1. Dr. Hugh Blanton ENTC 3331

2. Magnetostatics

3. Dr. Blanton - ENTC 3331 - Magnetostatics 3 Magnetostatics Magnetism Chinese—100 BC Arabs—1200 AD Magnetite—Fe 3 O 4 Found near Magnesia (now Turkey) Permanent magnet Not fundamental to magnetostatics. A permanent magnet is equivalent to a polar material in electrostatics. Equivalent to electrostatics The theoretical structure of magnetostatics is very similar to electrostatics. But there is one important empirical fact that accounts for all the differences between the theory of magnetostatics and electrostatics. There is no magnetic monopole!

4. Dr. Blanton - ENTC 3331 - Magnetostatics 4  + +  N S + +   N N S S A magnetic monopole does not exist—A magnetostatic field has no sources or sinks!

5. Dr. Blanton - ENTC 3331 - Magnetostatics 5 Elementary charge is a source Coulomb’s Law (Elementary) DC current is not a source Ampere’s Law + + I I

6. Dr. Blanton - ENTC 3331 - Magnetostatics 6 Current Density Moving charges  current. Charges move to the right with constant velocity, u. Over a period of time, the charges move distance,  l. u  l vv  s

7. Dr. Blanton - ENTC 3331 - Magnetostatics 7 The amount of charge through an area,  s, during  t: volume

8. Dr. Blanton - ENTC 3331 - Magnetostatics 8 Generalization: projection of projection of onto the surface normal

9. Dr. Blanton - ENTC 3331 - Magnetostatics 9 Current Density The definition of current density is: Therefore,

10. Dr. Blanton - ENTC 3331 - Magnetostatics 10 Electrical Current Without resistance Convection e.g. electron beam Typically in vacuum or dielectric medium With resistance Conduction e.g. copper wire Typically in a conducting medium Electrical Currents

11. Dr. Blanton - ENTC 3331 - Magnetostatics 11 Conducting Media Two types of charge carriers: Negative charges Positive charges + -

12. Dr. Blanton - ENTC 3331 - Magnetostatics 12 MediumNegative ChargesPositive Charges ConductorsFree electrons SemiconductorsElectronsHoles IonsNegative ionsPositive ions

13. Dr. Blanton - ENTC 3331 - Magnetostatics 13 Like mechanics, there is a resistance to motion. Therefore, an external force is required to maintain a current flow in a resistive conductor.

14. Dr. Blanton - ENTC 3331 - Magnetostatics 14 Since in most conductors, the resistance is proportional to the charge velocity. constant of proportionality (mobility)

15. Dr. Blanton - ENTC 3331 - Magnetostatics 15 In semiconductors: electron mobility electrons move against the direction hole mobility holes move in the same direction as

16. Dr. Blanton - ENTC 3331 - Magnetostatics 16 Since Ohm’s law conductivity

17. Dr. Blanton - ENTC 3331 - Magnetostatics 17 It follows that for a perfect dielectric  and for a perfect conductor  since current is finite. inside all conductors.

18. Dr. Blanton - ENTC 3331 - Magnetostatics 18 Since for all conductors. All conductors are equipotential, but may have surface charge.

19. Dr. Blanton - ENTC 3331 - Magnetostatics 19 Electrical Resistance For a conductor Show that for a conductor of cylindrical shape. A1A1 A2A2

20. Dr. Blanton - ENTC 3331 - Magnetostatics 20 Potential difference between A 1 and A 2. Current through A 1 and A 2.

21. Dr. Blanton - ENTC 3331 - Magnetostatics 21 The reciprocal of conductivity Resistivity (ohms/meter). Do not confuse charge distribution!

22. Dr. Blanton - ENTC 3331 - Magnetostatics 22 The electrical field can be expressed in terms of the charge density, . What is the equivalent expression for the magnetic field,.

23. Dr. Blanton - ENTC 3331 - Magnetostatics 23 Qualitatively, circular field lines

24. Dr. Blanton - ENTC 3331 - Magnetostatics 24 Jean-Baptiste Biot & Felix Savart developed the quantitative description for the magnetic field.

25. Dr. Blanton - ENTC 3331 - Magnetostatics 25 point of interest differential section of conductor contributes to field at field comes out of plane due to the cross product

26. Dr. Blanton - ENTC 3331 - Magnetostatics 26 Total field through integration over. The line integration is not convenient Wires are irregularly bent, but Wires typically have constant cross-sections,  s. magnetic field strength

27. Dr. Blanton - ENTC 3331 - Magnetostatics 27 Take advantage of: useful relationship Biot-Savart Law

28. Dr. Blanton - ENTC 3331 - Magnetostatics 28 What force does such a field exert onto a stationary current? What is equivalent to:

29. Dr. Blanton - ENTC 3331 - Magnetostatics 29 Experimental facts: Flexible wire in a magnetic field,. No current X X X X X X X X X X X X X X X X

30. Dr. Blanton - ENTC 3331 - Magnetostatics 30 Experimental facts: Flexible wire in a magnetic field,. Current up. X X X X X X X X X X X X X X X X right-handed rule

