1. Notes 6 ECE 3318 Applied Electricity and Magnetism Coordinate Systems
Spring 2019
Prof. David R. Jackson
Dept. of ECE
Notes 6
Coordinate Systems
Notes prepared by the EM Group University of Houston

2. Review of Coordinate Systems
An understanding of coordinate systems is
important for doing EM calculations.

3. Kinds of Integrals That Often Occur
We wish to be able to perform all of these calculations in various coordinates.

4. Rectangular Coordinates
Position vector:
Short hand notation:
Note:
We have the “tip to tail” rule when adding vectors.
Note:
A unit vector direction is defined by increasing one coordinate variable while keeping the other two fixed.
Note: Different notations are used for vectors in the books.

5. Rectangular Coordinates
Differentials
We increase x, y, or z starting from an initial point (blue dot).
Note:
dS may be in three different forms.

6. Rectangular (cont.) Path Integral (we need dr)
Note on notation: The symbol dl is often used instead of dr .

7. Cylindrical Coordinates.

8. . Cylindrical (cont.) Unit Vectors Note: and depend on (x, y)
A unit vector direction is defined by increasing one coordinate variable while keeping the other two fixed.
Note: and depend on (x, y)
This is why we often prefer to express them in terms of

9. Cylindrical (cont.) Expressions for unit vectors (illustrated for )
Assume
Similarly,
Solve for A1: so
Hence, we have

10. Cylindrical (cont.)
Summary of Results

11. cylindrical coordinates.
Cylindrical (cont.) .
Example:
Express the r vector in
cylindrical coordinates.
Substituting from the previous tables of unit vector transformations and coordinate transformations, we have

12. Cylindrical (cont.) .
Note:

13. Cylindrical (cont.) Differentials
Note:
dS may be in three different forms.
We increase ,  , or z starting from an initial point (blue dot).
Note: The angle  must be in radians here.

14. Note: The angle  must be in radians here.
Cylindrical (cont.)
Path Integrals
First, consider differential changes along any of the three coordinate directions.
Note: The angle  must be in radians here.

15. Note: A change in z is not shown, but is possible.
Cylindrical (cont.)
Note: A change in z is not shown, but is possible.
In general:
If we ever need to find the
length along a contour:

16. Spherical Coordinates. .
Note: 0    
Note:  = r sin 

17. Spherical (cont.) .
Note:  = r sin 

18. Spherical (cont.) Unit Vectors Note:
A unit vector direction is defined by increasing one coordinate variable while keeping the other two fixed.

19. Spherical (cont.)
Transformation of Unit Vectors

20. spherical coordinates.
Spherical (cont.)
Example:
Express the r vector in
spherical coordinates.
Substituting from the previous tables of unit vector transformations and coordinate transformations, we have:

21. Spherical (cont.)
After simplifying:

22. Spherical (cont.) Differentials
We increase r, , or  starting from an initial point (blue dot).
Note:
dS may be in three different forms (only one is shown). The other two are:
Note: The angles  and  must be in radians here.

23. Note: The angles  and  must be in radians here.
Spherical (cont.)
Path Integrals
Note: The angles  and  must be in radians here.

24. Note on dr Vector
Note that the formula for the dr vector never changes, no matter which direction we go along a path (we never add a minus sign!).
Example: Integrating along a horizontal radial path in cylindrical coordinates.
The limits take care of the sign of dr.
Note:
This form does not change, regardless of which limit is larger.

25. Example
Given:
Find the current I crossing a hemisphere (z > 0) of radius a, in the outward direction.
Hemisphere

26. Example (cont.)

27. Example (cont.)

28. Appendix
Here we work out some more examples.

29. This formula only works in rectangular coordinates!
Example
Cylindrical coordinates (, , z)
Given:
with distances in meters
Find d = distance between points
This formula only works in rectangular coordinates!
d = [m]

30. Example Find Q Solution: “A sphere with a hole in it” Note:
The integrand is separable and the limits are fixed.
Find Q
Solution:
“A sphere with a hole in it”

31. Example (cont.)
Note: The average value of cos2 is 1/2.

32. Example
Derive
Let
Dot multiply both sides with
Then

33. Example (cont.)
Result:

34. (We use a two-step process.)
Example
Derive
Let
Dot multiply both sides with
An illustration of finding the x component of
(We use a two-step process.)

35. Example (cont.)
Hence
Similarly,
Also,
Result:

36. Find VAB using path C shown below.
Example (Part 1)
(This is not an electrostatic field.)
Find VAB using path C shown below. .

37. Example (cont.)
Completing the calculus:

38. Example (cont.)
Alternative calculation (we parameterize differently):

39. Find VAB using path C shown below.
Example (Part 2)
(same field as in Part 1)
Find VAB using path C shown below.

40. Find VAB using an arbitrary path C in the xy plane.
Example
(This is a valid electrostatic field.)
Find VAB using an arbitrary path C in the xy plane.
Note:
The path does not have to be parameterized. Hence, only the endpoints are important.
The integral is path independent!

41. Example Note: If we have an electric field of the form:
(discussed later)
VAB is path independent.

42. Find VAB using path C shown below.
Example
Find VAB using path C shown below.

43. Example (cont.)
Note:
The angle  must change continuously along the path. If we take the angle  to be  / 2 at point B, then the angle  must be - at point A.

44. Example (cont.) Let’s examine this same electric field once again:
Question: Is this integral path independent?
Note:
The answer is yes because the curl of the electric field is zero, but we will talk about curl later.

45. Yes, it is path independent!
Example (cont.)
Question: Is this integral path independent?
Let’s find out from the calculus:
Yes, it is path independent!
Note: Even though the path was given as a circle, the calculus was easier in rectangular coordinates in this example!