8. magnetic forces, materials, and deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf ·...
TRANSCRIPT
![Page 1: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/1.jpg)
8. Magnetic Forces, Materials,and Device
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11
8.2 FORCES DUE TO MAGNETIC FIELDS
A. 하전입자에작용하는힘
Fe = QE 8.1
Fm = Qu × B 8.2
F = Fe + Fm
* Lorentz Force Equation
F = Q E + u × B 8.3
F = mdu
dt= Q E + u × B 8.4
𝐁𝐮
𝐐𝐮 × 𝐁
𝐁(유도 자장)𝐑~𝐚𝐫
𝐝 𝐋 I
P
외부 자장)
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22
Table 8.1 Force on a Charged Particle
State of particle 𝐄 field 𝐇 field Combined 𝐄 & 𝐇 field
Stationary
MovingQE
QE
-
Qu × BQE
Q(E + u × B)
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33
B. Force on a Current Element
J = ρv + u (8.5)
IdL = KdS = Jdv (8.6)
IdL =dQ
dtdL = dQu (8.7)
(cf) Fm = Qu × B
𝐁(유도 자장)𝐑~𝐚𝐫
𝐝 𝐋 I
P
외부 자장)
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44
K
a
1
IdL = K ∙ adL
= Ka ∙ dL
= KdS
dL
an
IdL = KdS = Jdv (8.6)
IdL = J ∙ AdL
= JA ∙ dL
= Jdv
L
an A
J
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55
dF = dQ u × B = IdL × B 8.8
F = IdL × B 8.9
dF = KdS × B or dF = Jdv × B 8.8a
F = KdS × B or F = Jdv × B (8.8b)
IdL = dQu 8.7
IdL = KdS = Jdv (8.6)
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66
Fig 7.1 IdL에 의한 자기장 dH
𝐁~𝐇
𝐱
𝐝 𝐋 I
P
𝛂
C. Force between Two Current Elements
)10.8(Bd Ld I )Fd(d
)11.8(R4
a LdI Bd
2111
221
R2202
21
LdI 11
LdI 22
2I1I
21R
)Fd(d 1
1 2
Fig 8.1 두 전류 Loop 사이의 힘
30
2R0
R4
RLdI
R4
aLdIBd)4.7(
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77
)13.8(R
)a Ld( Ld
4
II F
)12.8(R4
)a LdI( LdI )Fd(d
)10.8(Bd Ld I )Fd(d
)11.8(R4
a LdI Bd
1 2
21
21
21
L L 221
R212101
221
R221101
2111
221
R2202
LdI 11
LdI 22
2I1I
21R
)Fd(d 1
1 2
Fig 8.1 두 전류 Loop 사이의힘
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88
sec)1t(sec/m)3,15,22(
)3,0,4(t)0,15,18(
vtav
dt
vda
)b(
sec/m)0,15,18(
)0,10,12(2
3
Em
Qa
EQamF
)a(
0
2
속도입자의
가속도입자의
m)3,5.5,14(
)0,2,1()3,0,4()0,5.7,9(
Lt)3,0,4(2/t)0,15,18(L
)3,0,4(t)0,15,18(dt
dL
dt
dLv
)d(
J718
)31522(22
1
vm2
1E
)c(
02
222
2
위치입자의
운동에너지입자의
예제 8.1 전장 𝐄 = 12,10,3 volt/m 일 때 질량 m=2 kg, 전하 Q=3 C 인 전하가point(1,-2,0) 에서 초기속도 𝐯 = 4,0,3 m/sec의 속도로 출발하였다.t=1 sec에서 다음을 구하라.
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99
예제 8.2 m=2 kg, Q=1 C 인 하전입자가 원점에서 속도 v = 0,3, (w0 = 0) m/sec
로 출발하여 자장 B = 0,0,10 Wb/m2 안에서 움직일 때t=4 sec에서 다음을 구하라.
x
y
z
B
v
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1010
)0,v5,v5(
1000
vvv
aaa
2
1
BBB
vvv
aaa
m
Q
)v,v,v(dt
d
BvQdt
vdmamF
)a(
xy
zyx
zyx
zyx
zyx
zyx
zyx
가속도속도와입자의
sec/m)0,20cos3,20sin3(
)0,t5cos3,t5sin3(v
0C
3C
0C
0)0t(v
3)0t(v
0)0t(v
Cv
t5cosCt5sinCv
t5sinCt5cosCv
0dt
dv
v25dt
vd
v25dt
vd
0dt
dv
dt
dv5
dt
vd
dt
dv5
dt
vd
0dt
dv
v5dt
dv
v5dt
dv
0
2
1
z
y
x
0z
21y
21x
z
y2
y2
x2x
2
z
x2
y2
y
2x
2
z
xy
yx
2sec/m)0,20sin15,20cos15(
)0,t5sin15,t5cos15(
)0,t5cos3,t5sin3(dt
d
dt
vda
예제 8.2 m=2 kg, Q=1 C 인 하전입자가 원점에서 속도 v = 0,3, (w0 = 0) m/sec
로 출발하여 자장 B = 0,0,10 Wb/m2 안에서 움직일 때t=4 sec에서 다음을 구하라.