31. Dr. Blanton - ENTC 3331 - Magnetostatics 31 Experimental facts: Flexible wire in a magnetic field,. Current down. X X X X X X X X X X X X X X X X right-handed rule

32. Dr. Blanton - ENTC 3331 - Magnetostatics 32 The experimental facts also show that: and Thus, the magnetic force for a straight conductor is:

33. Dr. Blanton - ENTC 3331 - Magnetostatics 33 Important Consequences The force on a closed, current carrying loop is zero. closed loop = 0

34. Dr. Blanton - ENTC 3331 - Magnetostatics 34 Example Linear conductor Determine magnetic field. Determine the force,, on another conductor. Biot-Savart Law z xX

35. Dr. Blanton - ENTC 3331 - Magnetostatics 35 Substituting z xX  at P(x,z), points into the plane Note that for a small d , R is approximately unchanged when separated by d  which implies:

36. Dr. Blanton - ENTC 3331 - Magnetostatics 36 Note: z xX

37. Dr. Blanton - ENTC 3331 - Magnetostatics 37 Using the previous transformations: z xX

38. Dr. Blanton - ENTC 3331 - Magnetostatics 38 Note the following z x

39. Dr. Blanton - ENTC 3331 - Magnetostatics 39 For an infinitely long wire where

40. Dr. Blanton - ENTC 3331 - Magnetostatics 40 Now, what is the force on a parallel conductor wire carrying the current, I? z x y field by I 1 at location of I 2

41. Dr. Blanton - ENTC 3331 - Magnetostatics 41 z x y I 1 attracts I 2 Similarly I 2 attracts I 1 with the same force. Attraction is proportional to 1/distance.

42. Dr. Blanton - ENTC 3331 - Magnetostatics 42 Maxwell’s Magnetostatic Equations Experimental fact: An equivalent to the electrostatic monopole field does not exist for magnetostatics.  Charge is the source of the electrostatic field No equivalent in magnetostatics

43. Dr. Blanton - ENTC 3331 - Magnetostatics 43 Let’s apply Gauss’s theorem to an arbitrary field: Gauss’s law of Magnetostatics Mathematical expression of the experimental fact that a source of the magnetostatic field does not exist.

44. Dr. Blanton - ENTC 3331 - Magnetostatics 44 Experimental fact: The magnetostatic field is generally a rotational field. Apply Stoke’s theorem to any arbitrary field: Ampere’s Circuital Law

45. Dr. Blanton - ENTC 3331 - Magnetostatics 45 Mathematical expression of the experimental fact that the line integral of the magnetostatic field around a closed path is equal to the current flowing through the surface bounded by this path. X field vector of the magnetostatic field line differential surface contour current flowing through the surface

46. Dr. Blanton - ENTC 3331 - Magnetostatics 46 Long line Suppose we have an infinitely long line of charge: Recall that charge is the fundamental quantity for electrostatics

47. Dr. Blanton - ENTC 3331 - Magnetostatics 47 Long line Suppose we have an infinitely long line carrying current,I : What is. Orient wire along the z-axis Choose a circular Amperian contour about the wire. Ampere circuital law z

48. Dr. Blanton - ENTC 3331 - Magnetostatics 48 Symmetry implies that is constant on the contour and is always tangential to the contour. This implies that

49. Dr. Blanton - ENTC 3331 - Magnetostatics 49 is always tangential on circles about the wire and its magnitude decreases with 1/r.

50. Dr. Blanton - ENTC 3331 - Magnetostatics 50 What is inside the wire? Again, use an Ampere’s circuital law. z

51. Dr. Blanton - ENTC 3331 - Magnetostatics 51 is current through the Amperian surface The magnitude of increases linearly inside the conductor.

52. Dr. Blanton - ENTC 3331 - Magnetostatics 52 It is interesting to note that the comparison of part (a) and (b) of this problem shows that for a convective current, I, the electrostatic and magnetostatic fields are perpendicular to each other. This is generally true in electrodynamics!

53. Dr. Blanton - ENTC 3331 - Magnetostatics 53 The magnetostatic field is rotational without sources In electrostatics A scaler potential, V, exists, so that

54. Dr. Blanton - ENTC 3331 - Magnetostatics 54 Can any potential be defined in magnetostatics? Let’s take advantage of the general vector identity Define a vector potential,,so that It follows that in agreement with Maxwell equations

55. Dr. Blanton - ENTC 3331 - Magnetostatics 55 In a given region of space, the vector potential of the magnetostatic field is given by Determine

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57. Dr. Blanton - ENTC 3331 - Magnetostatics 57 Magnetic flux,,through an area S is given by the surface integral Use this equation and the solution to previous problem to calculate the magnetic flux,, for the field through a square loop. x y 0.25m

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60. Dr. Blanton - ENTC 3331 - Magnetostatics 60 Note that since, it follows from Stoke’s theorem that Calculate again using x 0.25m

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