30. 유도와 유도용량(Induction and Inductance)

B

V

dtd BΦ

−=ε

Faraday 법칙

iNL BΦ

≡

유도용량 (Inductance)

dtdIL

dtdN B

L −=Φ

−=ε L: Self-Inductance

유도기전력 (Induced emf)

지난 시간에 …

Ampere’s LawencisdB 0μ=⋅∫

rr

Biot-Savart Law3

0

4 rrsidBdrrr ×

=πμ

baba iidLF

πμ2

0=

niB 0μ=

Force by currents

Solenoid

(Coulomb’s Law)rrdq

πεEd

o

ˆ4

1 2=

o

enclqAdEε

=⋅∫rr

(Gauss’ Law)

30. 유도와 유도용량(Induction and Inductance)

Faraday의 두 가지 실험

도선 주위에서 자석을 움직여자기장을 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

( )mF q E v B= + ×r r rr

When the loop is not moving, there is neither E nor v, so Fm = 0.

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×

rr

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×rr

×

B-v I

×

B

Decreasing B↓

IDecrease 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).

dBIdt

∝

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 ∫ ⋅=Φ AdBB

r[Unit] weber (Wb), 1 Wb = 1 T·m2

dtd BΦ

−=ε Faraday 법칙

30-3. Faraday의 유도법칙

∫ ⋅=Φ AdBB

r

)cosAB(dtd

θ⋅−=ε

dtdN BΦ

−=εFor 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

dtd BΦ

−=εThe magnetic field due to the induced current has opposite sign to the change in the magnetic flux.

자기유도로 생기는 전류는 자기다발의 변화를방해하는 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

BS N

vB

N SNSN S

전기기타 (Electric guitar)

High frequency Low frequency

Toggle switch

확인문제 2. 자기장이 일정한 비율로증가/감소할 때, 고리에 유도되는 전류의크기가 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)

BLxB =Φ

BLvdtdxBL

dtd B −=−=Φ

−=ε

RBLv

Ri ==ε

RvLBBiLBLiFm

22

=⋅=×=rr

FFrr

−=1 32 FFrr

−=

RvLBF

22

=∴

RvLBFvP

222

==일률 (Power):

힘 F 가 한 일률 (power)

전류 i 에 의한 열에너지 방출률 (power)

RvLBRiP

2222 ==

전류고리를 당길 때힘 F가 한일은 모두열에너지로 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 wheelrotates magnetchanges fluxinduces emfdrives current

• “Dynamic” Microphones(E.g., some telephones)

–Soundoscillating pressure wavesoscillating [diaphragm + coil]oscillating magnetic fluxoscillating induced emfoscillating 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

generates changing flux–Faster swipe bigger signal

More Applications of Magnetic Induction

• Magnetic Levitation (Maglev) Trains– Induced surface (“eddy”) currents produce field in opposite direction

Repels magnet

Levitates train

–Maglev trains today can travel up to 310 mph

Twice the speed of Amtrak’s fastest conventional train!

–May eventually use superconducting loops to produce B-field

No power dissipation in resistance of wires!

NS

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 0≠Φ

−=εdt

d B

dtdsdE BΦ

−=⋅= ∫ε rrFaraday’s Law

0 qW ε=

∫∫ ⋅=⋅= sdEqsdFW rrrr0

∫ ⋅= sdE rrε

: 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 loopAn electric field is generated “ringing” the increasing magnetic fieldCirculating E-field will drive currents, just like a voltage difference

Loop integral of E-field is the “emf”

time

∫ •= ldEεrr

Note: The loop does not have to be a wire the emf exists even in vacuum!

When we put a wire there, the electrons respond to the emf 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=•∫ ldErr

정전기장(static E)과 유도전기장 (Induced E)이 다른 점

양(+) 전하에서 시작하여음(-) 전하에서 끝남.

경로에 무관 : conservative force

닫힌 고리에서는 V = 0 임.

닫힌 고리임에도 불구하고 변하는 자가다발에서는 V=0 이 아님.

경로에 의존 : nonconservative force

유도전기장에서는 전기퍼텐셜을정의할 수 없다.

Escher depiction of nonconservative emf

• The magnetic field in a region of space of radius 2Ris 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 -zdirection.

x x x xx x x x x x

x x x x x x x xx 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 2Ris 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 E2Rat radius 2R?

(a) E2R = ER (b) E2R = 2ER(c) E2R = 4ER

• The rate of change of the flux is proportional to the area: dt

dBπRdt

dΦB 2−=

RE ∝

• The path integral of the induced electric field is proportional to the radius:

)2( πREldE =•∫rr

x x x xx x x x x x

x x x x x x x xx x x x x x

x x x x

t

Bz

t1

x

y

R

확인 질문

확인문제 4.

