1. Biot-Savart Law for a Single Charge Electric field of a point charge: Moving charge makes a curly magnetic field: B units: T (tesla) = kg s -2 A -1

2. Observing magnetic field around copper wire: Can we tell whether the current consists of electrons or positive ‘holes’? In some materials current moving charges are positive: Ionic solution “Holes” in some materials (same charge as electron but +) Conventional Current The prediction of the Biot-Savart law is exactly the same in either case.

3. Metals: current consists of electrons Semiconductors: n-type – electrons p-type – positive holes Most effects are insensitive to the sign of mobile charges: introduce conventional current: Conventional Current Units: C/s  A (Ampere)

4. Typical electron current in a circuit is ~ 10 18 electrons/s. What is the drift speed of an electron in a 1 mm thick copper wire? Typical Mobile Electron Drift Speed

5. Superposition principle is valid The Biot-Savart law for a short length of thin wire The Biot-Savart Law for Currents

6. Moving charge produces a curly magnetic field B units: T (Tesla) = kg s -2 A -1 Single Charge: Biot-Savart Law The Biot-Savart law for a short length of thin wire Current:

7. Four-step approach: 1.Cut up the current distribution into pieces and draw  B 2.Write an expression for  B due to one piece 3.Add up the contributions of all the pieces 4.Check the result Magnetic Field of Current Distributions

8. Step 1: Cut up the current distribution into pieces and draw  B. Origin: center of wire Vector r: Magnitude of r: A Long Straight Wire

9. Step 2: Write an expression for  B due to one piece. Unit vector: :  B field due to one piece: A Long Straight Wire

10. need to calculate only z component A Long Straight Wire

11. Step 3: Add up the contribution of all the pieces. A Long Straight Wire

12. Special case: x<<L A Long Straight Wire What is the meaning of “x”?

13. Step 4: Check results direction  far away: r>>L  units:  A Long Straight Wire

14. For Infinite Wire Semi-infinite Straight Wire 0 For Semi-Infinite Wire Half the integral …

15. Right-hand Rule for Wire Conventional Current Direction

16. Question Current carrying wires below lie in X-Y plane.

17. Question

18. Step 1: Cut up the distribution into pieces Make use of symmetry! Need to consider only  B z due to one dl Magnetic Field of a Wire Loop

19. Step 2:  B due to one piece Origin: center of loop Vector r: Magnitude of r: Unit vector:  l: Magnetic field due to one piece: Magnetic Field of a Wire Loop

20. Step 2:  B due to one piece need only z component: Magnetic Field of a Wire Loop

21. Step 3: Sum the contributions of all pieces Magnetic field of a loop along its axis: Magnetic Field of a Wire Loop

22. Step 4: Check the results units:  direction:  Magnetic Field of a Wire Loop Check several pieces with the right hand rule Note: We’ve not calculated or shown the “rest” of the magnetic field

23. Magnetic field of a loop: Magnetic Field of a Wire Loop

24. Using general form (z=0) : Special case: center of the loop Magnetic Field of a Wire Loop

25. for z>>R: Magnetic Field of a Wire Loop Special case: far from the loop The magnetic field of a circular loop falls off like 1/z 3

26. For whole loop Magnetic Field of a Semicircle

27. What if we had a coil of wire? For N turns: single loop: A Coil of Wire

28. Vectors and Pseudovectors (Axial) Compare the electric field at the center of a uniformly charged ring to the magnetic field in a current- carrying ring. What do you notice about the field strength at the center of the ring? What breaks the symmetry in the current ring case?

29. Reflection of Current Ring yy xx z z

30. far from coil:far from dipole: magnetic dipole moment:  - vector in the direction of B Magnetic Dipole Moment

31. What is the magnetic dipole moment  of a 3000-turn 3  5 cm rectangular coil that carries a current of 2 A? for one turn for N turns Exercise

32. The magnetic dipole moment  acts like a compass needle! In the presence of external magnetic field a current-carrying loop rotates to align the magnetic dipole moment  along the field B. Twisting of a Magnetic Dipole

33. How does the magnetic field around a bar magnet look like? The Magnetic Field of a Bar Magnet NS

34. How do magnets interact with each other? Magnets interact with iron or steel, nickel, cobalt. Does it interact with charged tape? Does it work through matter? Does superposition principle hold? Similarities with E-field: can repel or attract superposition works through matter Differences with E-field: B-field only interacts with some objects curly pattern only closed field lines Magnets and Matter

35. Horizontal component of magnetic field depends on latitude Maine:~1.5. 10 -5 T Texas: ~2.5x10 -5 T Can use magnetic field of Earth as a reference to determine unknown field. Magnetic Field of Earth The magnetic field of the earth has a pattern that looks like that of a bar magnet

36. Current is flowing to the right in a wire. The magnetic field at the position P points 1.Choice One 2.Choice Two 3.Choice Three 4.Choice Four 5.Choice Five 6.Choice Six A.B. C.D.

