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Dr. Hugh Blanton ENTC 3331

ENTC 3331 RF Fundamentals. Dr. Hugh Blanton ENTC 3331. 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.

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Dr. Hugh Blanton ENTC 3331

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  1. ENTC 3331 RF Fundamentals Dr. Hugh Blanton ENTC 3331

  2. Magnetostatics

  3. Magnetism Chinese—100 BC Arabs—1200 AD Magnetite—Fe3O4 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! Magnetostatics Dr. Blanton - ENTC 3331 - Magnetostatics 3

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

  5. + + I I Dr. Blanton - ENTC 3331 - Magnetostatics 5

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

  7. The amount of charge through an area, Ds, during Dt: volume Dr. Blanton - ENTC 3331 - Magnetostatics 7

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

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

  10. Electrical Current Electrical Currents Dr. Blanton - ENTC 3331 - Magnetostatics 10

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

  12. Dr. Blanton - ENTC 3331 - Magnetostatics 12

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

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

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

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

  17. It follows that for • a perfect dielectric • s = 0  • and for a perfect conductor • s  • since current is finite. • inside all conductors. Dr. Blanton - ENTC 3331 - Magnetostatics 17

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

  19. Electrical Resistance • For a conductor • Show that for a conductor of cylindrical shape. A2 A1 Dr. Blanton - ENTC 3331 - Magnetostatics 19

  20. Potential difference between A1 and A2. • Current through A1 and A2. Dr. Blanton - ENTC 3331 - Magnetostatics 20

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  36. X z • Note: x Dr. Blanton - ENTC 3331 - Magnetostatics 36

  37. X z • Using the previous transformations: x Dr. Blanton - ENTC 3331 - Magnetostatics 37

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

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

  40. Now, what is the force on a parallel conductor wire carrying the current, I? z y field by I1 at location of I2 x Dr. Blanton - ENTC 3331 - Magnetostatics 40

  41. z • I1 attracts I2 • Similarly I2 attracts I1 with the same force. • Attraction is proportional to 1/distance. y x Dr. Blanton - ENTC 3331 - Magnetostatics 41

  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 Dr. Blanton - ENTC 3331 - Magnetostatics 42

  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. Dr. Blanton - ENTC 3331 - Magnetostatics 43

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

  45. X • 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. line differential field vector of the magnetostatic field surface current flowing through the surface contour Dr. Blanton - ENTC 3331 - Magnetostatics 45

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

  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 Dr. Blanton - ENTC 3331 - Magnetostatics 47

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

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

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

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