Most of the quantities that we deal in Electromagnetics are functions of both space and time. Variations in space are represented with the proper coordinate system.
Electromagnetics is the study of phenomena related to Electric field, Magnetic field, their interaction etc. These quantities are time-varying as well as spatial functions. In order to describe spatial variations of these quantities, one must be able to define all points and vectors uniquely in space in the appropriate manner.
A point can be represented using orthogonal or non-orthogonal coordinate system. An orthogonal system is one in which the coordinates arc mutually perpendicular. The non-orthogonal coordinate system is hard to work on and practically neglected.
Thre are different types of orthogonal coordinate systems- Cartesian (or rectangular), circular cylindrical, spherical, elliptic cylindrical, parabolic cylindrical, conical, prolate spheroidal, oblate spheroidal and ellipsoidal.
Cartesian Coordinate System
In the Cartesian coordinate system, any point of the space is represented using three coordinates that are x, y, and z and the point is represented as P (x,y,z). Basically, x, y, and z are the distances measured from reference planes formed by the three coordinate axes viz. X-axis, Y-axis, and Z-axis.
The x-coordinate is the perpendicular distance from the YZ plane. The y-coordinate is the perpendicular distance from the XZ plane, similarly, z-coordinate is the normal distance from XY plane.
The X, Y and Z axes are mutually perpendicular to each other and their selection must obey the right-hand screw rule.
The ranges for each coordinate is from +∞ to -∞ i.e. the value of x, y or z coordinate can be anything between +infinity to -infinity depending on the distance of the required point from each of the reference planes.
Constant Coordinate Surfaces and Lines
If only one coordinate of three is given to be constant and rest two are variables then the required locus comes out to be a plane or surface. In other words, x = constant (y, z variables), y = constant (x, z variables) or z = constant (x, y variables) represents the equations of planes each.
If two coordinates of three are given to be constant and remaining one is variable then the required locus becomes the straight line. For example, the graph for x = constant, y = constant and z is variable (not given) is a straight line parallel to Z-axis. Likewise, x = constant and z = constant gives us a line parallel to Y-axis and y = constant and z = constant leads to a line parallel to X-axis.
Finally, if all three coordinates are given or constants, then we get a specific point in the space.
Why is Cartesian Coordinate System called as Rectangular Coordinate System?
As seen from the initial diagram, if we draw all the distances i.e. x, y, and z pictorially then a rectangular block is formed with edges as coordinate values. Hence it is also named as Rectangular Coordinate System.
Vectors in Cartesian Coordinate System
Any vector in a Cartesian coordinate system is represented using three mutually perpendicular vectors.
The general form of any vector in the Cartesian coordinate system is as follows-Ax is the x-component, Ay is the y-component, and Az is the z-component of the given vector. The component of the vector along any axis is flux or effect of the given vector along that axis.