1. CH6 Static Magnetic Field

2. Electric force Magnetic force Elecmagnetic fore
(N) ~ Lorentz’s force equation

3. Free space Static Electric Field Static Magnetic Field Steady current
Permeability of free space

4. No magnetic flow sources
Magnetic flux lines always close
Low of conservation of magnetic flux
Each magnets has a
north
south
Magnetic poles cannot be isolated
pole

5. Ampere’s circuital law
summary

6. EX 6-1 Inside conductor Outside conductor
C2 outside conductor encloses I

7. If b
空心圓柱(柱座標)

8. EX 6-2 (Toroidal Coil) A circular contour C with radius r
(b - a) < r < (b + a)
(b - a) < r < (b + a)
(No source)

9. EX 6-3 (Solenoid Coil) (a) Direct application of Ampere Law
(b) Special case of torid
Ex 6-2,

10. 6-3 Vector Magnetic Potential
: magnetic potential [Vector]
c.f
: electric potential [Scalar] 取
Vector
Poisson’s equation
Coulomb gauge

11. In Cartesian coordinates,
Vector

12. Magnetic Flux through a given area S which is bounded by contour C

13. 6-4 Biot-Savart Law and applications
Vector source
Magnetic:
3-dim
Volume current density
2-dim
Surface current density
1-dim
current
0-dim
Current distribution
Current element

14. Biot-Savart Law：[Valid in steady current]
c.f.
因次分析

15. Biot-Sarvart Law：
Action at a distance： field

16. Gauss thm.
特殊式
Biot-Sarvart Law
Poission Eqe.
Stoke Thm.
Ampere’s Law

17. PF: =0 =0 N
封閉磁迴路 S
C.F
孤立電單極

18. From 其中

19. From Steady current Static magnetic field 其中： Coulomb Gauge
Steady state
Coulomb Gauge

20. 其中：
Ampere’ Law
Poission Equ.

21. Gauss
Thm.

22. Example 6-4
(a)

24. Example 6-5
From 6-4

25. Example 6-6
is canceled due to cylindrical sym.

26. 6-5 Magnetic Dipole
20 pole = 0
2’ pole = 0

27. c.f.

28. c.f.

30. Scalar Magnetic Potential
(not physical)
: Non conservative (path dependent)

31. 6-6 Magnetization and Equivalent Current Density
free electron
Conductor
polarized ion
Non-conductor
source
Components
磁性物質 定義

32. c.f.

34. EX : 6-8

35. 靜磁學 (包含導體與磁性材料)
C.F.

36. Static Magnetic Source : Conductor : Magnetic material :
(permeativity)
Conductor :
Magnetic material :
Copy 電學
數學上等效，無物理

38. Ex 6-9 電場
與（Dipole 類似）

39. 6-7
Ampere’s circuital law ;
relative permeability

40. 6-8 Magnetic Circuits
Electric circuit : Voltage / Current source ; V, I, …
Magnetic circuit : Transformer / Generator / Motor …
; closed path c to enclose N turns of I
(m.m.f) magnetomotive force [Amp]

41. Ex 6-10
Sol :
is identical in different material ;
f : ferromagnetic g : gap
Ampere law

42. Magnetic Flux ; S : cross-section
: length of ferromagnetic core.
:ferromagnetic core
:air gap
: Reluctance
Analog to : [Electric circuit]

43. Magnetic circuit
Electric circuit
mmf
mag. flux
reluctance
Permeability
emf
electric current ,
resistance ,
conductivity ,

44. An exact analysis of magnetic circuits is difficult
◎Leakage Fluxes
◎Fringing effect ◎
2 conditions must be satisfied
Similar to
Kirchhoff’s voltage Law
Kirchhoff’s current Law

45. EX.6-11
K.V.L. (Time Independent)
Loop1:
Loop2:

46. 6-9 Behavior of Magnetic Materials
◎Diamagetic:
◎Paramagnetic:
◎Ferromagnetic:

47. 6-10 Boundary Conditions for Magnetostatic Field

48. Ex 6-12 Bn component component or Similar to E-field Magnitude of
In ferromagnetic parallel interface
Originates in a ferromangeitc ,
Flux perpendicular to interface

49. Ex 6-13 Surface current Example 6-8 [p246] Center End
at interface quantity

50. 6-11 Inductances & Inductors
Mutual flux
： mutual inductance
between loops C1 and C2
If loop C2 has N2 turns ,
Generalizes to

51. Some of produced by I1 links only with C1 loop itself, not with C2
Self inductance of C1 loop
Procedure for Finding Inductance
1. Appropriate coordinate system 4.
2. Find 5. 3.

52. EX 6-14
flux linkage
total current

53. EX 6-15
Long solenoid
From(Ex6-3) p231
Inductance per unit length

54. EX 6-16
Current in annular ring
a) Inside inner conductor.
b) Between inner & outer conductors

55. EX 6-17
Internal
total
2 wires :
external :

56. Neumann Formula

57. EX：6-18
Outer coil has N2 turns,

58. EX：6-19
Find B2 is caused by long wire I2.

59. 6-12 Magnetic Energy Loop 1 Similary Total work at C2 Loop 2 : C1 & C2
Generalizing I1, I2, I3, … IN,

60. Consider Kth loop of N coupled loops
Magnetic energy
Total magnetic energy

61. 6-12.1 Wm in terms of Field Quantities其中
All space
Vector identity

62. Magnetic energy density Wmor
c.f.

63. Ex 6-20 (Ref. Ex 6-16)
Wm in inner conductor Wm between inner & outer
Hence,

64. 6-13 Magnetic forces & Torques
electron move toward to x-dir.
Creating a transverse -field.
Steady state, net force is Zero.

72. S.W.(OFF)
S.W.(ON)
system
s.w.

74. Two coils

76. Home Work #6 P6-2, P6-4, P6-5, P6-6, P6-10,P6-11,P6-12
David Cheng: Chapter6
P6-2, P6-4, P6-5, P6-6, P6-10,P6-11,P6-12
P6-13,P6-15,P6-18,P6-19,P6-22,P6-26,
P6-27,P6-29,P6-32,P6-37,P6-38,P6-39,
P6-40,P6-41,P6-42,P6-43,P6-44,P6-46,
P6-50,P6-53
Due: 2 weeks