ELECTROMAGNETICS THEORY (SEE 2523)

 An orthogonal system is one in which the coordinates are mutually perpendicular.  Examples of orthogonal coordinate systems include the Cartesian, cylindrical and spherical coordinates.  There must be three independent variables. e.g: u 1, u 2 and u 3. , and are unit vectors for each surface and the direction normal to their surfaces. 1.4 ORTHOGONAL COORDINATE SYSTEM

 The cross product between the unit vector is:  While the dot product is:

 Any vector can be represented as  The magnitude for is given by

 If and the vector operations:

1.4.1: CARTESIAN COORDINATE SYSTEM Defined by three variables x, y and z.  The ranges on the variables are:  A point P(x 1, y 1, z 1 ) in coordinate system is located at the intersection of the three surfaces which is determined by x = x 1, y = y 1 and z = z 1.  Most of the problems in electromagnetics only can be solved using line, surface and volume integral. -  < x < , -  < y <  and -  < z < 

 Fig shows the points P and Q whose coordinates are P(x, y, z) and Q(x+dx, y+dy, z+dz).  The movement from point P to point Q cause the variables vary from x to x+dx, y to y+dy and z to z+dz.  These changes will cause the differential volume elements in Cartesian coordinates given by : dv = dxdydz  Differential length, is given by :

Fig.1.10: Differential element in Cartesian coordinate dz dy dx

 Three differential surfaces generated, dz dy dx

 Defined by three variables : r,  and z and the unit vectors are, and. A variable r, at a point P is directed radially outward, normal to the z-axis.  is measured from the x-axis in the xy-plane to the r. z is the same as in the Cartesian system : CYLINDRICAL COORDINATE SYSTEM

 The ranges on the variables are: 0 < r <  0 <  < 2  -  < z <  Fig. 1.11: A view of a point in cylindrical coordinate system.

 A line, surface and volume will be generated when a single variable, two variables and three variables, respectively are varied.  When these changes are differential as shown in Fig.1.12, we generate the following differential lines, surfaces and volume. Fig 1.12: Differential elements of the cylindrical coordinate system

 Yield a differential volume when the coordinate increase from r,  and z to r+dr,  +d  dan z+dz.  When the angle vary from  to  +d , the changes in the distance is rd .

 A variable r is defined as a distance from the origin to any point.  is defined as an angle between the +z axis and the r line.  is an angle and exactly the same as in cylindrical coordinate system : SPHERICAL COORDINATE SYSTEM  Defined by three variables : r, and  and the unit vectors are, and.

 The ranges on the variables are: 0 < r <  0 <  <  0 <  < 2  Fig. 1.13: A view of a point in spherical coordinate system

 The changes in d  and d  will cause the distance change to rd  and rsin  d . Fig. 1.14: Differential elements in spherical coordinate system

 To reduce the length of certain equations found in electromagnetics : an operator,  called del or nabla.  In Cartesian coordinates for example:  The  operator itself has no physical meaning unless it is associated with scalars and vectors.  Should be noted that some  operations yields scalars while others yield vectors. 1.5: DEL OPERATOR

 The following operations involving  operator :