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Magnetostatics

Chapter 5. Magnetostatics. Department of Physics , ROCMA. The Lorentz Force Law. Magnetic Fields. The Lorentz Force Law. Magnetic Fields. The Lorentz Force Law. Magnetic Forces. => Lorentz Force. The Lorentz Force Law. Exp 1: cyclotron motion. The Lorentz Force Law.

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Magnetostatics

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  1. Chapter 5 Magnetostatics Department of Physics , ROCMA

  2. The Lorentz Force Law Magnetic Fields

  3. The Lorentz Force Law Magnetic Fields

  4. The Lorentz Force Law Magnetic Forces => Lorentz Force

  5. The Lorentz Force Law Exp 1: cyclotron motion

  6. The Lorentz Force Law Exp 2: A more exotic trajectory occurs if we include a uniform electric field, at right angles to the magnetic one. Suppose, for instance, that B points in the x-direction, and E in the z-direction. A particle at rest is released from the origin; what path will it follow?

  7. The Lorentz Force Law Exp 2: no force on x-direction

  8. The Lorentz Force Law Exp 2: Substitute to (1)

  9. The Lorentz Force Law Exp 2: The particle start from rest at origin

  10. The Lorentz Force Law Exp 2:

  11. The Lorentz Force Law Current :the charge per unit time passing a given point

  12. The Lorentz Force Law Current

  13. The Lorentz Force Law Exp 3: A rectangular loop of wire, supporting a mass m, hangs vertically with one end in a uniform magnetic field B, which points into the page in the shaded region of figure. For what current I, in the loop, would the magnetic force upward exactly balance the gravitational force downward?

  14. The Lorentz Force Law Exp 3: m

  15. The Lorentz Force Law Surface current density

  16. The Lorentz Force Law Volume current density

  17. The Lorentz Force Law Summarize

  18. The Lorentz Force Law Exp 4(a): A current I is uniformly distributed over a wire of circular cross section, with radius a. Find the volume current density J .

  19. The Lorentz Force Law Exp 4(b): suppose the current density in the wire is proportional to the distance from the axis.J=ks (k=constant) . Find the total current I in the wire.

  20. The Lorentz Force Law Equation of continuity e- e- e- e- e- e- e- e- total charge per unit time leaving a volume V

  21. The Biot-Savart Law The magnetic field of a steady current Biot-Savart law P permeability of free space units

  22. The Biot-Savart Law Exp 5: Find the magnetic field a distance s from a long straight wire carrying a steady current I . y x

  23. The Biot-Savart Law Exp 5: For an infinite wire

  24. The Biot-Savart Law

  25. The Biot-Savart Law Exp 6: Find the magnetic field a distance z above the center of a circular loop of radius a, which carries a steady current I .

  26. The Biot-Savart Law

  27. The Divergence and Curl of B

  28. The Divergence and Curl of B Ampere’s law in integral form from stoke’s theorem Ampere’s law in differential form

  29. The Divergence and Curl of B from Biot-Savart law

  30. The Divergence and Curl of B since The divergence of the magnetic field is zero

  31. The Divergence and Curl of B 0 0

  32. The Divergence and Curl of B since

  33. The Divergence and Curl of B 0 : for steady current 0 : for surface s → ∞

  34. The Divergence and Curl of B Ampere’s law –in differential form

  35. Application of Ampere’s law Ampere’s law (in differential form) Ampere’s law (in integral form)

  36. Comparison of Magnetostatics and Electrostatics

  37. Application of Ampere’s law Exp 7: Find the magnetic field a distance s from a long straight wire, carrying a steady current I.

  38. Application of Ampere’s law Exp 8: Find the magnetic field of an infinite uniform surface current , flowing over the xy plane.

  39. Application of Ampere’s law Exp 9: Find the magnetic field of a very long solenoid, consisting of n closely wound turns per unit length on a cylinder of radius R and carrying a steady current I .

  40. Application of Ampere’s law

  41. Application of Ampere’s law Exp 9:

  42. Application of Ampere’s law Exp 9:

  43. Application of Ampere’s law Exp 10: A toroidal coil consists of a circular ring, or ‘donut’, around which a long wire is wrapped. The winding is uniform and tight enough so that each turn can be considered a closed loop. The cross-sectional shape of the coil is immaterial. I made it rectangular in figure a for the sake of simplicity, but it could just as well be circular or even some weird asymmetrical form, as in figure b, just as long as the shape remains the same all the way around the ring. In that case it follows that the magnetic field of the toroid is circumferential at all points, both inside and outside the coil. (a) (b)

  44. Application of Ampere’s law Exp 10: From Biot-Savart law

  45. Application of Ampere’s law Exp 10:

  46. Application of Ampere’s law Exp 10:

  47. Application of Ampere’s law Exp 10: Ampere’s loop

  48. Application of Ampere’s law Exp 10:

  49. Magnetic Vector Potential

  50. Magnetic Vector Potential

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