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EE2030: Electromagnetics (I)

EE2030: Electromagnetics (I). Text Book: - Sadiku, Elements of Electromagnetics, Oxford University. References: - William Hayt, Engineering Electromagnetics, Tata McGraw Hill. Part 1: Vector Analysis. Associative Law:. Distributive Law:. Vector Addition.

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EE2030: Electromagnetics (I)

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  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. Associative Law: Distributive Law: Vector Addition

  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. 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) Vector Field

  15. Commutative Law: The Dot Product

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

  17. Projection of a vector on another vector

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

  19. Cross Product

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

  21. Vector Product or Cross Product

  22. Cylindrical Coordinate Systems

  23. Cylindrical Coordinate Systems

  24. Cylindrical Coordinate Systems

  25. Cylindrical Coordinate Systems

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

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

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

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

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

  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 rectangularcoordinate systems

  44. Differential elements in Cylindricalcoordinate systems

  45. Differential elements in Sphericalcoordinate 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

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