30. 유도와 유도용량
(Induction and Inductance)
B
V
dt d Φ
B−
ε =
Faraday 법칙
i L ≡ N Φ
B유도용량 (Inductance)
dt L dI dt
N d B
L = − Φ = −
ε
L: Self-Inductance유도기전력 (Induced emf)
지난 시간에 …
Ampere’s Law
i
encs d
B ⋅ = μ
0∫ r r
Biot-Savart Law
3 0
4 r
r s
B id
d r = r × r
π μ
b a
ba
i i
d F L
π μ
2
=
0ni B = μ
0Force by currents
Solenoid
(Coulomb’s Law)
r r dq E πε
d
o
4 ˆ 1
=
2o
qencl
A d
E ⋅ =
ε
∫
r r (Gauss’ Law)30. 유도와 유도용량
(Induction and Inductance)
Faraday의 두 가지 실험
도선 주위에서 자석을 움직여 자기장을 변화시킴
자석 대신 다른 도선 고리를 두고, 전류를 조절하여 자기장을 변화시킴
자기유도 (Magnetic induction) 유도전류 (유도기전력) 발생
Inside the shaded region, there is a magnetic field into the board.
– If the loop is stationary, the Lorentz force predicts:
(a) A Clockwise Current (b) A Counterclockwise Current (c) No Current
× B
( )
F r
m= q E v B r + × r r
When the loop is not moving,
there is neither E nor v, so Fm = 0.
Æ no motion of charge and no current.
유도효과 (induction effect)
Now the loop is pulled to the right at a velocity v.
– The Lorentz force will now give rise to:
×
B v
(a) A Clockwise Current
(b) A Counterclockwise Current (c) No Current, due to Symmetry
The charges in the wire all experience an upward force those in the end segment produce a clockwise current.
v B r × r
I
유도효과 (induction effect)
Fm
1831: Michael Faraday did the previous experiment, and a few others:
Move the magnet, not the loop.
Here there is no Lorentz force but there was still an identical current !!!
v B r × r
×
-v B I
× B
Decreasing B↓
I Decrease the strength of B.
Now nothing is moving,
but M.F. still saw a current !!
This “coincidence” bothered Einstein, and eventually led him to the
Special Theory of Relativity (all that matters is relative motion).
I dB
∝ dt
유도효과 (induction effect)
30-3. Faraday의 유도법칙
도선 고리를 지나는 자기력선의 수가 시간에 따라 변하면 고리에 기전력(
ε
: electromotive force)이 생긴다“Electric current is induced by the change of the Magnetic Flux”
Magnetic Flux ΦB =
∫
B⋅dAr [Unit] weber (Wb), 1 Wb = 1 T·m2dt d Φ
B−
ε =
Faraday 법칙30-3. Faraday의 유도법칙
∫
⋅=
ΦB B dAr
) cos A B dt (
d ⋅ θ
−
ε
=dt N dΦB
−
ε
=For N loops
고리를 지나는 자기력선의 수를 바꾸는 방법
▸코일 속 자기장 세기 B 를 바꾼다. ( B )
▸자기장 속에서의 코일의 넓이를 바꾼다. ( A )
▸자기장의 방향에 대한 코일의 방향을 바꾼다. (θ).
▸위 셋의 복합.
30-4. Lenz 법칙
dt d Φ
B−
ε =
The magnetic field due to the induced current has opposite sign to the change in the magnetic flux.
자기유도로 생기는 전류는 자기다발의 변화를 방해하는 방향으로 흐른다.
Lenz's Law and energy consideration
• Lenz's Law:
The induced current will appear in such a direction that it opposes the change in flux that produced it.
• Conservation of energy considerations:
Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated.
• Why???
– If current reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc..
v B
S N
v B
N S
N S S N
전기기타 (Electric guitar)
High frequency Low frequency Toggle switch
확인문제 2. 자기장이 일정한 비율로
증가/감소할 때,
고리에 유도되는 전류의 크기가 큰 순서는?
a = b > c=0
보기문제 30-3.
t = 0.10s 에서 전류고리에 생기는 유도기전력의 크기와 방향은?
크기: emf (t = 0.10s) = 14.4 V 방향: 반시계 방향
30-5. 유도와 에너지 전달
F
(v=일정)
B = BLx Φ
dt BLv BL dx
dt d B
−
=
− Φ =
−
ε
=R BLv i =
ε
R =R v L B B
iL B
L i Fm
2
= 2
⋅
=
×
= r r F Fr r
−
1 = Fr2 Fr3
−
= R
v L F B
2
= 2
∴
R v L Fv B
P
2 2
= 2
=
일률 (Power):
힘 F 가 한 일률 (power)
전류 i 에 의한 열에너지 방출률 (power)
R v L R B
i P
2 2 2
2 =
=
전류고리를 당길 때 힘 F가 한일은 모두 열에너지로 변환된다.
MRI 주사중 일어날 수 있는 화상
(산소농도계)
Magnetic resonance image(MRI) 장치에서 가해지는 변화하는 자기장
Loop current 발생 가능성 있음!
