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 1review of vector analysis2 Review of Vector AnalysisElectromagnetic field theory is the study of forces betweencharged particles resulting in energy conversion or signal transmis-sion and reception. These forces vary in magnitude and directionwith time and throughout space so that the theory is a heavy userof vector, differential, and integral calculus. This chapter presentsa brief review that highlights the essential mathematical toolsneeded throughout the text. We isolate the mathematical detailshere so that in later chapters most of our attention can be devotedto the applications of the mathematics rather than to itsdevelopment. Additional mathematical material will be presentedas needed throughout the text.1-1 COORDINATE SYSTEMSA coordinate system is a way of uniquely specifying thelocation of any position in space with respect to a referenceorigin. Any point is defined by the intersection of threemutually perpendicular surfaces. The coordinate axes arethen defined by the normals to these surfaces at the point. Ofcourse the solution to any problem is always independent ofthe choice of coordinate system used, but by taking advantageof symmetry, computation can often be simplified by properchoice of coordinate description. In this text we only use thefamiliar rectangular (Cartesian), circular cylindrical, andspherical coordinate systems.1-1-1 Rectangular (Cartesian) CoordinatesThe most common and often preferred coordinate systemis defined by the intersection of three mutually perpendicularplanes as shown in Figure 1-la. Lines parallel to the lines ofintersection between planes define the coordinate axes(x, y, z), where the x axis lies perpendicular to the plane ofconstant x, the y axis is perpendicular to the plane of constanty, and the z axis is perpendicular to the plane of constant z.Once an origin is selected with coordinate (0, 0, 0), any otherpoint in the plane is found by specifying its x-directed, y-directed, and z-directed distances from this origin as shownfor the coordinate points located in Figure 1-lb.ICoordinateSystems-3 -2 -12, 2)? I I II . i(b1(b)T(-2,2,3)-3 I2 3 4xdzdS, =Figure 1-1 Cartesian coordinate system. (a) Intersection of three mutually perpen-dicular planes defines the Cartesian coordinates (x,y, z). (b)A point is located in spaceby specifying its x-, y-and z-directed distances from the origin. (c) Differential volumeand surface area elements.By convention, a right-handed coordinate system is alwaysused whereby one curls the fingers of his or her right hand inthe direction from x to y so that the forefinger is in the xdirection and the middle finger is in the y direction. Thethumb then points in the z direction. This convention isnecessary to remove directional ambiguities in theorems to bederived later.Coordinate directions are represented by unit vectors i., i,and i2, each of which has a unit length and points in thedirection along one of the coordinate axes. Rectangularcoordinates are often the simplest to use because the unitvectors always point in the same direction and do not changedirection from point to point.A rectangular differential volume is formed when onemoves from a point (x, y, z) by an incremental distance dx, dy,and dz in each of the three coordinate directions as shown in3. -4 Review of VectorAnalysisFigure 1-Ic. To distinguish surface elements we subscript thearea element of each face with the coordinate perpendicularto the surface.1-1-2 CircularCylindrical CoordinatesThe cylindrical coordinate system is convenient to usewhen 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 theintersection of the three perpendicular surfaces of a circularcylinder of radius r, a plane at constant z, and a plane atconstant angle 4 from the x axis.The unit vectors i,, i6 and iz are perpendicular to each ofthese surfaces. The direction of iz is independent of position,but unlike the rectangular unit vectors the direction of i, and i6change with the angle 0 as illustrated in Figure 1-2b. Forinstance, 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 right-handed coordinate system so that curling the fingers of theright hand from i, to i4 puts the thumb in the z direction.A section of differential size cylindrical volume, shown inFigure 1-2c, is formed when one moves from a point atcoordinate (r,0, z) by an incremental distance dr, r d4, and dzin each of the three coordinate directions. The differentialvolume and surface areas now depend on the coordinate r assummarized in Table 1-1.Table 1-1 Differential lengths, surface area, and volume elements foreach geometry. The surface element is subscripted by the coordinateperpendicular to the surfaceCARTESIAN CYLINDRICAL SPHERICALdl=dx i+dy i,+dz i, dl=dri,+r d0 i#+dz i, dl=dri,+rdOis+ r sin 0 do i,dS. = dy dz dSr = r dO dz dS, = r9 sin 0 dO d4dS, = dx dz dS, = drdz dS@ = r sin Odr d4dS, = dx dy dS, = r dr do dS, = rdrdOdV=dxdydz dV= r dr d4 dz dV=r2 sin drdO d1-1-3 Spherical CoordinatesA spherical coordinate system is useful when there is apoint of symmetry that is taken as the origin. In Figure 1-3awe see that the spherical coordinate (r,0, 0) is obtained by theintersection of a sphere with radius r, a plane at constantCoordinateSystems 5(b)V= rdrdodz(c)Figure 1-2 Circular cylindrical coordinate system. (a) Intersection of planes ofconstant 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) Differentialvolume and surface area elements.angle 4 from the x axis as defined for the cylindrical coor-dinate system, and a cone at angle 0 from the z axis. The unitvectors i,, is and i# are perpendicular to each of these sur-faces and change direction from point to point. The triplet(r, 0, 4) must form a right-handed set of coordinates.The
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