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1111
)0,20sin30,20cos30(
)0,t5sin15,t5cos15(2
amF
)0,20sin30,20cos30(
)0,t5sin30,t5cos30(
1000
0t5cos3t5sin3
aaa
1
BvQF
)b(
zyx
힘작용하는입자에
twz
6.0y)6.0x(
1y3
51x
3
5
1t5sint5cos
twz
t5sin6.0y
t5cos6.06.0x
tw,t5sin5
3,t5cos
5
3
5
3x
)0,0,5/3(C0C)0,0,5
3()0t(x
C)tw,t5sin5
3,t5cos
5
3(
Cdt)w,t5cos3,t5sin3(
Cdtvx
)d(
0
222
22
22
0
0
0
0
궤적운동입자의
예제 8.2 m=2 kg, Q=1 C 인 하전입자가 원점에서 속도 v = 0,3, (w0 = 0) m/sec
로 출발하여 자장 B = 0,0,10 Wb/m2 안에서 움직일 때t=4 sec에서 다음을 구하라.
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1212
x
y
z
B
v
twz
6.0y)6.0x(
0
222
x
y
z
B
+하전입자의 궤적
0.6
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1313
00
22
2
w4twz
20sin5
3t5sin
5
3y
20cos5
3
5
3t5cos
5
3
5
3x
)e(
9
)t5cos9t5sin9(
v2
mKE
)c(
위치
없다관계자장과운동에너지는
위치운동에너지와입자의
tw,t5sin5
3,t5cos
5
3
5
3
tw,t2
Bsin
5
3,t
2
Bcos
5
3
5
3x
)w,t5cos3,t5sin3(
w,t2
Bcos3,t
2
Bsin3v
0
000
0
000
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1414
Figure 8.2 Magnetic focusing of a beam of electrons:
(a) helical paths of electrons, (b) end view of paths.
0z
6.0y)6.0x( 222
궤적운동입자의
문제의문제점: 전자총에서 방출된 전자는 각각 나선형 궤적을 따라 진행하다가같은 초점에 도달한다.
cathode ray tube와 하전입자 거동에 대한 이해.
ω=QB/m
r=v0/ω=mv0/QB
B가 매우 크면 r은 매우 작다. 궤적을 point에 가깝게 할 수 있다.
B가 매우 크면 입자는 자력선을 따라 움직인다.
x
y
v0B=(0,0,10)
zw0
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1515
- +
전자총
전자총
전자
CPT: Brown 관
B
Shadow mask
형광체
Viewer
potential
전자
![Page 17: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/17.jpg)
1616
예제 8.3 입자가 속도 v = 4,0,0 m/sec 로 전장 E = 0,20,0 volt/m 와
자장 B = 0,0, B0 Wb/m2 안에서 움직일 때입자의 속도가 상수가 되기 위한 B0 는?
Figure 8.3 A velocity filter for charged particles.
T5
m/Wb5B
B
)0,0,B20()0,0,4(
velocitydriftBEB
BEv
0vBEB
0)Bv(BEB
0)]BvE(Q[B
)BvE(Q0amF
20
20
0
2
2
B
E
)0,0,B20(
00
)00(
0B20
B00
0200
aaa
BE 0
0
0
zyx
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1717
E × B drift
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1818
예제 8.4 전류 I2가 흐르는 사각 Loop에서 전류 I1이 흐르는무한히 긴 직선 도선과 나란히 있다. Loop에 작용하는 힘이다음과 같음을 유도하라.
Figure 8.4 For example 8.4:
forces acting on the loop and wire.
newtonaa
11
4
bII F
00
210
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19190FF
a)a(2
bIIF
a2
bII
)a(Ld2
II
LdI
LdAJ
LdAJ
xdJ
aLd2
II
xdBJF
a2
IB
a)cos(cos4
IH)12.7(
BdvJF)a9.8(
42
0
2103
0
210
b
00
210
2
2
2
32
b
00
210
3121
0
101
12
v
Figure 8.4 For example 8.4:
forces acting on the loop and wire.
1
2
P
I
newtonaa
11
4
bII
FFFFF
00
210
4321
B ⊗
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2020
8.3 Magnetic Torque and Moment
The Torque 𝐓 on the loop is the vector product of the force 𝐅 and The Moment arm 𝐫 .
Figure 8.5 (a) Rectangular planar loop in a uniform
magnetic field. (b) cross-sectional view of part (a).
(b)(a)
α
𝐫𝟐 = 𝐰𝐬𝐢𝐧𝛂𝐚𝐧
𝐑𝟐
α
𝐚𝐧
α
𝐅𝟐
𝐅𝟏
𝐁
α
F0
F0L
wA
B
C D
I𝐁
y
x z
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2121
momentdipolemagnetic:
)18.8(aISmwhere
)19.8(Bm
)17.8()a(sinBIS
)16.8()a(sinBILw
)a(sin]BIL][w[T
)15.8(0FFF
aBILF
BILF
aBIL
)a(BdzI
BLdI
xdBJF
)14.8(FRT
n
z
z
z
21
x2
1
x
xC
B
31
LId
LdAJ
LdAJ
xdJ 3
α
𝐫𝟐 = 𝐰𝐬𝐢𝐧𝛂𝐚𝐧
𝐑𝟐
α
𝐚𝐧
α
𝐅𝟐
𝐅𝟏
𝐁
α
w
F0
F0L
AB
C D
I𝐁
y
x z
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2222
8.4 A Magnetic Dipole
)22.8()asinacos2(r4
m
AB
)b21.8(r4
am A
aISm)18.8(ar4
sinIS
arwhere
)a21.8(ar4
sinaI
)20.8(R
Ld
4
I A
R
Ld
4
I A(7.41)
r30
2r0
n20
2
20
0
0
Figure 8.6 Magnetic field
at P due to a current loop.
