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**CH6 Static Magnetic Field**

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**Electric force Magnetic force Elecmagnetic fore**

(N) ~ Lorentz’s force equation

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**Free space Static Electric Field Static Magnetic Field Steady current**

Permeability of free space

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**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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**Ampere’s circuital law**

summary

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**EX 6-1 Inside conductor Outside conductor**

C2 outside conductor encloses I

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If b 空心圓柱(柱座標)

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**EX 6-2 (Toroidal Coil) A circular contour C with radius r**

(b - a) < r < (b + a) (b - a) < r < (b + a) (No source)

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**EX 6-3 (Solenoid Coil) (a) Direct application of Ampere Law**

(b) Special case of torid Ex 6-2,

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**6-3 Vector Magnetic Potential**

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

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**In Cartesian coordinates,**

Vector

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

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**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

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**Biot-Savart Law：[Valid in steady current]**

c.f. 因次分析

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Biot-Sarvart Law： Action at a distance： field

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

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PF: =0 =0 N 封閉磁迴路 S C.F 孤立電單極

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From 其中

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**From Steady current Static magnetic field 其中： Coulomb Gauge**

Steady state Coulomb Gauge

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其中： Ampere’ Law Poission Equ.

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Gauss Thm.

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Example 6-4 (a)

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Example 6-5 From 6-4

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Example 6-6 is canceled due to cylindrical sym.

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

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c.f.

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c.f.

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**Scalar Magnetic Potential**

(not physical) : Non conservative (path dependent)

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**6-6 Magnetization and Equivalent Current Density**

free electron Conductor polarized ion Non-conductor source Components 磁性物質 定義

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c.f.

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EX : 6-8

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靜磁學 (包含導體與磁性材料) C.F.

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**Static Magnetic Source : Conductor : Magnetic material :**

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

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Ex 6-9 電場 與（Dipole 類似）

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6-7 Ampere’s circuital law ; relative permeability

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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]

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

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**Magnetic Flux ; S : cross-section**

: length of ferromagnetic core. :ferromagnetic core :air gap : Reluctance Analog to : [Electric circuit]

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

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**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

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

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**6-9 Behavior of Magnetic Materials**

◎Diamagetic: ◎Paramagnetic: ◎Ferromagnetic:

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**6-10 Boundary Conditions for Magnetostatic Field**

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**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

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**Ex 6-13 Surface current Example 6-8 [p246] Center End**

at interface quantity

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**6-11 Inductances & Inductors**

Mutual flux ： mutual inductance between loops C1 and C2 If loop C2 has N2 turns , Generalizes to

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**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.

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EX 6-14 flux linkage total current

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

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

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EX 6-17 Internal total 2 wires : external :

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Neumann Formula

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EX：6-18 Outer coil has N2 turns,

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

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**6-12 Magnetic Energy Loop 1 Similary Total work at C2 Loop 2 : C1 & C2**

Generalizing I1, I2, I3, … IN,

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**Consider Kth loop of N coupled loops**

Magnetic energy Total magnetic energy

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**6-12.1 Wm in terms of Field Quantities**

其中 All space Vector identity

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**Magnetic energy density Wm**

or c.f.

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

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**6-13 Magnetic forces & Torques**

electron move toward to x-dir. Creating a transverse -field. Steady state, net force is Zero.

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

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Two coils

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**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

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Fundamental Physics II PETROVIETNAM UNIVERSITY FACULTY OF FUNDAMENTAL SCIENCES Vungtau, 2013 Pham Hong Quang

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