll U I V
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