x
y
z ),,r(P
Ld
I
r
am R
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2323
2
0 22
2
2
20
2
0 22
0
2
0 22
0y
2
0 22
xy0
2
0 22
0
0
cossinar2ar
cosd
a
r
r4
Ia
cossinar2ar
cosd
4
Ia
cossinar2ar
cosad
4
I A
cossinar2ar
)asina(cosad
4
I
cossinar2ar
aad
4
I
)20.8(R
Ld
4
I A
방정식 (8.21a) 유도
vector: 뒷장
r: 뒷장
x
y
z ),,r(P
Ld
I
r
am R
x
y
z
)0,,r(P
LdI
ra
R
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2424
dcos)cossinr
a1(
a
r
r4
Ia
cossinr
a21
cosd
a
r
r4
Ia
cossinr
a2
r
a1
cosd
a
r
r4
Ia
cossinar2ar
cosd
a
r
r4
Ia A
2
02
20
2
02
20
2
0 22
20
2
0 22
2
2
20
y
)a21.8(ar4
sinIaA
2
20
2
20
y
2
20
2
02
20
2
02
20
2
02
2
20
r4
sinIaA
r4
sinIa
d2
)2cos(
r4
sinIa
d2
)2cos(1
r4
sinIa
dcossinr4
Ia
Taylor series
방정식 (8.21a) 유도
![Page 26: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/26.jpg)
2525
cossinar2ar
)cosr()sina()cosasinr(
)cosr()sinasinsinr(
)cosacossinr(R
)0,sina,cosa(
)cosr,sinsinr,cossinr(R
22
222
22
22
방정식 (8.21a) 유도
x
y
z ),,r(P
Ld
I
r
am R
x
y
z
)0,,r(P
LdI
ra
R
![Page 27: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/27.jpg)
2626
xy
xy
asinacos
a)2/(coaacosa
x
y
a
ya
xa
xa
방정식 (8.21a) 유도
![Page 28: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/28.jpg)
2727
cossinr
a1
cossinr
a2)1(f
!1
1)1(f
cossinr
a21)xx(f
cossinr
a2x
2
1)1(fx
2
1)x(f
1)1(fx)x(f
)x)(x(f!2
1x)x(f
!1
1)x(f)xx(f
2/1
2/3
2/1
2
Taylor series
방정식 (8.21a) 유도
![Page 29: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/29.jpg)
2828
막대 자석도 magnetic dipole.
![Page 30: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/30.jpg)
2929
Electric Magnetic
V =Q
4πε0r
E =Qar
4πε0r2 Monopole does not exist.
V =Qcosθ
4πε0r2
E =Qd
4πε0r3 (2cosθar + sinθaθ)
A =μ0msinθar
4πr2
B =μ0m
4πr3 (2cosθar + sinθaθ)
r~a r
Q Monopole(point charge)
P
Dipole(two point charge)
aθ
arθ r
d
+Q
−Q
Dipole
(small current loop
or bar magnet)
aθ
arθ r
d
+Qm
+Qm
Table 8.2 Comparison between Electric and Magnetic Monopoles and Dipoles
![Page 31: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/31.jpg)
3030
T = m × B
= QmL × B
= r × F (8.23)
(*) Qm=isolated magnetic charge.
아직 존재하지 않음.
L= the length of the magnetic bar
F = QmB (8.24)
T = QmLB=ISB (8.25)
where m=ISan
QmLB=ISB (8.26)
electric dipole과 형태가 같다.Im = ISan
)18.8(aISmwhere
)19.8(Bm
)17.8()a(sinBIS
)16.8()a(sinBILwT
)14.8(FrT
n
z
z
Fig 8.8 A bar magnet in an
external magnetic field.
N
S
+Qm
L-QmF
F
𝐁
![Page 32: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/32.jpg)
3131
예제 8.5 Fig 8.9의 삼각형 Loop로 만들어진 전기회로의 magnetic dipolemoment를 구하라.
Figure 8.9 Triangular loop of
Example 8.5.
02zyx
1C
1B
2D
0DC2)2,0,0(f
0DB2)0,2,0(f
0D2)0,0,2(f
0DCzByx)z,y,x(f
2
n
n
mA)10,10,10(
3
)1,1,1(325m
)(3
)1,1,1(
f
fa
32
3sin)22)(22(
2
1
sinba2
1
ba2
1S
aISm)18.8(
방향
a
b
![Page 33: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/33.jpg)
3232
예제 8.6 magnetic dipole moment 𝐦𝟏 = 𝟓𝐚𝐳 𝐀 ∙ 𝐦𝟐 인 미소전류 Loop L1이원점에 놓여 있다. magnetic dipole moment 𝐦𝟐 = 𝟑𝐚𝐲 𝐀 ∙ 𝐦𝟐 인
미소전류 Loop L2가 (4,-3,10)에 놓여있다. L2에 작용하는 Torque를 구하라.
