Electromagnetic Field Theory A Problem Solving Approach (Zahn)

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.

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 xdirected, y directed, and z-directed distances from this origin as shown for the coordinate points located in Figure 1-1b.

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 i , 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 Figure 1-1c. To distinguish surface elements we subscript the area element of each face with the coordinate perpendicular to the surface.

A scalar quantity is a number completely determined by its magnitude, such as temperature, mass, and charge, the last being especially important in our future study. Vectors, such as velocity and force, must-also have their direction specified and in this text are printed in boldface type. They are completely described by their components along three coordinate directions as shown for rectangular coordinates in Figure 1-4. A vector is represented by a directed line segment in the direction of the vector with its length proportional to its magnitude.