37. What is the direction of the magnetic field inside the solenoid? 1.Choice One 2.Choice Two 3.Choice Three 4.Choice Four 5.Choice Five 6.Choice Six A.B. C. D. Current upward on side nearest you

38. A current in the loop has created the magnetic field, B, shown below. What is the current direction in this loop if you look from the top? And which side of the loop is the north pole? (To get the pole, you need to replace the loop with a bar magnet that has the same field direction) A.Current clockwise; north pole on top B.Current counterclockwise, north pole on top C.Current clockwise; north pole on bottom D.Current counterclockwise, north pole on bottom B

39. Magnetic Field of Earth Before 1600 the connection between the deflection of a compass needle and the magnetic field of the earth was unknown. René Descartes (1596-1650) William Gilbert (1544-1603)

40. Exercise Magnetic dipole moment of a bar magnet The pattern of directions of the magnetic field around a bar magnet is very similar to the pattern of directions of the magnetic field around a current loop and to the pattern of the electric field around a permanent electric dipole.

41. Question The bar magnet produces a magnetic field at the compass location Whose strength is comparable to that of the Earth. The needle of the compass points in what direction? A) B) C) D) E) N S Compass N B earth A B C D E

42. How many turns must be in a coil of the same radius as your magnet bar to produce comparable magnetic field? Assume that the battery can supply current of ~5 A. Exercise

43. An electric dipole consists of two opposite charges – monopoles N S Break magnet: S N There are no magnetic monopoles! Magnetic Monopoles

44. The magnetic field of a current loop and the magnetic field of a bar magnet look the same. What is the direction? S N What is the average current I? current=charge/second: One loop: The Atomic Structure of Magnets Electrons

45. Magnetic dipole moment of 1 atom: Method 1: use quantized angular momentum Orbital angular momentum: Quantum mechanics: L is quantized: If n=1: Magnetic Dipole Moment

46. Magnetic dipole moment of 1 atom: Method 2: estimate speed of electron Momentum principle: Circular motion:  – angular speed Magnetic Dipole Moment

47. Magnetic dipole moment of 1 atom: Mass of a magnet: m~5g Assume magnet is made of iron: 1 mole – 56 g 6. 10 23 atoms number of atoms = 5g/56g. 6. 10 23 ~ 6. 10 22 Magnetic Dipole Moment

48. 1. Orbital motion There is no ‘motion’, but a distribution Spherically symmetric cloud (s-orbital) has no  Only non spherically symmetric orbitals (p, d, f) contribute to  There is more than 1 electron in an atom Modern Theory of Magnets

49. Alignment of atomic dipole moments: most materialsferromagnetic materials: iron, cobalt, nickel Modern Theory of Magnets

50. 2. Spin Electron acts like spinning charge - contributes to  Electron spin contribution to  is of the same order as one due to orbital momentum Neutrons and proton in nucleus also have spin but their  ‘s are much smaller than for electron same angular momentum: NMR, MRI – use nuclear  Modern Theory of Magnets

51. Nuclear Magnetic Resonance Felix Bloch (1905 -1983) Edward Purcell (1912-1997) B field S N Proton spinMagnet

52. Magnetic Resonance Imaging B

53. Magnetic domains Very pure iron – no residual magnetism spontaneously disorders Hitting or heating can also demagnetize Modern Theory of Magnets

54. Magnetic domains Why are there Multiple Domains?

55. Multiplier effect: Electromagnet: Iron Inside a Coil

56. Step 1: Cut up the distribution into pieces B origin: center of the solenoid Step 2: Contribution of one piece one loop: Number of loops per meter: N/L Number of loops in  z: (N/L)  z Field due to  z: Magnetic Field of a Solenoid

57. Step 3: Add up the contribution of all the pieces B Magnetic field of a solenoid: Magnetic Field of a Solenoid

58. Special case: R<<L, center of the solenoid: in the middle of a long solenoid Magnetic Field of a Solenoid