nNma901.0a536.1a384.0
0625/10625/410
5
12
55
18
55
9
aaa
BmT
a5
4a
55
6a
55
33
acosasincosasinsina3m
)aa4(625
10)a
5
1a
5
4(
56254
5104B
)asinacos2(r4
mB)22.8(
BmT:TorqueLBL
r
77
r
122
r
ry2
r
7
r
7
1
r30
122211
받는가의해에생긴의해에
x
y
z
)10,3,4(P
55r
1m
2m5
![Page 34: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/34.jpg)
3333
)24.2(
acosasina
asin
asincos
acoscosa
acos
asinsin
acossina
)23.2(
asinacosa
acos
asincos
asinsina
asin
acoscos
acossina
yx
z
y
x
z
y
xr
rz
ry
rx
ρ
z
x
y
θ
φ
rz = rcosθ
y = ρsinφx = ρcosφ
ar
aθ
aφ
![Page 35: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/35.jpg)
3434
)22.2(
cosrz
sinsinry
cossinrx
)21.2(
x
ytan
z
yxtan
zyxr
1
221
222
ρ = rsinθ = x2 + y2
z
x
y
θ
φ
rz = rcosθ
y = ρsinφx = ρcosφ
![Page 36: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/36.jpg)
3535
x
y
z
)10,3,4(P
55r
1m
y2 a3m
5
a5
4a
5
6a
55
33
a5
43a
5
2
5
33a
5
1
5
33
acos)acosa(sinsin3
a3m
5
4xcos
5
3ysin
5
1
55
5
rsin
5
2
55
10
r
zcos
r
r
r
y2
y2 a3m
x
y
asin3
acos3
z
a
rasin
acos
![Page 37: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/37.jpg)
3636
)asinacos2(r4
mB)22.8( r3
0
x
y
z ),,r(P
I
r
am
![Page 38: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/38.jpg)
3737
8.5 Magnetization in Materials
Figure 8.10 (a) Electron orbiting around
the nucleus. (b) Electron spin.Figure 8.11 Circular current
loop equivalent.
)27.8(v
m
limM
N
1kk
0
* Magnetization 𝐌 is the magnetic dipole moment per unit volume
)18.8(aISmwhere
)19.8(BmT)(
n
naISm)18.8(
𝐁
𝐚𝐧 𝐈
area=S
contour C
![Page 39: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/39.jpg)
3838
Fig 8.12 Magnetic dipole moment in a volume Δv
(a) before B is applied, (b) after B is applied.
B0 = 0 M = 0
∆v
(a) (b) diamagnetic case
M ∆v
B0
![Page 40: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/40.jpg)
3939
M ≡ χmH ≡ χmB0/μ0 8.34where χm= magnetic susceptibility
B = B0 + μ0M 8.33
B = μ0 1 + χm H
≡ μH (8.35)
≡ μ0μrH (8.36)where μr = 1 + χm = μ/μ0
= relative permeability
𝛻 ×B
μ0=
𝜕E
𝜕t+ J
= 𝛻 × H + 𝛻 × M
𝛻 × H ≡ Jf or 𝛻 ×B0
μ0≡ Jf 8.32
(b) diamagnetic case
M ∆v
B0
* Bound (or magnetization)
volume current density
Jb ≡ 𝛻 × M (8.30)
* Bound surface current density
Kb ≡ M × an (8.31)
![Page 41: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/41.jpg)
4040
xx
SdKvd
xx
J
4A
xx
SdaMvd
xx
M
4
AfAf)Af(vdxx
M
xx
1)M(
4
vdxx
1M
4A
vdxx
1M
4
vdxx4
)xx(M
vdR4
aMAd
r4
amA(8.21b)
0n0
0
0
0
30
2R0
2r0
)42.7()43.7(
v
m
limM(8.27)
N
1kk
0
v0
R4
dvJA)43.7(
참고
![Page 42: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/42.jpg)
4141
8.6 자성재료의 분류
Fig 8.13 Classification of materials.
Materials
Diamagnetic
𝛘𝐦 < 𝟎, 𝛍𝐫 ≤ 𝟏. 𝟎Paramagnetic
𝛘𝐦 > 𝟎, 𝛍𝐫 ≥ 𝟏. 𝟎Feromagnetic
𝛘𝐦 ≫ 𝟎, 𝛍𝐫 ≫ 𝟏. 𝟎
Linear Nonlinear
![Page 43: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/43.jpg)
4242
반자성체 (diamagnetic substance)
- 외부 자기장에 의해서 자기장과 반대 방향으로 자화되는 물질
- 금, 은 등의 금속, 산소를 제외한 대부분의 기체, 유기물질·염류·물·유리 등
상자성체 (paramagnetic substance):
- 자기장 안에 넣으면 자기장 방향으로 약하게 자화하고, 자기장이 제거되면 자화하지 않는 물질.
- 알루미늄·주석·백금·이리듐 외에 산소·공기 등.
강자성체 (ferromagnetic substance)
- 외부에서 강한 자기장을 걸어주었을 때 그 자기장의 방향으로 강하게 자화된 뒤 외부 자기장이 사라져도 자화가 남아 있는 물질.
- Ni, Fe, Co 등.
![Page 44: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/44.jpg)
4343
Figure 8.14 Magnetic screening:
(a) iron shield protecting a small compass,
(b) compass gives erroneous reading without the shield.
