2. Chapter 10 Magnetic Field of a Steady Current in Vacuum

3. § 10-2 Magnetic Field Gauss’law in Magnetic Field § 10-1 Magnetic Phenomena Ampere’s Hypothesis §10-3 Boit-Savart Law & Its Application § 10-4 Ampere’s Law & Its Application § 10-6 Magnetic Force on Current-carrying Conductors § 10-5 Motion of Charged Particles in Magnetic § 10-7 The Hall Effect § 10-7 Magnetic Torque on a Current Loop

4. 1. Magnetic Phenomena (1) the earliest magnetic phenomena that human knew: the permanent magnet (Fe 3 O 4 ) has N, S poles. Same poles repel each other and different poles attract each other. § 10-1 Magnetic Phenomena Ampere’s Hypothesis § 10-1 Magnetic Phenomena Ampere’s Hypothesis

5. N pole S pole Magnetic monopole ？ N poleS pole Never be seen!

6. (2) The magnetic field surrounding the earth

7. (3)The interaction between current and magnet

8. attraction repellen t

9. The motion of electron in M-field

10. 2. Ampere’s Hypothesis Each molecule of the matter can be equated with a closed current –called molecular current. When the molecular currents arrange in same direction, the matter appears magnetism in a macroscopic size. All Magnetic phenomena result from the motion of the charge.

11. 3. Magnetic field M- field M- field Moving charge M- field M- field magne t curren t

12. 1. Magnetic field  each point in the M-field has a special direction. when the q moves along this direction( or opposite the direction), no force acts on it. The direction of M-field at this point § 10-2 Magnetic Field Gauss’law in Magnetic field § 10-2 Magnetic Field Gauss’law in Magnetic field Take a moving charge( and q) as a test charge. The characters of the force on the moving charge by the magnetic field:

13. Definition --the magnitude of M-field tesla(T) or  M-force depends on q,v and angle  between and M-field direction. along the direction of  The direction of the M-force acting on q always perpendicular to and M-field direction.

14. Superposition principle of M-field

15.  the magnitude of 2. Magnetic field line ( line)  tangential direction of line--M-field direction. -line is different with line ： -lines are always closed lines linked with electric current. They have neither origin nor termination. -lines are always closed lines linked with electric current. They have neither origin nor termination.

17. 3. Magnetic flux and Gauss’ Law in magnetics unit ： weber(Wb)T·m 2 For any closed surface S, Gauss’ Law in magnetics ---- Gauss’ Law in magnetics ----M-field is non-source field M-flux ： The number of -lines through a given surface.

18. § 10-3 Calculation of the magnetic field set up by a current 1. Boit-Savart Law The magnetic field set up by at point P is --current element --permeability of vacuum --B-S Law

19. For any long current, -- superposition principle of M-field 2. Application of B-S Law

20. [Example 1]Calculate the M-field of a straight wire segment carrying a current I. solution direction ：  all set up by all have same direction Choose any set up at P :

22. discussion  for infinite long current  semi-infinite current  on the prolong line of current

23. [Example 2] Calculate the M-field on the axis of a circle with radius R and carrying current I. Solution ： choose any choose any

24. ---- and I satisfy Right Hand Rule ---- and I satisfy Right Hand Rule

25. discussion  at the center ， x =0  the magnetic moment of the circular current

26. [Example 3] Calculate the M-field on the axis of a solenoid with radius R. The number of turns per length of solenoid is n, its carrying current I. Solution The number of turns on length dl, the current on dl, Take dl along axis and its distance to P is l.

27. direction ： 

28. Discussion Solenoid “infinite long” ： Solenoid “infinite long” ： -- the M-field on the axis of a solenoid with infinite length. ’s direction: satisfy right-hand rule with I.

29. [Example 4]A long straight plate of width L carrying I uniformly. P and plate current are at same plane. Find B=? of P. Solution I  dI Straight line current

30. Direction :  All have same direction.

31. [Example 5]A half ring with radius R, uniform charge Q and angular speed . Find at O. Solution It charge, Take an any dl,

32. When dQ is rotating, it equates with dI set up M-field at O,

33. All dB have same direction Direction : 

34. 3. M-field set up by moving charge Take ， it set up Take ， it set up The number of moving charges in dl,, same direction

35. set up by each moving charge ( q, ): set up by each moving charge ( q, ):

36. 1. Ampere’s Law Special example, infinite straight line current I Question ： §10-4 Ampere’s Law

37.  Choose a circle L is just along B-line. -- does not depend on r  choose is any closed line L surrounding I and in the plane perpendicular to I

38.  Any L surrounding I

39.  L does not surround I

40. --Ampere’s Law Conclusion Notes ： is the algebraic sum of all currents closed by L.  is the algebraic sum of all currents closed by L. I >0 when it satisfy right-hand rule with L. otherwise, I 0 when it satisfy right-hand rule with L. otherwise, I <0. I has not contribution to if it is outside L  I has not contribution to if it is outside L  is non-conservative field. Set up by all I (inside or outside L) Amperian loop

41. [Example1]A long straight wire with R,uniform I. Find B=? inside and outside it.. Solution 2. Application of Ampere’s Law Analyze the symmetry of B --Axis symmetry Take L to be a circle with r, same direction with B, r>R： r>R： r>R： r>R：

42. r<R:the current closed by L, L

43. [Example 2] Find the M-field of a infinite solenoid. The number of turns per length of it is n, its carrying current I. --uniform field exterior ： Choose a rectangular loop abcda Solution ： analyze the distribution of

44. [Example 3] A  straight cylinder conductor with R. A hole with radius a is far b from the central axis of cylinder. The conductor has current I ， Find B=? at point P. Solution P Current density ： l Compensatory method ： Imagine there are and in the hole. Imagine there are and in the hole. Assume a current I in conductor 

45. P The set by the conductor with a hole = the set up by one without hole + the set up by the hole’s negative current Direction :see Fig.

