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Electromagnetic 1 CH6 Static Magnetic Field

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

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

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Electromagnetic 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

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Electromagnetic 5 Ampere’s circuital law summary

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Electromagnetic 6 EX 6-1 Inside conductor Outside conductor C 2 outside conductor encloses I

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Electromagnetic 7 b If 空心圓柱 ( 柱座標 )

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

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

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

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Electromagnetic 11 In Cartesian coordinates, Vector

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

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

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Electromagnetic 14 Biot-Savart Law ： [Valid in steady current] c.f. 因次分析 0 0

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

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

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Electromagnetic 17 PF: =0 C.F 封閉磁迴路 孤立電單極 N S

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Electromagnetic 18 From 其中 0

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

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Electromagnetic 20 其中： Ampere’ Law Poission Equ.

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Electromagnetic 21 Gauss Thm.

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Electromagnetic 22 Example 6-4 (a)

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Electromagnetic 23

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Electromagnetic 24 Example 6-5 From 6-4

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

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Electromagnetic Magnetic Dipole 2 0 pole = 0 2’ pole = 0

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Electromagnetic 27 c.f.

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Electromagnetic 28 c.f.

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Electromagnetic 29

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

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Electromagnetic Magnetization and Equivalent Current Density free electronConductor polarized ion Non-conductor sourceComponents 磁性物質 定義

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Electromagnetic 32 c.f.

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Electromagnetic 33

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Electromagnetic 34 EX : 6-8

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Electromagnetic 35 靜磁學 ( 包含導體與磁性材料 ) C.F.

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

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Electromagnetic 37

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Electromagnetic 38 Ex 6-9 與（ Dipole 類似） 電場

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Electromagnetic Ampere’s circuital law ; relative permeability

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Electromagnetic 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]

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

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

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

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

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

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Electromagnetic Behavior of Magnetic Materials ◎ Paramagnetic: ◎ Diamagetic: ◎ Ferromagnetic:

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Electromagnetic Boundary Conditions for Magnetostatic Field

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

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

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Electromagnetic Inductances & Inductors Mutual flux ： mutual inductance between loops C 1 and C 2 If loop C 2 has N 2 turns, Generalizes to

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Electromagnetic 51 Some of produced by I 1 links only with C 1 loop itself, not with C 2 Self inductance of C 1 loop Procedure for Finding Inductance 1. Appropriate coordinate system 2. Find

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Electromagnetic 52 EX 6-14 total current flux linkage

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

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

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

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Electromagnetic 56 Neumann Formula

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Electromagnetic 57 EX ： 6-18 Outer coil has N 2 turns,

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Electromagnetic 58 EX ： 6-19 Find B 2 is caused by long wire I 2.

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Electromagnetic Magnetic Energy Loop 1 Loop 2 : C 1 & C 2 Similary Total work at C 2 Generalizing I 1, I 2, I 3, … I N,

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Electromagnetic 60 Consider K th loop of N coupled loops Magnetic energy Total magnetic energy

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Heng-Ming Hsu Electromagnetic Wm in terms of Field Quantities Vector identity 其中 All space

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Electromagnetic 62 or c.f. Magnetic energy density Wm

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

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

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Heng-Ming Hsu Electromagnetic 65

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Heng-Ming Hsu Electromagnetic 66

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Heng-Ming Hsu Electromagnetic 67

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Heng-Ming Hsu Electromagnetic 68

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Heng-Ming Hsu Electromagnetic 69

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Heng-Ming Hsu Electromagnetic 70

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Heng-Ming Hsu Electromagnetic 71

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Heng-Ming Hsu Electromagnetic 72 S.W.(OFF)S.W.(ON) s.w. system

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Heng-Ming Hsu Electromagnetic 73

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Heng-Ming Hsu Electromagnetic 74 Two coils

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Heng-Ming Hsu Electromagnetic 75

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Heng-Ming Hsu Electromagnetic 76 Home Work #6 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

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