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S J P N Trust's Hirasugar Institute of Technology, Nidasoshi.

Explore vector analysis in electromagnetics, covering scalar and vector fields, coordinate systems, dot and cross products, Gauss's law, symmetrical charge distributions, divergence, energy in fields, and more. Ideal for Electronics & Communication Engineering students.

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S J P N Trust's Hirasugar Institute of Technology, Nidasoshi.

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  1. S J P N Trust's Hirasugar Institute of Technology, Nidasoshi. Inculcating Values, Promoting Prosperity Approved by AICTE, Recognized by Govt. of Karnataka and Affiliated to VTU Belagavi Department of Electronics & Communication Engg. Course : Engineering Electromagnetics -15EC36. Sem.: 3rd(2017-18) Course Coordinator: Prof. S. S. KAMATE

  2. Vector Analysis • Scalars and Vectors • Scalar Fields (temperature) • Vector Fields (gravitational, magnetic) • Vector Algebra

  3. There are three co-ordinate systems • Cartesian or rectangular co-ordinate system • Cylindrical co-ordinate System • Spherical co-ordinate System

  4. The Cartesian Coordinate System : vertices are x,y,z

  5. Vector Components and Unit Vectors

  6. Constant planes

  7. Contd… Distance vector - dl Differential surfaces - dv =

  8. The Dot product B in the direction of A You need to normalize a before the dot product. A.B = |A||B| cos qAB

  9. The Cross Product A x B = aN|A||B| sin qAB Example

  10. Circular Cylindrical Coordinate System

  11. Differential volume

  12. Circular Cylindrical Coordinate System Dot Product

  13. Spherical co-ordinate system

  14. The Spherical Coordinate System

  15. Sperical Co-ordinate System

  16. The Spherical Coordinate System

  17. The Experimental Law of Coulomb

  18. Electric Field Intensity

  19. Electric Field Intensity

  20. Field of a Line Charge

  21. Field of a Line Charge (neglect symmetry)

  22. Field of a Line Charge (neglect symmetry)

  23. Field of a Sheet of Charge This is a very interesting result. The field is constant in magnitude and direction. It is as strong a million miles away from the sheet as it is right of the surface.

  24. Streamlines and Sketches of Fields Cross-sectional view of the line charge. Lengths proportional to the magnitudes of E and pointing in the direction of E

  25. Streamlines and Sketches of Fields

  26. 3.1 Electric Flux Density • Faraday’s Experiment

  27. Electric Flux Density, D • Units: C/m2 • Magnitude: Number of flux lines (coulombs) crossing a surface normal to the lines divided by the surface area. • Direction: Direction of flux lines (same direction as E). • For a point charge: • For a general charge distribution,

  28. D3.1 Given a 60-uC point charge located at the origin, find the total electric flux passing through: • That portion of the sphere r = 26 cm bounded by • 0 < theta < Pi/2 and 0 < phi < Pi/2

  29. Gauss’s Law • “The electric flux passing through any closed surface is equal to the total charge enclosed by that surface.”

  30. The integration is performed over a closed surface, i.e. gaussian surface.

  31. We can check Gauss’s law with a point charge example.

  32. Symmetrical Charge Distributions • Gauss’s law is useful under two conditions. • DS is everywhere either normal or tangential to the closed surface, so that DS.dS becomes either DSdS or zero, respectively. • On that portion of the closed surface for which DS.dS is not zero, DS = constant.

  33. Gauss’s law simplifies the task of finding D near an infinite line charge.

  34. Infinite coaxial cable:

  35. Differential Volume Element • If we take a small enough closed surface, then D is almost constant over the surface.

  36. Divergence Divergence is the outflow of flux from a small closed surface area (per unit volume) as volume shrinks to zero.

  37. Water leaving a bathtub • Closed surface (water itself) is essentially incompressible • Net outflow is zero • Air leaving a punctured tire • Divergence is positive, as closed surface (tire) exhibits net outflow

  38. Mathematical definition of divergence Surface integral as the volume element (v) approaches zero D is the vector flux density - Cartesian

  39. Divergence in Other Coordinate Systems Cylindrical Spherical

  40. 4.1 Energy to move a point charge through a Field • Force on Q due to an electric field • Differential work done by an external source moving Q • Work required to move a charge a finite distance

  41. Line Integral • Work expression without using vectorsEL is the component of E in the dL direction • Uniform electric field density

  42. Potential • Measure potential difference between a point and something which has zero potential “ground”

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