Iron shield
![Page 45: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/45.jpg)
4444
강자성체의 𝐁 = 𝐁𝟏 + 𝐁𝟎
𝐁𝟎~𝐈 (주어진 외부 자장)
Hysteresis loop 𝐁𝟎, 𝐁𝟎 = 𝛍𝐍𝐈/𝐋
𝐁𝟏 𝐁 = 𝐁𝟏 + 𝐁𝟎
Gaussmeter
FeI
![Page 46: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/46.jpg)
4545
I
B
I
B
전자
V
![Page 47: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/47.jpg)
4646
예제 8.7 0≤z≤2인 무한 slab (μr= 2.5) 이 있다.
B = 10yax − 5xay mWb/m2 이라 할 때 z=0 에서 다음을 구하라.
(a) J (b) Jb (c) M (d) Kb
풀이
23
7
r0
3
zyx
r0
r0
m/A10)775.4,0,0(
)15,0,0(5.2104
1
)15,0,0(1
10
0x5y10
zyx
aaa
1
BHJ(a)
m/kA)0,775.4,387.2(
)1,0,0()0,387.2,775.4(
aMaMK)31.8()d(
m/kA)0,387.2,775.4(
105.2104
)0,5,10(5.1
BHM)c(
m/kA)163.7,0,0(
)4775,0,0)(15.2(
J)1(J)H(MJ(b)
znb
3
7
r0mm
2
rmmb
![Page 48: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/48.jpg)
4747
θ2
1
2
B1n
B1t
B1
θ1
B2n
B2t
μ2
μ1
Δh
a12
Δw
K
B2
θ2
1
2
B1n
B1t
B1
θ1
B2n
B2t
μ2
μ1
Δh
ΔS
B2
8.7 Magnetic Boundary Conditions
)44.8(KBB
)43.8()0h(KHH
)42.8(2
hH
2
hHwH
2
hH
2
hHwHwK
)41.8(HHBB
)40.8()0h(0SBSB
)39.8(wKI Ld Hxd)JH(
)38.8(0xd Bxd)0B(
2
t2
1
t1
t2t1
n1n2t2
n2n1t1
n22n11n2n1
n2n1
S2
23v
![Page 49: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/49.jpg)
4848
θ2
1
2
B1n
B1t
B1
θ1
B2n
B2t
μ2
μ1
Δh
a12
Δw
K
B2)48.8(tan
tan
)47.8/()48.8(
)48.8(sinB
HHsinB
)47.8(cosBBBcosB
)46.8(BB
orHH
0Kfor
)45.8(Ka)HH(
Ka)HHHH(
)43.8(KHH
2
1
2
1
22
2t2t11
1
1
22n2n111
2
t2
1
t1t2t1
12n21
12nn2t2n1t1
t2t1
θ2
1
2
B1n
B1t
B1
θ1
B2n
B2t
μ2
μ1
Δh
ΔS
B2
![Page 50: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/50.jpg)
4949
t2t1
n2n1
t2t1
n2n1
EE
DD
HH
BB
1
2
μ2 μ1
01-2
1
2
y
x
an
![Page 51: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/51.jpg)
5050
예제 8.8 y-x-2≤0 인 영역에서 자기장 𝐇𝟏 = −2,6,4 A/m 이다. 𝛍𝐫𝟏 = 5 이다.
2
7
11r01
11r
1m1
m
11
m/Wb)13.25,7.37,57.12(
)4,6,2(5104
HB
m/A)16,24,8(
)4,6,2)(15(
H)1(
HM
HM)34.8(
B&M)a(
Fig 8.17 For Example 8.8.
1
2
μ2 = 2μ0 μ1 = 5μ0
01-2
1
2
y
x
an
![Page 52: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/52.jpg)
5151
222r02
n2t22
n12
1n2n1n2
t1t2
n11t1
nn1n1
t2t1
n2n1
n
222r
m/Wb)05.10,1.30,11.20(HB
m/A)4,12,8()0,10,10()4,2,2(HHH
)0,10,10()0,4,4(2
5HHBB
)4,2,2(HH
)4,2,2()0,4,4()4,6,2(HHH
)0,4,4(2
)0,1,1(
2
)0,1,1()4,6,2(
a)aH(H
HH)46.8(
BB)41.8(
2
)0,1,1(
f
fa
B&H02xy2)b(
영역에서인때일
예제 8.8 y-x-2≤0 인 영역에서 자기장 𝐇𝟏 = −2,6,4 A/m 이다. 𝛍𝐫𝟏 = 5 이다.
1
2
μ2 = 2μ0 μ1 = 5μ0
01-2
1
2
y
x
an
Fig 8.17 For Example 8.8.
![Page 53: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/53.jpg)
5252
2
)0,1,1(
f
fa
2f
)0,1,1(f
xyfor
2xyf
n
![Page 54: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/54.jpg)
5353
예제 8.9 Fig 8.18에서 𝐊 = 0,1/μ0 , 0 mA/m 이고 𝐁𝟐 = 5,0,8 mWb/m2 일 때
𝐇𝟏과 𝐁𝟏을 구하라.
