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Electromagnetic Theory

Electromagnetic Theory. G. Franchetti , GSI CERN Accelerator – School Budapest, 2-14 / 10 / 2016. Mathematics of EM. Fields are 3 dimensional vectors dependent of their spatial position (and depending on time). Products. Scalar product. Vector product. The gradient operator.

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Electromagnetic Theory

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  1. Electromagnetic Theory G. Franchetti, GSI CERN Accelerator – School Budapest, 2-14 / 10 / 2016 G. Franchetti

  2. Mathematics of EM Fields are 3 dimensional vectors dependent of their spatial position (and depending on time) G. Franchetti

  3. Products Scalar product Vector product G. Franchetti

  4. The gradient operator Is an operator that transform space dependent scalar in vector Example: given G. Franchetti

  5. Divergence / Curl of a vector field Divergence of vector field Curl of vector field G. Franchetti

  6. Relations G. Franchetti

  7. Flux Concept Example with water v v v G. Franchetti

  8. v v Volume per second or v θ v L G. Franchetti

  9. Flux θ A G. Franchetti

  10. Flux through a surface G. Franchetti

  11. Flux through a closed surface: Gauss theorem Any volume can be decomposed in small cubes Flux through a closed surface G. Franchetti

  12. Stokes theorem Ex(0,D,0) y Ey(D,0,0) Ey(0,0,0) x Ex(0,0,0) G. Franchetti

  13. for an arbitrary surface G. Franchetti

  14. How it works G. Franchetti

  15. Electric Charges and Forces Two charges + + + + - - - - Experimental facts G. Franchetti

  16. Coulomb law G. Franchetti

  17. Units System SI Newton Coulomb C2 N-1 m-2 Meters permettivity of free space G. Franchetti

  18. Superposition principle G. Franchetti

  19. Electric Field G. Franchetti

  20. force electric field G. Franchetti

  21. By knowing the electric field the force on a charge is completely known G. Franchetti

  22. Work done along a path work done by the charge G. Franchetti

  23. Electric potential For conservative field V(P) does not depend on the path ! Central forces are conservative UNITS: Joule / Coulomb = Volt G. Franchetti

  24. Work done along a path B work done by the charge A G. Franchetti

  25. Electric Field  Electric Potential In vectorial notation G. Franchetti

  26. Electric potential by one charge Take one particle located at the origin, then G. Franchetti

  27. Electric Potential of an arbitrary distribution set of N particles origin G. Franchetti

  28. Electric potential of a continuous distribution Split the continuous distribution in a grid origin G. Franchetti

  29. Energy of a charge distribution it is the work necessary to bring the charge distribution from infinity More simply More simply G. Franchetti

  30. In integral form Using and the “divergence theorem” it can be proved that is the density of energy of the electric field G. Franchetti

  31. Flux of the electric field θ A G. Franchetti

  32. Flux of electric field through a surface G. Franchetti

  33. Application to Coulomb law On a sphere This result is general and applies to any closed surface (how?) G. Franchetti

  34. On an arbitrary closed curve q2 q1 G. Franchetti

  35. First Maxwell Law integral form for a infinitesimal small volume differential form (try to derive it. Hint: used Gauss theorem) G. Franchetti

  36. Physical meaning If there is a charge in one place, the electric flux is different than zero One charge create an electric flux. G. Franchetti

  37. Poisson and Laplace Equations and As combining both we find Poisson Laplace In vacuum: G. Franchetti

  38. Magnetic Field There exist not a magnetic charge! (Find a magnetic monopole and you get the Nobel Prize) G. Franchetti

  39. Ampere’s experiment I1 I2 G. Franchetti

  40. Ampere’s Law dl all experimental results fit with this law I2 I B F G. Franchetti

  41. Units [Tesla] From From follows To have 1T at 10 cm with one cable I = 5 x 105 Amperes !! G. Franchetti

  42. Biot-Savart Law dB I From analogy with the electric field G. Franchetti

  43. Lorentz force A charge not in motion does not experience a force ! B v F + No acceleration using magnetic field ! G. Franchetti

  44. G. Franchetti

  45. Flux of magnetic field There exist not a magnetic charge ! No matter what you do.. The magnetic flux is always zero! G. Franchetti

  46. Second Maxwell Law Integral form Differential from G. Franchetti

  47. Changing the magnetic Flux... Magnetic flux h Following the path L G. Franchetti

  48. Faraday’s Law integral form valid in any way the magnetic flux is changed !!! (Really not obvious !!) G. Franchetti

  49. for an arbitrary surface G. Franchetti

  50. Faraday’s Law in differential form G. Franchetti

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