Lec 5-6 Cartesian Coordinate Systems PDF

Electromagnetic Theory  Represented a vector at a point in space in terms of its component vectors along a set of three mutually orthogonal directions defined by three mutually orthogonal unit vectors at that point.  Coordinate system: to relate vectors at one point in space to vectors at anot...

Electromagnetic Theory  Represented a vector at a point in space in terms of its component vectors along a set of three mutually orthogonal directions defined by three mutually orthogonal unit vectors at that point.  Coordinate system: to relate vectors at one point in space to vectors at another point in space, we must define the set of three reference directions at each and every point in space.  Cartesian, cylindrical, and spherical coordinate systems Cartesian/Rectangular Coordinate System(RCS)  Cartesian coordinate system is defined by a set of three mutually orthogonal planes  The point at which the three planes intersect is known as the origin O.  The origin is the reference point relative to which any other point in space is located.  Each pair of planes intersects in a straight line.  The three planes define a set of three straight lines that form the coordinate axes x, y, & z.  the positive x-, y-, and z-directions are chosen such that they form a right- handed system, ie. ax x ay = az  In the yz-plane, the value of x is constant and equal to zero, its value at the origin, since movement on this plane does not require any movement in the x-direction. Similarly, on the zx-plane, the value of y is constant and equal to zero, and on the xy-plane, the value of z is constant and equal to zero.  Any point other than the origin is given by the intersection of three planes x=constant, y=constant, z=constant; i.e. P(2,5,4) where x=2, y=5, z=4 planes intersect.  In EMT we have to work with line, surface, and volume integrals. These involve differential lengths, surfaces, and volumes, obtained by incrementing the coordinates by infinitesimal amounts.  In RCS, the three coordinates represent lengths, the differential length elements obtained by incrementing one coordinate at a time, keeping the other two coordinates constant, are dxax, dyay, and dzaz for the x-, y-, and z-coordinates, respectively. Differential length vector: The differential length vector dl is the vector drawn from a point P(x, y, z) to a neighboring point Q(x+dx, y+dy, z+dz) obtained by incrementing the coordinates of P by infinitesimal amounts.  To find the unit vector normal to a surface at a point on that surface  Differential surface vector. Two differential length vectors dl1 and dl2 originating at a point define a differential surface whose area dS is that of the parallelogram having dl1 and dl2 as two of its adjacent sides. and differential surface vector dS whose magnitude is the area dS and whose direction is normal to the differential surface. Normal vector can be directed from either side of surface. ds associated with x=constant, y=constant, z=constant plane respectively  Differential volume. Three differential length vectors dl1, dl2 and dl3 originating at a point define a differential volume which is that of the parallelepiped having dl1, dl2 and dl3 as three of its contiguous edges.