)0,12,0()1,0,0()8,B2,15B2(
)0,1
,0()1,0,0()8,0,5(4
1)8,B,B(
6
1
Ka)HH()45.8(
)8,B,B(6
1BH
)8,0,5(4
1/BH
)8,B,B(B
)8,0,0(BB
Ka)HH()45.8(orKHH)43.8(
BB)41.8(
)1,0,0(a
y1x1
00y1x1
0
12n21
y1x101
11
0222
y1x11
n2n1
n21t2t1
n2n1
12n
Fig 8.18
an12 = a𝑧
1
2
μr2 = 4
μr1 = 6
z
y
𝐁𝟐 = 5,0,8 mWb/m2
𝐊 = 0,1/μ0 , 0 mA/m
![Page 55: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/55.jpg)
5454
m/mA)2,0,25.1(1
/BH
m/mWb)8,0,5.1(B
)0,12,0()0,15B2,B2(
)0,12,0(
100
8B215B2
aaa
)0,12,0()1,0,0()8,B2,15B2(
0111
21
x1y1
y1x1
zyx
y1x1
Fig 8.18
an12 = a𝑧
1
2
μr2 = 4
μr1 = 6
z
y
𝐁𝟐 = 5,0,8 mWb/m2
𝐊 = 0,1/μ0 , 0 mA/m
![Page 56: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/56.jpg)
5555
8.8 Inductors and Inductances
cetaninducmutual:
I/
dI
dMwhere
dt
dIM
dt
d
V
cetaninducself:
IdI
dL
dSnBSdBwhere
dt
dIL
dt
dI
dI
d
dt
d
121
1
2121
121
2121
전압기전력
![Page 57: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/57.jpg)
5656
S
N
𝐁
𝐚𝐧 𝐈
area=S
contour C
![Page 58: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/58.jpg)
5757
𝐁~𝐇
𝐑~𝐚𝐑
𝐝 𝐋I
P
𝛂
IdI
d
cstS
I~B
(7.8)current) (volume R4
advJH
(7.7)current) (surface R4
adSKH
(7.6)current) (line R4
aLdIH
)5.7( dvJdSKdLI
R4
RLdI
R4
aLdIHd)4.7(
V 2R
S 2R
L 2R
32R
![Page 59: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/59.jpg)
5858
Fig 8.19 Magnetic field B
produced by a circuit.
I I
𝐁
N=권선수
)54.8(I
W2L
energyinductorWwhere
)53.8(]Joule[LI2
1W
)52.8(]HA/Wb[I
N
IL
cetanInduc:Lwhere
)51.8(LI
I~
:Nwhere
)50.8(N
SdB
IdI
dL
2m
m
2m
저장된에
권선수
0692
![Page 60: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/60.jpg)
5959
Fig 8.20 Magnetic interaction
between two circuits.
)59.8(I
N
IL
)58.8(I
N
IL
)b57.8(MM
)a57.8(I
N
IM
)56.8(I
N
IM
)55.8(SdB
2
22
2
22
1
11
1
111
2112
1
212
1
2121
2
121
2
1212
S 2121
Neumann 공식
![Page 61: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/61.jpg)
6060
Neumann 공식
그림 5.7 Coil C1, C2가 있고 각 Coil에는 전류 I1, I2가흐른다. ψ1, ψ2는 각 Coil에 대한 Magnetic Flux이다. Coil C1, C2가 둘러 싼 면적은 각각 S1, S2이다. 전류 I1
에 의해 C2에 Magnetic Flux ψ21이 생겼다.
Coil 1의 전류 I1에 의한 Coil 2의 Magnetic Flux를 ψ21, 기전력을 φ21이라 하고 Mutual Inductance M21를 정의 한다.
)4(I/
)3(dI
dMwhere
)2(dt
dIM
)1(dt
d
121
1
2121
121
2121
![Page 62: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/62.jpg)
6161
Neumann 공식을 유도하기 위해 전류가흐르는 Coil C2의 I2가 생성하는 자장의Vector Potential을 구한다.
)8(r
Ld
4
I
)7(xdxx
)x(J
4)x(A
)6(xdxx
)x(J
4)x(A
)5(xdxx
)x(J
4
LawtvarSaBiotxdxx
)xx()x(J
4)x(B
2C22
322
322
32
3
322
![Page 63: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/63.jpg)
6262
Neumann 공식에서 첨자 1과 2를 바꾸어도 값이 같다. 다음의결론을 얻는다.
)10(r
LdLd
4Mwhere
)9(IM
r
LdLd
4
I
Ldr
Ld
4
I
LdA
SdA
SdB
1 2
1 2
1 2
1
1
1
C C12
12
212
C C122
C 1C22
C 12
S 2
S 212
방정식 (8)을 이용하여 Mutual Inductance를 구한다. Coil C2에 의해 Coil C1을 통과하는 Magnetic Flux는 다음과 같다. 방정식 (10)을Neumann 공식이라고 한다. C1, C2
는 Integral Path이다.