소용돌이 전류 (Eddy current, or Foucault current)
Roller coaster의 braking
Vending machines (detection of coins)
http://www.ndt-ed.org/EducationResources/HighSchool/Electricity/eddycurrents.htm
Applications of Magnetic Induction
• AC Generator
– Water turns wheel Æ rotates magnet Æ changes flux Æ induces emf Æ drives current
• “Dynamic” Microphones
(E.g., some telephones) – Sound
Æ oscillating pressure waves Æ oscillating [diaphragm + coil]
Æ oscillating magnetic flux Æ oscillating induced emf Æ oscillating current in wire
Question: Do dynamic microphones need a battery?
More Applications of Magnetic Induction
• Tape / Hard Drive / ZIP Readout
– Tiny coil responds to change in flux as the magnetic domains (encoding 0’s or 1’s) go by.
• Credit Card Reader
– Must swipe card
Æ generates changing flux – Faster swipe Æ bigger signal
More Applications of Magnetic Induction
• Magnetic Levitation (Maglev) Trains
– Induced surface (“eddy”) currents produce field in opposite direction Æ Repels magnet
Æ Levitates train
– Maglev trains today can travel up to 310 mph
Æ Twice the speed of Amtrak’s fastest conventional train!
– May eventually use superconducting loops to produce B-field Æ No power dissipation in resistance of wires!
N S
rails
“eddy” current
30-6. 유도 전기장
(Faraday 법칙)
변하는 자기장은 전류고리에 유도전류를 만든다
변하는 자기장은 전기장을 만든다
Assume the magnetic field changes in time
ε
= − Φ ≠ 0 dtd B
dt s d
d
E⋅ = − ΦB
=
∫
ε
r r Faraday’s Lawq0
W =
ε
∫
∫
⋅ = ⋅= F ds q E ds
W r r r r
0
∫
⋅= E rr ds
ε
: 전하 q0 가 한 바퀴 돌 때 유도기전력이 한 일 : 전하 q0 가 한 바퀴 돌 때 유도전기장이 한 일
Electro-Motive Force or emf
A magnetic field, increasing in time, passes through the blue loop An electric field is generated “ringing” the increasing magnetic field
Circulating E-field will drive currents, just like a voltage difference Loop integral of E-field is the “emf”
time
∫ •
= E d l
ε r r
Note: The loop does not have to be a wire Æ the emf exists even in vacuum!
Æ When we put a wire there, the electrons respond to the emf Æ current.
Note: In electrostatics , so we could define a potential
independent of path. This holds only for charges at rest (electrostatics).
Forces from changing magnetic fields are nonconservative, and no potential can be defined!
= 0
∫
Er • dlr정전기장(static E)과 유도전기장 (Induced E)이 다른 점
양(+) 전하에서 시작하여 음(-) 전하에서 끝남.
경로에 무관 : conservative force 닫힌 고리에서는 V = 0 임.
닫힌 고리임에도 불구하고 변하는 자가 다발에서는 V=0 이 아님.
경로에 의존 : nonconservative force Æ유도전기장에서는 전기퍼텐셜을
정의할 수 없다.
Escher depiction of nonconservative emf
• The magnetic field in a region of space of radius 2R is aligned with the z-direction and changes in time as shown in the plot.
– What is the direction of the induced electric field around a ring of radius R at time t=t1?
• There will be an induced emf at t=t1 because the magnetic field (and therefore the magnetic flux) is changing.
It makes NO DIFFERENCE that at t=t1 the magnetic field happens to be equal to ZERO!
• The magnetic field is increasing at t=t1 (actually at all times shown!) which induces an emf which opposes the corresponding change in flux. Electric field must be induced in a clockwise sense so that the current it would drive would create a magnetic field in the -z direction.
x x x x x x x x x x x x x x x x x x
x x x x x x x x x x
t Bz
t1
x y
R
(a) E is ccw (b) E= 0 (c) E is cw
• Note:
There does not need to be an actual wire –
the electric field exists anyway
확인 질문
• The magnetic field in a region of space of radius 2R is aligned with the z-direction and changes in time as shown in the plot.
– What is the relation between the magnitudes of the induced electric fields ER at radius R and E2R at radius 2R?
(a) E2R= ER (b) E2R= 2ER (c) E2R= 4ER
• The rate of change of the flux is proportional to the area:
dt πR dB dt
dΦ
B= −
2R E ∝
• The path integral of the induced
electric field is proportional to the radius:
∫ E r • d l r = E ( πR 2 )
x x x x x x x x x x x x x x x x x x
x x x x x x x x x x
t Bz
t1
x y
R
확인 질문
확인문제 4.
