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ELECTROMAGNETICS THEORY (SEE 2523)
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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
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The cross product between the unit vector is: While the dot product is:
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Any vector can be represented as The magnitude for is given by
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If and the vector operations:
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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 <
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Fig. 1.10 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 :
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Fig.1.10: Differential element in Cartesian coordinate dz dy dx
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Three differential surfaces generated, dz dy dx
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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. 1.4.2: CYLINDRICAL COORDINATE SYSTEM
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The ranges on the variables are: 0 < r < 0 < < 2 - < z < Fig. 1.11: A view of a point in cylindrical coordinate system.
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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
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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 .
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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. 1.4.2: SPHERICAL COORDINATE SYSTEM Defined by three variables : r, and and the unit vectors are, and.
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The ranges on the variables are: 0 < r < 0 < < 0 < < 2 Fig. 1.13: A view of a point in spherical coordinate system
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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
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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
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The following operations involving operator :
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