)11(MM 1221
![Page 64: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/64.jpg)
6363
)60.8(IIM IL2
1 IL
2
1
W W W W
21122
222
11
1221m
)62.8(CL
RC)35.6(
)61.8(L L L
ext
extin
* Step to find self inductance- Choose a suitable coordinate system, Let the inductor carry current I- Determine B (Biot-Savart’s law or Ampere’s law)
- Calculate 𝛙 = 𝐁 ∙ 𝐝 𝐒 Find 𝐋 = 𝛌/𝐈 = 𝐍𝛙/𝐈
도체 내부 자속에 의해 생기는 Inductance
도체 외부 자속에 의해 생기는 Inductance
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6464
TABLE 8.3 A Collection of Formulas for Inductance of Common Elements.
a
bln
2L
conductorCoaxial.4
ad,d
a
dln
2L
wiresParallel.3
a
1a
2ln
2L
cylinderHollow.2
8L
Wire.1
0
0
0
0
b2 a2
5.0tb
2ln2L
Sheet.8
a[NL
Torus.7
SNL
Solenoid.6
d,2
45.2a
4ln
2L
loopCircular.5
0
2200
20
20
00
0
0
d
a2
t
a b
a
0
a2
a2
d
a2
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6565
)66.8(dvH2
1dvHB
2
1W
)65.8(B
2
1
H2
1
v
Wlimw
vH2
1
zyxH2
1IL
2
1W
yHI
)63.8(I
yxH
IL
)53.8(]Joule[LI2
1W
dv E2
1 dv ED
2
1 W)96.4(
2m
2
2m
0vm
2
22m
2m
2E
Δx
Δy
Δz
x
y
z
B
ΔI
ΔI도체판
Fig 8.21 A differential volume
in a magnetic field.
8.9 Magnetic Energy
![Page 67: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/67.jpg)
6666
yHI
yH
ILdH
Sd)JH(
JH
Δx
Δy
Δz
x
y
z
B
ΔI
ΔI도체판
Δy
y
zΔI
B
![Page 68: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/68.jpg)
6767
예제 8.10 무한히 긴 solenoid의 단위 길이당 inductance를 구하라.길이 w, 반경 a, 권선수 N 인 solenoid에 전류 I가 흐른다.
m/HSnw/SNw
LL
w/SNI
L
w/SINN
w/INSBS
:
w/Nnwhere
w/INInHB
222
2
2
권선수길이당단위
InB
wB
wInLdB
Sd)JB(
JB
Fig 7.9 Cross section of solenoid.
θ2
θ
P
dz I
w
z
a θ1
![Page 69: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/69.jpg)
6868
예제 8.11 내부 반경이 a이고 외부 반경이 b 인 동축 선로의self-inductance를 구하라.
Fig 8.22 예제 8.11의 동축 케이블의 단면(a) 영역 1, 0<ρ<a (b) 영역 1, a<ρ<b
(b)(a)
1
2
ab
ρI
I
z-axis
a
b
ρ
dρII
𝐁
도체
절연체
도체
dρ
![Page 70: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/70.jpg)
6969
d
z
1
2
z-axis
2
1
![Page 71: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/71.jpg)
7070)2.11.8(]m/H[
8w
LL
)1.11.8(8
w
IL
8
Iwdzd
a2
I
dzdaa2
I
dzda
Ba
dI
Idd
I:
a0dzda2
IdzBdd
baa2
I
a0aa2
I
B)29.7(
inin
1in
a
0
w
0z 4
3
1
2
2
2
2
2
2
2enc
1
2
2
분포균일는단면에서가정
(방법 𝟏)
𝐁 𝐛𝐲 𝐈𝐞𝐧𝐜
2
2
enca
II
dzd
a
z-axis
a
b
ρ
dρII
1
2
Nρ에 의해 생긴 자장이
자기자신을 통과하는 비율.
![Page 72: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/72.jpg)
7171
]m/H[a
bln
4
1
2
w
LL
w
LL
a
bln
2w
LL
a
bln
2
w
IL
a
bln
2
Iw
dzd2
I
extin
extext
2ext
b
a
w
0z22
baa2
I
a0aa2
I
B)29.7(2
1
2
ab
ρ
dρ
I
I
z-axis
![Page 73: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/73.jpg)
7272
a
d
2
2enc
2
2
enc
2
2
aI
I
N
N
aII
aNN
NNa
d dzd
a2
IdzdBd
211
z
baa2
I
a0aa2
I
B)29.7(2
1
2
![Page 74: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/74.jpg)
7373
(방법 𝟐)
8
w
d4
I1ddz
I
1
dB1
ddzI
1
xdHBI
1L
I
W2L)54.8(
xdHB2
1LI
2
1W
baa2
I
a0aa2
I
B)29.7(
a
03
2
222
0
w
02
a
022
0
w
02
3
2in
2m
32m
2
a
bln
4
1
2
wLLL
a
bln
2
w
d4
Iddz
I
1
dBddzI
1
xdB
I
2L
extin
b
a 22
222
0
w
02
b
a22
0
w
02
32
2ext
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7474
예제 8.12 간격이 d 인 2선식 선로의 단위 길이당 self-inductance를 구하라.
a
adln
4
1
w
LL
2~)2()1(byfluxmagnetic
LIm/Ha
adln
4
1
2
Iw
)1(byfluxmagnetic
a
adln
2
Iwdzd
2
I
SdB
11.8exfrom8
Iw
ad
a
w
0z
22
1
baa2
I
a0aa2
I
B)29.7(2
(방법 𝟏)
IdI
dL
![Page 76: 8. Magnetic Forces, Materials, and Deviceiirc.khu.ac.kr/uploads/6/3/4/3/63434825/sadiku_ch08.pdf · 2019-11-23 · 22 Table 8.1 Force on a Charged Particle State of particle field](https://reader033.vdocuments.site/reader033/viewer/2022042018/5e764f6644e9c06d605f75b3/html5/thumbnails/76.jpg)
7575
(방법 𝟐)
a
adln
4
1w)LL(2L
2
a
bln
4
1
2
wLLL
a
bln
2
wL
8
wL
11.8exfrom
extin
extin
ext
in
개도선이
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7676
Fig 8.23 Two coaxial circular wires;
for example 8.13.
