1. Darshan Institute of Engg. & Tech. Subject:- EM Topic:- Coordinate Systems Prepared By:  Pruthvirajsinh Jadeja (130540111013)

2.  Purpose of a Coordinate System: The purpose of a coordinate system is to uniquely determine the position of an object or data point in space. By `space' we may literally mean in physical space, but in general it simply refers to what we might call `variable-space', where each dimension corresponds to one variable. A graph of stock prices probably has variables of `time' and `value', so we are in time-value space, and our coordinate system and any equations we might write need to specify the time and value of each data point. How many parameters does our coordinate system need to have? As many parameters as we have variables. This is related to the idea of dependent and independent vectors. If we only Coordinate Systems

3. had two independent vectors, we couldn't get to all of the points in a three dimensional object; likewise if we only have two parameters, we can't specify data in terms of three variables. However, we saw that any three independent variables will let us get to all of 3D space (will span 3-space, more formally). This also has an analogy in coordinate space. The choice of parameters is not in general unique | any two independent parameters will do, although there are only a few standard choices.

4.  Orthogonal coordinate systems i.The Cartesian (rectangular) coordinate system ii.The cylindrical coordinate system iii.The spherical coordinate system

5. A coordinate system consists of four basic elements: (1) Choice of origin (2) Choice of axes (3) Choice of positive direction for each axis (4) Choice of unit vectors for each axis  Cartesian coordinate system (1) Choice of Origin Choose an originO. If you are given an object, then your choice of origin may coincide with a special point in the bo.dy. For example, you may choose the mid-point of a straight piece of wire.

6. (2) Choice of Axis Now we shall choose a set of axes. The simplest set of axes are known as the Cartesian axes, X-axis, Y-axis, and the Z-axis. Once again, we adapt our choices to the physical object. For example, we select the -axis so that the wire lies on the -axis,

7. Then each point in space our PS can be assigned a triplet of values (xP,yP,zP), the Cartesian coordinates of the point. The ranges of these values are: −∞P< Xp <+∞, −∞<yP<+∞, −∞<zP<+∞. The collection of points that have the same the coordinatePy is called a level surface. Suppose we ask what collection of points in our space S have the same value of Pyy=. This is the set of points {}(,,)suchthatPyPSxyzSyy= ∈ =. This set Py S is a plane, the plane (Figure B.1.2), called a level set for constant- xz Py. Thus, the - coordinate of any point actually describes a plane of points perpendicular to the yy-axis.

8. (3) Choice of Positive Direction Our third choice is an assignment of positive direction for each coordinate axis. We shall denote this choice by the symbol + along the positive axis. Conventionally, Cartesian coordinates are drawn with the -xyplane corresponding to the plane of the paper. The horizontal direction from left to right is taken as the positive -axis, and the vertical direction from bottom to top is taken as the positive -axis. In physics problems we are free to choose our axes and positive directions any way that we decide best fits a given problem. Problems that are very difficult using the conventional choices may turn out to be much easier to solve by making a thoughtful choice of axes. The endpoints of the wire now have the coordinates and xy (/2,0,0)a(/2,0,0)a−.

9. (4) Choice of Unit Vectors We now associate to each point in space, a set of three unit directions vectors. A unit vector has magnitude one: P(PPPˆˆˆ,,ijk1Pˆ=i, 1Pˆ=j, and 1Pˆ=k. We assign the direction of to point in the direction of the increasing -coordinate at the Pˆix ) point P. We define the directions for Pˆj and in the direction of the increasing -coordinate and -coordinate respectively.ˆkyz

10. Cartesian coordinate system x y z dl  

11. Infinitesimal Line Element Consider a small infinitesimal displacement dsbetween two points P 1 and P 2. In Cartesian coordinates this vector can be decomposed into

12. Infinitesimal Area Element An infinitesimal area element of the surface of a small cube is given by

13. Area elements are actually vectors where the direction of the vector is perpendicular to the plane defined by the area. Since there is a choice of direction, we shall choose the area vector to always point outwards from a closed surface, defined by the right-hand rule. So for the above, the infinitesimal area vector is

14. Cylindrical coordinate system We first choose an origin and an axis we call the -axis with unit vector pointing in the increasing z-direction. The level surface of points such that z= zp.We shall choose coordinates for a point in the plane as z= zp follows. The coordinate measures the distance from the -axis to the point. Its value ranges from zP0ρ≤<∞. In Figure B.2.1 we draw a few contours that have constant values of ρ. These “level contours” are circles. On the other hand, if z were not restricted to Pzz=, as in Figure B.2.1, the level surfaces for constant values ofρ would be cylinders coaxial with the z-axis.

15. Our second coordinate measures an angular distance along the circle. We need to choose some reference point to define the angular coordinate. We choose a “reference ray,” a horizontal ray starting from the origin and extending to +∞ along the horizontal direction to the right. (In a typical Cartesian coordinate system, our reference ray is the positive x-direction).

16. We define the angle coordinate for the point as follows. We draw a ray from the origin to. We define PPφ as the angle in the counterclockwise direction between our horizontal reference ray and the ray from the origin to the point All the other points that lie on a ray from the origin to infinity passing through have the same value of Pφ. For any arbitrary point, φ can take on values from02φπ≤<. In Figure B.2.3 we depict other “level surfaces” for the angular coordinate.

17. When referring to any arbitrary point in the plane, we write the unit vectors as and, keeping in mind that they may change in direction as we move around the plane, keeping unchanged. If we need to make a reference to this time changing property, we will write the unit vectors as explicit functions of time. If you are given polar coordinates of a point in the plane, the Cartesian coordinates xy can be determined from the coordinate transformations

19. Cylindrical coordinate system z x y

20. the differential areas and volume z x y

21. Spherical coordinate system Like cylindrical coordinates, spherical coordinates can be viewed as a 3D extension of polar coordinates. In this case, the third parameter is another angle,, measured from the `north pole', and r refers to the total distance of the point from the origin, not the distance in one plane. The earth's lines of latitude and longitude are a familiar system of spherical coordinates. Longitude is the, spanning 360 degrees or 2 radians, latitude is the, spanning 180 degrees or radians, and we don't usually bother about the r, since that is assumed to be radius of the earth. The other dierence is that spherical coordinate systems in mathematics usually use colatitude, measured from the north pole, rather than latitude, measured from the equator. This means that

22. spans 0{ rather than =2{=2. N.B.: There are dierent conventions on whether is longitude and colatitude or vice versa. I have used for longitude to make the relationship between polar and spherical coordinate clearer, but it's a good idea to check which convention is being used before you start a problem in spherical coordinates.

24. Spherical coordinate system

25. differential volume in Spherical coordinate system

26. Examples (1) Find the area of the strip (2) A sphere of radius 2cm contains a volume charge density Find the total charge contained in the sphere