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Understanding Multipole Magnets from Maxwell’s Equations

Explore multipole fields in magnets from Maxwell’s Equations, covering solutions, physical interpretations, generating techniques, iron-core magnets, and more.

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Understanding Multipole Magnets from Maxwell’s Equations

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  1. Multipole Magnets from Maxwell’s Equations Alex Bogacz USPAS, Hampton, VA, Jan. 17-28, 2011

  2. Maxwell’s Equations for Magnets - Outline • Solutions to Maxwell’s equations for magneto static fields: • in two dimensions (multipole fields) • in three dimensions (fringe fields, insertion devices...) • How to construct multipole fields in two dimensions, using electric currents and magnetic materials, considering idealized situations. • A. Wolski, Academic Lectures, University of Liverpool , 2008 USPAS, Hampton, VA, Jan. 17-28, 2011

  3. Basis • Vector calculus in Cartesian and polar coordinate systems; • Stokes’ and Gauss’ theorems • Maxwell’s equations and their physical significance • Types of magnets commonly used in accelerators. • following notation used in: A. Chao and M. Tigner, “Handbook of Accelerator Physics and Engineering,” World Scientific (1999). USPAS, Hampton, VA, Jan. 17-28, 2011

  4. Maxwell’s equations USPAS, Hampton, VA, Jan. 17-28, 2011

  5. Maxwell’s equations USPAS, Hampton, VA, Jan. 17-28, 2011

  6. Physical interpretation of USPAS, Hampton, VA, Jan. 17-28, 2011

  7. Physical interpretation of USPAS, Hampton, VA, Jan. 17-28, 2011

  8. Linearity and superposition USPAS, Hampton, VA, Jan. 17-28, 2011

  9. Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

  10. Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

  11. Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

  12. Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

  13. Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

  14. Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

  15. Generating multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  16. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  17. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  18. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  19. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  20. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  21. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  22. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  23. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  24. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  25. Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

  26. Superconducting quadrupole - collider final focus USPAS, Hampton, VA, Jan. 17-28, 2011

  27. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  28. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  29. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  30. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  31. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  32. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  33. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  34. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  35. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  36. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  37. Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  38. Generating multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  39. Generating multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  40. Generating multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  41. Generating multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

  42. Maxwell’s Equations for Magnets - Summary USPAS, Hampton, VA, Jan. 17-28, 2011

  43. Maxwell’s Equations for Magnets - Summary USPAS, Hampton, VA, Jan. 17-28, 2011 Maxwell’s equations impose strong constraints on magnetic fields that may exist. The linearity of Maxwell’s equations means that complicated fields may be expressed as a superposition of simpler fields. In two dimensions, it is convenient to represent fields as a superposition of multipole fields. Multipole fields may be generated by sinusoidal current distributions on a cylinder bounding the region of interest. In regions without electric currents, the magnetic field may be derived as the gradient of a scalar potential. The scalar potential is constant on the surface of a material with infinite permeability. This property is useful for defining the shapes of iron poles in multipole magnets.

  44. Multipoles in Magnets - Outline Deduce that the symmetry of a magnet imposes constraints on the possible multipole field components, even if we relax the constraints on the material properties and other geometrical properties; Consider different techniques for deriving the multipole field components from measurements of the fields within a magnet; Discuss the solutions to Maxwell’s equations that may be used for describing fields in three dimensions. USPAS, Hampton, VA, Jan. 17-28, 2011

  45. Previous lecture re-cap USPAS, Hampton, VA, Jan. 17-28, 2011

  46. Previous lecture re-cap USPAS, Hampton, VA, Jan. 17-28, 2011

  47. Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

  48. Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

  49. Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

  50. Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

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