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Department of Semiconductor Systems Engineering SoYoung Kim
Engineering Electromagnetics- 1 Lecture 16: Static Magnetic Field
SoYoung Kim
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