경로 : (1) ε, (2) 2 ε, (3) 3 ε, (4) 0 일 때, 각 구역에 걸린 자기장의 방향은?

∫ ⋅= sdE rrε

(4) c 와 e 는 반대 방향, (2) d와 e 는 같은 방향, (3) b와 c는 a와 같은 방향

a : out

b, c : outd, e : into

유도기유도기 ((InductorInductor))

An An inductorinductor is a twois 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 iΦΦ = = LiLi L= L= ΦΦ //iiLL is the inductanceis the inductanceUnit: Henry (H) or (VUnit: Henry (H) or (V••s/A)s/A)

i

+

_v N

NφφBB

30-7. 유도기(inductor)와 유도용량 (inductance)

유도용량 (inductance)

iNL BΦ

≡

인덕터의 유도용량 (Inductance)

[1 Henry = 1 H = 1 T⋅m2/A]

AinnlBAnlN B ))(()( 0μ==Φ

솔레노이드의 중심 부근 길이 l 인 부분의 자기다발

lAni

NL B 20μ=

Φ≡

AnlL 2

0μ= : 단위 길이당 유도용량

30-8. 자체유도 (Self-inductance)

B-field: niB 0μ=

dtdi

lAN

AdtdinN

dtNd B

L

20

0

)(

μ

μ

ε

−=

⋅−=

Φ−=

dtdiL

dtdN B

L −=Φ

−=ε L: Self-Inductance

자체유도 기전력 (εL) : Lε

εL : Against the increase of the current

AnlL 2

0μ=

유도기의 연결

Inductance in Series: 21 LLL +=

Inductance in Parallel:

L1

L2

i1

i2

i i

dtdiL

dtdiL

dtdiLL

−=

−=

−=ε

22

11

21

21

LLdtdi

dtdi

dtdi

LLLL εεε

−−=+==−

21

111LLL

+=⇒

21 iii +=

(저항연결인 경우와 같음)

30-9. RL 회로

0

0

=+−−

=+−−

ε

εε

dtdiLiR

iR L

ε=+ RidtdiL

)1( / LteR

i τε −−=RL, L =τ

(a) 연결시 전류의 증가

(b) 단절시 전류의 감소

0=+ RidtdiL

LL tt eieR

i ττε /0

/ −− ==

i

ε/R

t

c/te τ−− 11c/te τ− 2

t1 t2

εR εL

ε=0

εL

i

확인문제 6.

(a) 스위치가 닫힌 직후전류가 큰 순서는?

(a) 2 > 3 > 1 (=0)

(b) 2 > 3 > 1

(b) 오랜 시간 후전류가 큰 순서는?

30-10. 자기장에 저장된 에너지

In LR Circuit

ε +

−

R

LdtdiLiR +=ε

dtdiLi

dtdUB =

2

21 LiU B =

dtdiLiRii +=ε 2Power:

Cf ) Energy stored in a capacitor

2

21 CVUC =

Inductor의 일률

30-11. 자기장의 에너지 밀도

In a solenoid (Magnetic Energy) A

lAlnL ⋅= 2

0μ

( ) 2202

1 iAlnU B ⋅⋅= μn

BiniB0

0 μμ =⇒=

lB VBAlBU 00

2

0

2

22 μμ=⋅=

0

2

2μBuB = (Magnetic energy density)

Cf) 202

1 EuE ε= (Electric energy density)

2

magnetic0

12

Buμ

=… energy

density ...

Energy stored in an inductor …. B

Energy stored in a capacitor ...

2electric 0

12

u Eε=… energy density…

E

How we can store the energy?How we can store the energy?

dielectric+ + + + + + + +

- - - - - - - -

)()( 221 tLitM =E

)()( 221 tCvtE =E

30-12. 상호유도 (Mutual induction)

상호유도용량 (mutual inductance) : M

1

21221 i

NM Φ≡

전류 i1 이 흐르는 코일-1 에 의한 코일-2 의 상호유도용량

221

21

21212121 ε−=Φ=⇒Φ=

dtdN

dtdiMNiM

전류 i2 이 흐르는 코일-2 에 의한 코일-1 의 유도용량

dtdiM 2

121−=ε

dtdiM 1

212−=ε

MMM ≡= 1212

dtdiM 2

1−=ε

dtdiM 1

2−=ε

30. Summary

dtdsdE BΦ

−=⋅= ∫ε rrFaraday’s Law (Lenz’s Law)

Eddy current

AnlL 2

0μ= : 단위 길이당 유도용량

2

21 LiU B =

0

2

2μBuB =

Magnetic energy

dtdiL

dtdN B

L −=Φ

−=ε L: Self-Inductance

εL : Against the increase of the current