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Chapter 10

Chapter 10. Magnetic Field of a Steady Current in Vacuum. § 10-1 Magnetic Phenomena Ampere’s Hypothesis . § 10-2 Magnetic Field Gauss’law in Magnetic Field. §10-3 Boit-Savart Law & Its Application. § 10-4 Ampere’s Law & Its Application. § 10-5 Motion of Charged Particles in Magnetic.

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Chapter 10

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

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

  3. § 10-1 Magnetic Phenomena Ampere’s Hypothesis 1. Magnetic Phenomena (1) the earliest magnetic phenomena that humanknew:the permanent magnet (Fe3O4) has N , S poles.Same poles repel each other and different poles attract each other.

  4. N pole S pole N pole S pole Magnetic monopole？ Never be seen!

  5. (2) The magnetic field surrounding the earth

  6. (3)The interaction between current and magnet

  7. attraction repellent

  8. The motion of electron in M-field

  9. 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.

  10. magnet magnet M- field current current Moving charge Moving charge M-field 3. Magnetic field

  11. Take a moving charge( and q) as a test charge. The direction of M-field at this point § 10-2 Magnetic Field Gauss’law in Magnetic field 1. Magnetic field The characters of the force on the moving charge by the 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.

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

  13. Superposition principle of M-field

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

  15. M-flux：The number of -lines through a given surface. 3. Magnetic flux and Gauss’ Law in magnetics unit：weber(Wb)T·m2 For any closed surface S , ---- Gauss’ Law in magnetics ----M-field is non-source field

  16. §10-3 Calculation of the magnetic field set up by a current --current element The magnetic field set up by at point P is 1. Boit-Savart Law --B-S Law --permeability of vacuum

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

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

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

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

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

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

  23. Solution [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. Take dl along axis and its distance to P is l. The number of turns on length dl , the current on dl,

  24. direction：

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

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

  27. All have same direction. Direction :

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

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

  30. All dB have same direction Direction : 

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

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

  33. §10-4 Ampere’s Law Question： 1. Ampere’s Law Special example, infinite straight line current I

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

  35. Any L surrounding I

  36. L does not surround I

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

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

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

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

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

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

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

  44. [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.

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

  46. §10-5Motion of Charged Particles in M-field 1. Lorentz force --Magnetic force acting on the moving charge.

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

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

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