1. EE2030: Electromagnetics (I)
Text Book:
- Sadiku, Elements of Electromagnetics, Oxford University
References:
- William Hayt, Engineering Electromagnetics, Tata McGraw Hill

2. Part 1:
Vector Analysis

3. Vector Addition
Associative Law:
Distributive Law:

4. Rectangular Coordinate System

5. Point Locations in Rectangular Coordinates

6. Differential Volume Element

7. Summary

8. Orthogonal Vector Components

9. Orthogonal Unit Vectors

10. Vector Representation in Terms of Orthogonal Rectangular Components

11. Summary

12. Vector Expressions in Rectangular Coordinates
General Vector, B:
Magnitude of B:
Unit Vector in the
Direction of B:

13. Example

14. Vector Field We are accustomed to thinking of a specific vector:
A vector field is a function defined in space that has magnitude
and direction at all points:
where r = (x,y,z)

15. The Dot Product
Commutative Law:

16. Vector Projections Using the Dot Product
B • a gives the component of B
in the horizontal direction
(B • a) a gives the vector component
of B in the horizontal direction

17. Projection of a vector on another vector

18. Operational Use of the Dot Product
Given
Find
where we have used:
Note also:

19. Cross Product

20. Operational Definition of the Cross Product in Rectangular Coordinates
Therefore:
Or…
Begin with:
where

21. Vector Product or Cross Product

22. Cylindrical Coordinate Systems

23. Cylindrical Coordinate Systems

24. Cylindrical Coordinate Systems

25. Cylindrical Coordinate Systems

26. Differential Volume in Cylindrical Coordinates
dV = dddz

27. Point Transformations in Cylindrical Coordinates

28. Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems

29. Example Transform the vector, into cylindrical coordinates:
Start with:
Then:

30. Example: cont.
Finally:

31. Spherical Coordinates

32. Spherical Coordinates

33. Spherical Coordinates

34. Spherical Coordinates

35. Spherical Coordinates

36. Spherical Coordinates
Point P has coordinates
Specified by P(r)

37. Differential Volume in Spherical Coordinates
dV = r2sindrdd

38. Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems

39. Example: Vector Component Transformation
Transform the field,
, into spherical coordinates and components

40. Constant coordinate surfaces- Cartesian system
If we keep one of the coordinate variables constant and allow the
other two to vary, constant coordinate surfaces are generated in rectangular, cylindrical and spherical coordinate systems.
We can have infinite planes:
X=constant,
Y=constant,
Z=constant
These surfaces are perpendicular to x, y and z axes respectively.

41. Constant coordinate surfaces- cylindrical system
Orthogonal surfaces in cylindrical coordinate system can be generated as
ρ=constnt
Φ=constant
z=constant
ρ=constant is a circular cylinder,
Φ=constant is a semi infinite plane with its edge along z axis
z=constant is an infinite plane as in the
rectangular system.

42. Constant coordinate surfaces- Spherical system
Orthogonal surfaces in spherical coordinate system can be generated as
r=constant
θ=constant
Φ=constant
r=constant is a sphere with its centre at the origin,
θ =constant is a circular cone with z axis as its axis and origin at the vertex,
Φ =constant is a semi infinite plane as in the cylindrical system.

43. Differential elements in rectangular coordinate systems

44. Differential elements in Cylindrical coordinate systems

45. Differential elements in Spherical coordinate systems

46. Line integrals
Line integral is defined as any integral that is to be evaluated along a line. A line indicates a path along a curve in space.

47. Surface integrals

48. Volume integrals

49. DEL Operator DEL Operator in cylindrical coordinates:
DEL Operator in spherical coordinates:

50. Gradient of a scalar field
The gradient of a scalar field V is a vector that represents the
magnitude and direction of the maximum space rate of increase of V.
For Cartesian Coordinates
For Cylindrical Coordinates
For Spherical Coordinates

51. Divergence of a vector In Cartesian Coordinates:
In Cylindrical Coordinates:
In Spherical Coordinates:

52. Gauss’s Divergence theorem

53. Curl of a vector

54. Curl of a vector In Cartesian Coordinates: In Cylindrical Coordinates:
In Spherical Coordinates:

56. Stoke’s theorem

57. Laplacian of a scalar

58. Laplacian of a scalar