Department of Semiconductor Systems Engineering SoYoung Kim

Engineering Electromagnetics- 1 Lecture 16: Static Magnetic Field

SoYoung Kim

ksyoung@skku.edu

Department of Semiconductor Systems Engineering

College of Information and Communication Engineering

Sungkyunkwan University

Department of Semiconductor Systems Engineering SoYoung Kim

Outline

Magnetic Field

Fundamental Properties of Magnetic Field

Biot-Savart’s Law

Department of Semiconductor Systems Engineering SoYoung Kim

Introduction

charge electricE

current electricB

Hans Christian Oersted 1777-1851 While preparing for an evening lecture on 21 April 1820, Oersted developed an experiment which provided evidence that surprised him. As he was setting up his materials, he noticed a compass needle deflected from magnetic north when the electric current from the battery he was using was switched on and off. This deflection convinced him that magnetic fields radiate from all sides of a wire carrying an electric current, just as light and heat do, and that it confirmed a direct relationship between electricity and magnetism. At the time of discovery, Oersted did not suggest any satisfactory explanation of the phenomenon, nor did he try to represent the phenomenon in a mathematical framework. However, three months later he began more intensive investigations. Soon thereafter he published his findings, proving that an electric current produces a magnetic field as it flows through a wire.

Department of Semiconductor Systems Engineering SoYoung Kim

Introduction

Motors

Generators

On-chip inductors

Hard disk drive

MRI

Department of Semiconductor Systems Engineering SoYoung Kim

Introduction

I

Department of Semiconductor Systems Engineering SoYoung Kim

Introduction

Department of Semiconductor Systems Engineering SoYoung Kim

Magnetic Flux Density

Unit : T Wb/m2 gauss

Introduction

B

Department of Semiconductor Systems Engineering SoYoung Kim

Two null identities ( Review )

The curl of the gradient of any scalar field is zero

The divergence of the curl of any vector field is zero

Department of Semiconductor Systems Engineering SoYoung Kim

Fundamental Postulates of Magnetostatics in Free Space

Two fundamental postulates of magnetostatics in free space

Permeability of free space

Divergence of the curl

(5-44) for steady currents

No magnetic monopole ρ

mH /

Existence of magnetic vector potential

Department of Semiconductor Systems Engineering SoYoung Kim

Fundamental Postulates of Magnetostatics in Free Space

Two fundamental postulates of magnetostatics in free space - Integral form

No magnetic source

The law of conservation of magnetic flux

Department of Semiconductor Systems Engineering SoYoung Kim

Fundamental Postulates of Magnetostatics in Free Space

Two fundamental postulates of magnetostatics in free space - Integral form

C : bounding the surface S

S를 지나는 전류

Ampere’s circuital law

Amperian closed path

* Symmetry를 갖는 구조에서 B를 구하는 가장 쉬운 방법

Department of Semiconductor Systems Engineering SoYoung Kim

Analogy between Electric and Magnetic Field (I)

Electric Field

Electric field intensity

E Magnetic field intensity

H

Electric flux density D Magnetic flux density

B

Electric potential V Magnetic scalar potential

Vm

Magnetic vector potential A

Coulomb’s law Biot-Savart’s law

Gauss’s law Ampere’s law

Magnetic Field

Department of Semiconductor Systems Engineering SoYoung Kim

Analogy between Electric and Magnetic Field (II)

Department of Semiconductor Systems Engineering SoYoung Kim

Biot-Savart’s Law

Biot-Savart’s law dictates the relation between differential magnetic field intensity dH and differential current element Idl

Directions in Biot-Savart’s law

2

2 3

sin

4

4 4

l a l RH R

I dldH

R

I d I dd

R R

Department of Semiconductor Systems Engineering SoYoung Kim

Biot-Savart’s Law for Distributed Current

Biot-Savart’s law can be applied to line current, surface current, volume current

2

2

2

(line current)4

(surface current)4

(volume current)4

l aH

K aH

J aH

R

L

R

S

R

v

I d

R

dS

R

dv

R

Department of Semiconductor Systems Engineering SoYoung Kim

Magnetic Field due to Finite Line Current

3

,

3/22 2

4

,

4

l RH

l a R a a l R a

H a

z z

I dd

R

d dz z d dz

I dz

z

2 2

1 1

2

2 2

3 3

2 1

Letting cot , cosec ,

cosecsin

4 cosec 4

(cos cos )4

H a a

H a

z dz d

I d Id

I

Department of Semiconductor Systems Engineering SoYoung Kim

Magnetic Field due to Infinite Line Current

Half infinite line current

Infinite line current

Direction

1 2(0,0,0), (0,0, ) or 90 , 0 .A B

1 2(0,0, ), (0,0, ) or 180 , 0 .A B

4H a

I

2H a

I

a a a

Department of Semiconductor Systems Engineering SoYoung Kim

Example 1

Calculate magnetic field due to side 1 of triangle given below

1 2

1 2 1

2cos cos90 0, cos , 5

29

10 2(cos cos ) 0 ( )

4 4 (5) 29

a a a a

H a a

x z y

y

I