46. P For hole’s -j : Direction: see Fig. Direction: 

47. [Example 4] A  conductor flat carries current The current density is j per unit length along the direction of perpendicular to j. Find the distribution of B outside the flat.

48. Solution At the two side of the flat, M-field has same magnitude and opposite direction. Take a rectangle path abcda

49. 1. Lorentz force --Magnetic force acting on the moving charge. §10-5 §10-5 Motion of Charged Particles in M-field

50. Notes   there are E-field + M-field in the space ， a moving charge q sustains: a moving charge q sustains: does not do work to q. does not do work to q. --Change ’s direction, don’t change ’s magnitude. don’t change ’s magnitude.

51. l Let q goes into M-field with initial velocity 2. Moving charge in uniform M-field ：  ： --straight line motion with uniform velocity.

52. :  : period --Circle motion with uniform speed in the plane of

53. Application: mass spectrometer （质谱仪）  A charged particle from S is speeded up by  U  Enter M-field UUUU S2S2S2S2 2R B  (1)  (2) Combine (1) and (2)

54. Application: cyclotron （回旋加速器） E: speed up q B: change the velocity direction of q do not depend on R

55.  and with any angle ---- // uniform, straight line ----  uniform, circle Revolving radius helical distance Moving path --- helix Moving path --- helix

56. Application: magnetic focusing （磁聚焦） The particles have same v // B A A ·· h same h They focus on point again They focus on point A again

57. 3. Moving charge in non-uniform M-field R, h are different when B is not constant. R, h are different when B is not constant. As Magnetic restraint ---Magnetic bottle plasma

58. M-field of the earth Van Allen radiation belts Van Allen radiation belts beautiful aurora beautiful aurora

59. is in M-field is in M-field § 10-6 Magnetic force on a current-carrying conductor 1. Ampere’s Law The force acting on each electron is The numbers of electron in is

60. The resultant force acting on the dN electrons is --Ampere’s Law of M-force l for any shape of current-carrying wire,

61. Take force  The M-force acting on L is I L direction ： 2. The application of Ampere’s Law [Example 1] A straight wire with length L carrying I is in a uniform. Find =?

62. Set up a coordinate system, take any take any [example 2] A curved wire segment with I is in the plane which . Suppose AB=L is known.Find =?

63. Similar, Vector express: Same as the straight wire from A to B.

64. Conclusion  in a uniform, the M-force acting on any shape wire = the M-force acting on the equivalent straight wire.  for a closed wire, F=0 in uniform

65. [Example 3] I 1  I 2. AB=L. Find =? acting on AB. I1I1I1I1 I2I2I2I2 d L A B L  dl the force acting on dl isx

66. 1 set up at 2, 1 set up at 2, 3. The interaction between two parallel currents The M-force acting on,

67. Magnitude direction ： 2  1 Similarly, The force for per unit length wire,, have opposite direction., have opposite direction.

68. 1. The Hall effect l Experiment result,  H ： Hall coefficient. §10-7 The Hall effect --there is an electric potential difference on the direction of  when a current-carrying plate is put in M-field. Depends on the material.

69. 2. Theoretical explanation l Let the velocity of free electrons is, number density is n In equilibrium state,

70. Hall E-P-difference, And

71. For moving positive charges, For moving positive charges,

72. Notes ：  n has large magnitude in conductors (~10 29 /m 3 ). The Hall effect is not obvious. The Hall effect is obvious in semiconductor The Hall effect is obvious in semiconductor F n type semiconductor ： electron conduction. F p type semiconductor ： hole conduction.  to measure  H (or V H ) can judge the moving charges and find n. Positive charge

73. The normal direction of loop ： § 10-8 Magnetic torque on a current loop Same magnitude, opposite direction, locate on a line. 1. The torque acting on a loop by M-field  Satisfy right hand rule with I

74. Do not locate a line.  Set up a torque direction ： 

75. Vector express: Define --M-moment of a current loop --can be used for any shape plane loop in uniform M-field For the loop with N turns,

76. M=0  =0 ： M=0 Discussion  =  : M =0 --stable equilibrium position. -- unstable equilibrium position When suffers disruption, it turns  =0   The resultant force acting on loop=0 in uniform M-field. But the torque  0 --only rotation, not translation  In non-uniform M-field, M  0, F  0. --rotation and translation

77. 2. Potential energy of current loop A current loop has I It suffers ： M makes  decreasing Increase  1 to  2 ， external force does work: ’s direction:

78. A loop with M-moment is put in, the A loop with M-moment is put in, the potential energy of the system ( loop + M-field) is = The increment of potential energy of the loop in