예제 8.13 Fig 8.23의 2 coil의 mutual inductance를 구하라. (h ≫ a, b)
3
22
1
2121
3
221
3
21
21212121
3
21
2/322
21
1
2
2
h2
ba
IM
h2
baI
b2h4
baI
LdASdASdB
ah4
baI
a)bh(4
baI A
arwherear4
sinIa A(8.21a)
1
21
1
2121
IdI
dM
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8.10 MAGNETIC CIRCUIT
* Magnetomotive force (기자력) F [ampere-turns]
* Reluctance (자기저항) R [ampere-turns/weber], Permeance P=1/R
* Apply Ohm’s law (V=IR)
H
Ld,I
)67.8(LdHNI
NILdHxd)JH(
JHt
DJH
2
)68.8(S
R
)69.8(R
NINI
BS
BSR
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Fig 8.24 (a) 전기회로와 (b) 자기회로의유사성
(a)
RV
I
ψ
N turn
I
(b)
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)70.8(n321
)71.8(n21
)72.8(n321
* For n magnetic circuit elements in series
* For n magnetic circuit elements in parallel
)73.8(n21
ψ
Φ1
Φ2
Φ3
Φ
ψ1 ψ2 ψ3
ψ
ψ
ΔΦ
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8.11 FORCE ON MAGNETIC MATERIALS
)75.8(2
SB 2F
)74.8(dL SB
2
1 2dWdL F
0
2
0
2
m
)77.8(BH2
1
2
B
S
Fp
)76.8(2
SBF
0
2
0
2
Figure 8.25 An electromagnet.
공극이 2 개
1 개의 공극에 미치는 힘
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예제 8.14 ρ0=10 cm, a=1 cm, core는 강철 (μ=1000μ0), coil turns=200 일 때magnetic flux ψ가 0.5 mWb가 되기 위한 전류는?
A100
Na
2I
2
aNIBS
2
NI
w
NI B
1)(
20r
0
0
20r
0
0r
방법
A100
Na
2I
a
2
SRNI
2)(
20r
0
20r
0
방법
(b) 등가 유사 전기 회로
Ra+R3Φ
ψ
ψ
2a
200 turn
I
ρ0
(a) 예제 8.14의 Toroid core
Fig 8.26
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예제 8.15 공극의 자속밀도 B=1.5 Wb/m2 이 되기 위한 coil 전류를 구하라.μ=50μ0, 모든 가지의 단면적은 10 cm2, 400 turns.
Figure 8.27 Magnetic circuit for Example 8.15.
Figure 8.28 Electric circuit analog of the
magnetic circuit in Figure 8.27.
(b)
R1//R2Ra+R3
Φ
경로123R2
경로143R1
경로56Ra
Φ
경로35+16R3
(a)
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A16.44N
RSBI
SB
RNI:
20
104.7RRRRR:
20
105.1
2
R
RR
RRRR
20
105
001.0104
01.0
S
w R
20
109.0
001.010450
09.0
S
w R
20
103
001.010450
3.0
S
w RR
Ta
aa
Ta
8
213aT
81
21
2121
8
70r
a
8
70r
3
8
70r
21
기자력
총자기저항
Figure 8.28 Electric circuit
analog of the magnetic circuit
in Figure 8.27.
Figure 8.27 Magnetic circuit for Example 8.15.
(b)
R1//R2Ra+R3
Φ
R2R1 Ra
Φ
R3
(a)
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213aT RRRRR: 총자기저항
R2
R3
R1 Ra
Φ
경로123R2
경로143R1
경로56Ra
Φ
경로35+16R3
R1//R2Ra+R3
Φ
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예제 8.16 400 kg을 들어 올리기 위한 coil turn N은?μ=3000μ0, 모든 가지의 단면적은 40 cm2,
평균 길이 50 cm, 공극 0.1mm, 전류 1 A.
NI11
6
RR
R
48
105
004.01043000
5.0
S
wR
48
106
004.0104
102
S
w2R
)RR(NI
m/Wb11.1B
004.0
1048.9400
S
mgB
mg2
SB2F
2
SBF)76.8(
ia
aa
6
7a
i
6
7
4a
a
ia
2a
72a
2a
2
기자력
Figure 8.29 U-shaped
electromagnet; for
example 8.16.
길이 0.5 m
weight 400 kg
I= 1 A
10-4 m
N turn
Iron(μ=3000μ0)
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NI11
6
RR
R
48
105
004.01043000
5.0
S
wR
48
106
004.0104
0001.02
S
w2R
)RR(NI
m/Wb11.1B
004.0
1048.9400
S
mgB
mg2
SB2F
2
SBF)76.8(
ia
aa
6
7a
i
6
7a
a
ia
2a
72a
2a
2
기자력
162
9.161
B
6
11N
BHNI)67.8(
BH
NI11
6
RR
R
m/Wb11.1B
0
aa
a
aaaa
0
aaaa
ia
aa
2a
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8.12 응용 노트-자기 부상
Figure 8.30 A levitated object.
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Figure 8.31 Maglev train.