경로 : (1)
ε
, (2) 2ε
, (3) 3ε
, (4) 0 일 때, 각 구역에 걸린 자기장의 방향은?∫
⋅= E rr ds
ε
(4) c 와 e 는 반대 방향, (2) d와 e 는 같은 방향, (3) b와 c는 a와 같은 방향
a : out
b, c : out
d, e : into
유도기 유도기 ( ( Inductor Inductor ) )
An inductorAn inductor is a two-is a two-terminal device that consists of terminal device that consists of a a coiled conducting wire wound around a core coiled conducting wire wound around a core
A current flowing through the device produces A current flowing through the device produces a magnetic flux
a magnetic flux φφBB forms closed loops threading its coils forms closed loops threading its coils
Total flux linked by Total flux linked by NN turns of coils, turns of coils, total flux istotal flux is ΦΦ = = NNφφBB
For a linear inductor, For a linear inductor, ΦΦ is proportional to iis proportional to i ΦΦ = Li= Li ÎÎ L= L= ΦΦ /i/i
LL is the inductanceis the inductance
Unit: Henry (H) or (V•Unit: Henry (H) or (V•s/A)s/A)
i +
_
v N
NφφBB
30-7. 유도기(inductor)와 유도용량 (inductance)
유도용량 (inductance)
i L N Φ
B≡
인덕터의 유도용량 (Inductance)
[1 Henry = 1 H = 1 T⋅m2/A]
A in nl
BA nl
NΦB = ( ) = ( )(
μ
0 )솔레노이드의 중심 부근 길이 l 인 부분의 자기다발
lA i n
L ≡ NΦB =
μ
0 2A l n
L 2
μ
0= : 단위 길이당 유도용량
30-8. 자체유도 (Self-inductance)
B-field: B = μ
0ni
dt di l
A N
dt A n di N
dt N
d B
L
2 0
0
) (
μ μ
ε
−
=
⋅
−
=
− Φ
=
dt L di dt
N d B
L = − Φ = −
ε
L: Self-Inductance자체유도 기전력 ( ε
L) : ε
Lε
L : Against the increase of the currentA l n
L 2
μ
0=
유도기의 연결
Inductance in Series: L = L1 + L2
Inductance in Parallel:
L1
L2 i1
i2
i i
dt L di
dt L di
dt L di
L
−
=
−
=
−
ε
=2 2
1 1
2 1
2 1
L L
dt di dt
di dt
di L
L L
L
ε ε
ε
= = + = − −−
2 1
1 1
1
L L
L = +
⇒
2
1 i
i i = +
(저항연결인 경우와 같음)
30-9. RL 회로
0 0
= +
−
−
= +
−
−
ε ε ε
dt L di iR
iR
Lε
= dt + Ri
L di
) 1
( e t/ L i =
ε
R − − τR ,τL = L
(a) 연결시 Æ 전류의 증가
(b) 단절시 Æ 전류의 감소
= 0 dt + Ri
L di
L
L t
t i e
R e
i
ε
τ /τ0
/ −
− =
=
i
ε
/Rt
/ c
e−t τ
− 1 1
/ c
e−t2 τ
t1 t2
ε
Rε
Lε
=0ε
Li
확인문제 6.
(a) 스위치가 닫힌 직후 전류가 큰 순서는?
(a) 2 > 3 > 1 (=0)
(b) 2 > 3 > 1
(b) 오랜 시간 후
전류가 큰 순서는?
30-10. 자기장에 저장된 에너지
In LR Circuit
ε
+−
R
dt L
L di iR+
ε
=dt Li di dt
dUB =
2
1 Li 2 U
B=
dt Li di R
i
i
ε
= 2 +Power:
Cf ) Energy stored in a capacitor
2
1 CV2 UC =
Inductor의 일률
30-11. 자기장의 에너지 밀도
In a solenoid (Magnetic Energy) A l
A l n L =
μ
0 2 ⋅(
0 2)
22
1 n l A i UB =
μ
⋅ ⋅n i B
ni B
0
0
μ
μ
⇒ ==
l
B B V
A B l
U 0
0 2
0 2
2 2
μ
⋅ =μ
=
0 2
2
μ
uB = B (Magnetic energy density)
Cf) 0 2
2
1 E
uE = ε (Electric energy density)
2 magnetic
0
1 2 u B
= μ
… energy density ...
Energy stored in an inductor ….
B Energy stored in a capacitor ...
2 electric 0
1
u = 2 ε E
… energy density…
E
How we can store the energy?
How we can store the energy?
dielectric
+ + + + + + + +
- - - - - - - -
) ( )
( t
21Li
2t
M
=
E
) ( )
( t
21Cv
2t
E
=
E
30-12. 상호유도 (Mutual induction)
상호유도용량 (mutual inductance) : M
1 21 2
21
i
M N Φ
≡
전류 i1이 흐르는 코일-1 에 의한 코일-2 의 상호유도용량
2
21 2
1 21 21
2 1
21
= Φ ⇒ = Φ = − ε
dt N d
dt M di
N i
M
전류 i2 이 흐르는 코일-2 에 의한 코일-1 의 유도용량
dt M
12di
21
= −
ε
dt M
21di
12
= −
ε
M M
M
12=
12≡
dt M di
21
= −
ε
dt M di
12
= −
ε
30. Summary
dt s d
d
E ⋅ = − Φ
B= ∫
ε r r Faraday’s Law (Lenz’s Law) Eddy current
A l n
L 2
μ
0= : 단위 길이당 유도용량
2
1 Li 2 U
B=
0 2
2 μ u
B= B
Magnetic energy
dt L di dt
N d B
L = − Φ = −